Isothermal quantum dynamics: Investigations for the harmonic oscillator

Transcription

1 Isothermal quantum ynamics: Investigations for the harmonic oscillator Dem Fachbereich Physik er Universität Osnabrück zur Erlangung es Graes eines Doktors er Naturwissenschaften vorgelegte Dissertation von Dipl. Phys. Detlef Mentrup aus Georgsmarienhütte Osnabrück, Januar 2003

2 Betreuer : PD Dr. J. Schnack Zweitgutachter : Prof. Dr. M. Luban

3 The key principle of statistical mechanics is as follows: If a system in equilibrium can be in one of N states, then the probability of the system having energy E n is (1/Q) e En/kBT, where Q = N n=1 e En/kBT. (...) If we take i as a state with energy E i an A as a quantum mechanical operator for a physical observable, then the expecte value of the observable is A = 1/Q i i A i e E i/k B T. This funamental law is the summit of statistical mechanics, an the entire subject is either the slie-own from this summit, as the principle is applie to various cases, or the climb-up to where the funamental law is erive an the concepts of thermal equilibrium an temperature clarifie. R. P. Feynman, Statistical mechanics [1, ch. 1]

4 Contents 1. Introuction 4 2. Methos of isothermal classical ynamics The classical Langevin equation Deterministic temperature control methos The Nosé-Hoover metho The emon metho The chain thermostat Non-Hamiltonian phase space as a manifol Note on eterministic chaos an ergoicity Coherent states Introuction General properties of coherent states Determination of ensemble averages with coherent states One particle Two ientical particles Case of N fermions Isothermal quantum ynamics for the harmonic oscillator A quantum Langevin equation The quantum Nosé-Hoover thermostat One particle The Nosé-Hoover thermostat an the Nosé-Hoover chain The emon metho Two ientical particles The simple Nosé-Hoover thermostat The emon metho Case of N fermions Results One particle Nosé-Hoover metho Nosé-Hoover chain metho The emon metho Convergence spee Mean values of selecte observables Two particles Bose-attraction an Pauli-blocking Ergoicity investigations for two fermions

5 Contents Nosé-Hoover metho The emon metho Ergoicity investigations for two bosons Nosé-Hoover metho The emon metho Convergence spee Mean values of typical observables Summary an iscussion 74 A. The Grilli-Tosatti thermostat 76 B. Position representation of the canonical ensity operator 80 C. Sinusoial oscillations of isplace harmonic oscillator eigenfunctions 82 Bibliography 85 Acknowlegements 87 2

6 List of Figures 5.1. Time averaging with a simple Nosé-Hoover scheme, T = Time averaging with a simple Nosé-Hoover scheme, T = 0.25, Time averaging with Nosé-Hoover chains, T = Short-time behaviour of a Nosé-Hoover chain ynamics Time averaging with the cubic coupling scheme, T = 0.2, Convergence spee of histograms Mean values of selecte observables Dynamical illustration of Pauli-blocking Dynamical illustration of Bose-attraction Time averaging with a simple Nosé-Hoover scheme, T = 0.1, Density plots an marginal istributions for a Nosé-Hoover scheme, only one particle is couple to a thermostat Short-time behaviour of a Nosé-Hoover scheme, only one particle is couple to a thermostat Marginal istributions for the emon metho, fermions Marginal istributions for the emon metho, bosons Convergence spee of histograms, two fermions Results of time averages for the internal energy an its variance for two fermions Results of time averages for the two-particle ensity Results of time averages for the mean occupation numbers for a two-fermion system 73 3

7 1. Introuction In a macroscopic system, the number of particles or egrees of freeom is extremely large, such that it is impossible to obtain a complete physical escription of the system, both experimentally an theoretically. For example, the microscopic state of a classical gas inclues all positions an momenta of particles. However, the etaile behaviour of the constituents is not reflecte on a macroscopic scale, where one is only intereste in a few properties of a system, e. g., we only require that a given system has N particles, a volume V an an energy in a small interval aroun the value E. These macroscopic conitions are met by a large number of microscopic states. The mental collection of systems that are in these states is calle an ensemble. The typical problem in statistical physics is the etermination of averages over such ensembles. The three ensembles that are usually ealt with in stanar textbooks on statistical physics are the microcanonical (constant number of particles N, volume V, energy E), the canonical (constant N, V, temperature T ) an the gran-canonical (constant chemical potential µ, V, T ) ensemble. Each of the ensembles is characterise by a istribution function that escribes the probability for a macroscopically prepare system to be in a given microscopical state, which is, in classical physics, given by a point in the phase space of the system, whereas in quantum physics it is given by a state vector in Hilbert space. The etermination of this istribution function is the funamental question that is answere by statistical mechanics. The istribution function permits the calculation of ensemble averages an, more generally, of the partition function, which is of outstaning importance in statistical mechanics since it contains the whole thermoynamics of a system. If f(q, p) enotes the classical unnormalise istribution function on the phase space Γ (with configuration coorinates (q 1,... q N ) = q an conjugate momenta (p 1,... p N ) = p), a classical statistical ensemble average B of an observable B(q, p) (which in classical mechanics is a function on phase space) is given by the phase space integral B = 1 qp B(q, p)f(q, p), (1.1) Z cl Γ i. e. the phase space function corresponing to the macroscopic observable has to be average over properly weighte microstates. The quantity Z cl = qp f(q, p), (1.2) Γ which is use to normalise the above expression, is calle the classical partition function. A irect evaluation of the high-imensional integrals (1.1) an (1.2) is possible only in very particular cases, such as the ieal gas. In more general cases, e. g. for interacting many-particle systems, it is usually impossible. In orer to be able to investigate the wie variety of physically interesting phenomena of these systems, a large number of techniques has been evelope for the evaluation of ensemble averages, which can basically be ivie into two groups: The stochastic or Monte Carlo methos that make use of ranom numbers, an the eterministic methos. 4

8 The latter are use in the context of molecular ynamics simulations an will be in the focus of the present work. The basic iea of molecular ynamics (MD) simulations is to solve Hamilton s equations of motion for the particles of a given system numerically to obtain the temporal trajectories q(t), p(t) of all particles in phase space. This is, espite the large number of particles an possibly complicate interactions, usually possible given the availability of both accurate numerical methos an fast computers. Now, in orer to calculate an ensemble average, one averages over the time evolution, an hopes that the ergoic hypothesis is satisfie. Loosely speaking, this means that the trajectory runs through the allowe region of phase space 1 with the correct weight such that the average over the time evolution of the system is equivalent to the average over the corresponing statistical ensemble: 1 lim τ τ τ 0 t B(q(t), p(t)) = 1 Z cl Γ qp B(q, p)f(q, p). (1.3) In Monte Carlo (MC) methos, one introuces an artificial ynamics on phase space which is base on ranom numbers. MC simulations are very powerful an popular for static properties. The most common type of MC simulations, the Metropolis Monte Carlo metho [2], is naturally aapte to the canonical ensemble. It allows a irect sampling of the Boltzmann istribution by generating a Markov chain with a suitable transition rule an using rejections to achieve etaile balance. The ynamics obtaine in an MC simulation also nees to be ergoic in the sense that the ranom walk nees to sample the entire allowe phase space such that MC averages an ensemble averages coincie. However, since in MC one oes not rely on a real (i. e., physically meaningful) ynamics, this is less of a problem. In case of non-ergoic behaviour, one has to invent more elaborate transition rules. Usually molecular ynamics calculations are performe with a fixe number of particles in a given volume of constant shape. In aition, as a consequence of Hamilton s equations, the energy of the system is conserve uring time evolution. Therefore, if the trajectory passes uniformly through all parts of phase space that have the specifie energy, the time average one obtains from an MD simulation correspons to a microcanonical ensemble average. Only phase space points which lie on the hypersurface escribe by the conition of constant energy H(q, p) = E contribute to the equilibrium ensemble average, an they contribute with equal weight accoring to the principle of equal a priori probability. The corresponing probability ensity in phase space is therefore given by f(q, p) = δ(h(q, p) E). (1.4) As a conclusion, the microcanonical ensemble may be consiere as the natural ensemble for MD simulations. However, if this approach to the calculation of ensemble averages were limite to the microcanonical ensemble, it woul be practically useless. Experiments are usually carrie out at constant temperature (an pressure), an therefore it is esirable to have techniques to realise ifferent types of thermoynamic ensembles in MD simulations. More specifically, in orer to obtain a canonical ensemble average, the above technique is inaequate, since in the canonical ensemble, the conition of constant energy is replace by a conition of constant temperature, an the istribution function (1.4) is replace by a Boltzmann istribution 2, f(q, p) = exp( βh(q, p)). (1.5) 1 The term allowe refers to the constraints impose on the system. 2 In this work, we will use the notation T for the temperature, an β = 1/k BT in parallel, k B being Boltzmann s constant. 5

9 1. Introuction The total energy of the system is allowe to fluctuate aroun its mean value by thermal contact with an external heat reservoir which allows for energy exchange. The physical effect of the heat bath upon the system of interest is to impose a constant temperature conition, while the etails of the thermal interactions are usually unknown. How can this moifie situation be escribe on the level of the equations of motion, such that the temperature value can be given beforehan as a fixe parameter? Clearly, a mechanism is neee that introuces suitable energy fluctuations. We will call such a mechanism a thermostat. The constant temperature conition is certainly fulfille for a Brownian particle which is a macroscopic particle immerse into a liqui of a given temperature. Energy transfer takes place via the ceaseless ranom collisions between the Brownian particle an the constituents of the liqui. The motion of the particle is escribe by the so-calle Langevin equation, an it has been shown in 1945 that uner appropriate conitions, the long-time limit of a time average over the solution of the Langevin equation correspons to a canonical ensemble average [3]. The Langevin equation involves a stochastic force that mimics irectly the collisions mentione above, therefore we call this approach a stochastic thermostat. In contrast to pure MC sampling, this phenomenological equation is base on Newton s equation of motion an inclues only a moerate amount of ranomness. It is surprising that beyon this irect moelling of the heat bath interaction, a ifferent technique which is completely eterministic has been initiate by Nosé in 1984 [4]. His original metho was base on the iea of a scaling of the particle momenta, allowing energy fluctuations an thereby temperature control by a control of the kinetic energy. Although formally correct, the original formulation feature ergoicity problems an therefore was not applie very much in practice. Later on, numerous extensions an refinements have been ae that turne out to be more efficient an easier to hanle. These moifie techniques are known as extene system methos. Their common unerlying iea is to appen aitional egrees of freeom to the original physical system that act as pseuofriction terms, thereby estroying energy conservation an, moreover, the overall Hamiltonian structure of the ynamics. The equations of motion of the enlarge system are esigne in such a way that in the subspace belonging to the original physical system, the temporal average correspons to a canonical average. To ensure this, the equipartition theorem of classical statistical mechanics is implicitly exploite. Extene system methos are commonly use nowaays in classical MD simulations [5, 6] an have turne out to be extremely powerful. The main avantage a molecular ynamics approach has over Monte Carlo is that in the course of an MD simulation, physically reasonable ynamical equations are integrate. This makes ynamical information available, even though one can argue that the particular constant temperature methos appear somewhat artificial. Consequently, ynamical properties such as time correlation functions may be calculate. Monte Carlo simulations are not suitable for the etermination of ynamical physical properties an allow only the calculation of static properties, unless one accepts that the ranom walk generate by MC is an interesting physical ynamical moel. In the fiel of finite-temperature simulations of quantum systems, the most successful approach is base on the path-integral formulation by Feynman [1]. The power of the metho is ue to the fact that it allows to relate the quantum ensity matrix at arbitrary temperature, e βh, where H is the Hamiltonian of the system, to integrals over paths in coorinate space, R e βh R = R 1... R M exp( S(R 1,... R M )). (1.6) S is the so-calle action of the path an is real, an thus (1.6) involves a basically classical 6

10 istribution function, such that one can use classical molecular ynamics [7] or Monte Carlo techniques [8] to evaluate the integral. The latter have been applie very successfully to the interacting boson system 4 He that unergoes the famous λ-transition at T = 2.18 K [8], an to bosons in a magnetic trap [9]. However, for fermions, the contributions of even an o permutations to the ensity matrix involve opposite signs ue to the require antisymmetry of the wave function. The cancellation of contributions usually causes the signal/noise ratio to approach zero rapily an rules out a straightforwar MC evaluation of the integran. This is known as the fermion sign problem. The main obstacle to an approach to the calculation of ensemble averages via time averaging over the quantum time evelopment is the fact that the solution of the Schröinger equation is not reaily available for complicate systems. Contrary to the classical case, quantum ynamics itself is a very har computational problem. In essence, the quantum time evolution implies to calculate an exponential of the Hamiltonian, which is practically equivalent to treating the canonical ensity operator irectly. On this level, the ansatz of time averaging oes not lea to substantial computational avantages. Nevertheless, the question whether it is possible to etermine canonical averages for a quantum system by averaging over trajectories generate by an appropriate ynamics is challenging. The interest for techniques comparable to the classical extene system methos in the realm of finite-temperature quantum MD simulations has ifferent sources. On the one han, even for relatively simple quantum systems constisting of a very small number of particles, it is usually impossible to etermine the full set of eigenfunctions an eigenvalues. Given the availability of various efficient approximate quantum MD schemes (for a review of techniques for fermions, see [10]), the following question is interesting from a methoological point of view: Does a generalisation of the classical methos to quantum ynamics permit to make the power of approximate quantum MD schemes available for the calculation of equilibrium averages without iagonalising the full many-boy Hamiltonian? The present work may be consiere a first step towars an affirmative answer to this emaning question. Beyon, given the fact that quantum MC has a non-physical time evolution, it is highly esirable to have an isothermal quantum MD metho at han that epicts the physical ynamics of the system at a given finite temperature more realistically. This woul enlarge the variety of techniques available in many-boy theory an possibly exten their overall range of applicability. Apart from permitting the etermination of equilibrium ensemble averages, the classical extene system methos also offer a moel scenario for the ynamical evolution of a nonequilibrium state towars thermal equilibrium. Although it is not clear whether the specific approach using pseuofriction terms is a faithful physical picture of the real ynamical evolution of a system enroute to thermal equilibrium, this view is a natural interpretation of the methos that permit cooling an heating of non-equilibrium initial states. Consequently, one may hope that a quantum analogue of the classical methos moels the approach of a quantum system to equilibrium. On the other han, real physical systems such as ultracol trappe gases are in the focus of moern research for which such a metho appears to be tailor-mae an may allow for the irect theoretical analysis of systems in a constant temperature conition. Recently, investigations of ultracol magnetically trappe atomic gases have le to the iscovery of intriguing physical phenomena, among them the spectacular evience for Bose-Einstein conensation in weakly interacting Bose gases [11, 12]. On the fermionic sie, researchers stuy the large impact of Fermi-Dirac statistics on the behaviour of so-calle egenerate Fermi gases [13, 14]. It is remarkable that these systems constitute ilute gases in which the interparticle interactions are weak. For fermions, the quantum statistical suppression of s-wave interactions makes an 7

11 1. Introuction ultracol trappe gas of fermionic atoms even an excellent realisation of an ieal gas. In aition, we note that the confinement by the magnetical trap may approximately be consiere as harmonic [15]. Beyon, from a theorist s point of view, the ieal gas forms the starting point for perturbative treatments of interacting many-particle assemblies, an motion in a common harmonic oscillator potential is of special interest because of its importance for lowexcitation ynamics. Therefore, it appears reasonable to start with this tractable case. In interacting fermion systems, a number of other fascinating effects is iscusse, e. g., theorists have stuie the prospects of observing a superflui phase base on the BCS concept of Cooperpair formation [16]. First ieas for a translation of classical constant temperature MD methos to quantum mechanics have been iscusse by Grilli an Tosatti in 1989 [17], but their approach has turne out to have serious shortcomings [18] some of which are iscusse in appenix A. An alternate metho ue to Kusnezov [19] is limite to quantum systems of finite imensionality an has only been applie to a two-level system. Schnack has investigate a quantum system at constant temperature using a thermometer an a feeback mechanism to rive the system to the esire temperature value by complex time steps [20]. The main rawback of this ansatz is the fact that an interaction is neee to equilibrate the system of interest an the thermometer, which leas to a perturbation of the original system an thereby exclues simulations at very low temperatures. This illustrates a main ifficulty encountere in quantum mechanics: While in classical mechanics, the equipartition theorem provies a irect an infallible thermometer, such a useful a priori relation between the average value of some observable an temperature is not reaily at han in quantum mechanics. Instea, it nees to be implemente in a sophisticate manner, involving a number of ifficulties, like the perturbation ue to the interaction an the insecurity about correct equilibration between system an thermometer. In view of this unsatisfactory situation, the major goal of the present work has been to evise a quantum thermostat following the lines of the methos successfully employe in classical mechanics. We will show that in the case of an ieal quantum gas enclose in an external harmonic oscillator potential, the framework of coherent states permits new, far-reaching methoological evelopments. For a single quantum particle, the analogy to classical physics is very close, whereas for two non-interacting inistinguishable quantum particles genuine quantum features have been foun an investigate. It turns out that the approach base on the classical Langevin equation is not suitable for ientical particles, whereas the extene system methos can successfully be translate to quantum mechanics. The outline of the present work is as follows. Chapter 2 eals with temperature control methos in classical mechanics, namely the Langevin equation an the extene system metho of Nosé an Hoover along with its refinements. Chapter 3 gives a brief introuction to coherent states an their properties. Chapter 4 contains the main outcome of this work, presenting the unpreceente quantum thermostat methos for one an two particles, an N fermions in an external harmonic oscillator potential. In chapter 5, the results obtaine with the new methos are presente. Chapter 6 summarises the work an gives a critical analysis of the chances an limits of the quantum thermostat metho evise in this work. 8

12 2. Methos of isothermal classical ynamics 2.1. Stochastic temperature control: The classical Langevin equation The Langevin equation may be regare as a phenomenological approach to temperature control in classical mechanics. It has been evelope to escribe the irregular motion of a macrosopic (so-calle Brownian) particle immerse into a liqui of absolute temperature T. The main iea is to escribe the action of the liqui that acts as a heat bath upon the particle by two aitional forces that are introuce in Newton s equation of motion: Firstly, a slowly varying frictional force proportional to the velocity of the particle, mγ tq, where m is the mass of the particle an γ is a constant frictional coefficient, an seconly, a rapily fluctuating ranom force F (t) that escribes the isorere collisions of the particles of the liqui with the Brownian particle an that vanishes on average. If in aition the particle moves in an external potential V (q), the resulting equation escribing its motion on the spatial q-axis reas m 2 q t 2 = V q Equivalently, one can stuy the set of equations of first orer in time, q mγ + F (t). (2.1) t q t = p m, p t = V q γ p + F (t). (2.2) The time average of F (t) vanishes, 1 τ lim F (t)t = 0, (2.3) τ τ 0 an F (t) shall be purely ranom, which means that it has a vanishingly short correlation time 1. Moreover, the amplitue of the ranom force is relate to the temperature T an the friction coefficient γ by the secon fluctuation-issipation theorem, which together is expresse as F (t 1 )F (t 2 ) = 2 mγk B T δ(t 1 t 2 ). (2.4) In aition, for technical reasons, one must assume that the ranom force is Gaussian, i. e. one assumes that the coefficients of the Fourier series of F (t) (which are ranom variables) are istribute accoring to a Gaussian istribution. Equation (2.4) guarantees that the temperature is kept at a constant value by the balance between the thermal agitation ue to the ranom force an the slowing own ue to the friction. Uner the assumptions mae above, it has been shown that in the limit t, the probability ensity P (q, p; t q 0, p 0 ) at the phase space point (q, p) given that at time t = 0 the particle was 1 As a result of the Wiener-Khintchine-theorem that relates the correlation function of a stochastic function to its power spectrum [21], the spectral ensity of the fluctuating force is constant uner this conition. Therefore, the spectrum of F (t) is frequently sai to be white. 9

13 2. Methos of isothermal classical ynamics situate at q 0 with initial momentum p 0 is the canonical Maxwell-Boltzmann ensity [3]. This limiting probability istribution is inepenent of the initial state of the system. Therefore, a time average over a sufficiently long perio will be equivalent to a canonical ensemble average. The proof that the limiting probability istribution of the Langevin equation (2.1) is a Maxwell-Boltzmann istribution uses a Fokker-Planck equation associate to (2.1) [3, 21]. In general, for a one-imensional Markov process y(t), the Fokker-Planck equation is a partial ifferential equation for the probability ensity P (y, t y 0 ) that is erive from the obvious conition P (y, t + t y 0 ) = z P (y, t z)p (z, t y 0 ), (2.5) which is calle the Smoluchowski equation, in the limit of small t an a small ifference y y 0. The resulting Fokker-Planck equation for a one-imensional Markov process, P t = y (M 1(y)P ) y 2 (M 2(y)P ), (2.6) contains the first an secon moment of the change of the ranom variable y, the nth moment being efine as 1 M n (y) = lim z (z y) n P (z, t y). (2.7) t 0 t Equation (2.6) is erive assuming that the moments M n vanish for n > 2 which expresses the fact that in a short perio of time, the spatial coorinate can only change by small amounts. The Fokker-Planck equation for an n-imensional Markov process y = (y 1,..., y n ) reas P n t = (M 1 (y)p ) + 1 y i 2 i=1 n k,l=1 2 y k y l (M 2kl (y)p ). (2.8) The case of the harmonic oscillator, V (q) = 1 2 mω2 q 2, is particularly simple. The moments require in the Fokker-Planck equation can be etermine, an the resulting equation for the probability ensity P (q, p; t) reas explicitly P t = p P m q + p ( ( γ p + mω 2 q) P an may be solve analytically with the initial conitions ) + mγk B T 2 P p 2 (2.9) P (q, p, t=0) = δ(q q 0 )δ(p p 0 ). (2.10) The result is a two-imensional Gaussian istribution in q an p with time-epenent average values an withs. In the limit t, one obtains the Maxwell-Boltzmann istribution (C is a normalisation constant), ( lim P (q, p; t q 0, p 0 ) = C exp 1 ( p 2 t k B T 2m mω2 q 2)), (2.11) which is a Gaussian istribution both in positions an momenta, inepenent of the initial conitions. Note that the amplitue of the ranom force (2.4) which contains the temperature T etermines the with of the limiting Gaussian istribution. 10

14 2.2. Deterministic temperature control methos We summarise the main points of this section. The long-time limit of a time average over the solution of the classical Langevin equation correspons to a canonical ensemble average. In the simplest case of the harmonic oscillator, the solution of the Langevin equation provies an average with Gaussian istribution functions. This statement is verifie by an explicit solution of the associate Fokker-Planck equation. The with of the Gaussians is etermine by the amplitue of the fluctuating force via the fluctuation-issipation relation Deterministic temperature control methos While the Langevin approach is reaily comprehensible an physically intuitive, it is less evient that a completely eterministic time evelopment may also provie canonical averages. Such an alternate technique has originally been propose by Nosé [4], an has been refine by Hoover [22], Kusnezov, Bulgac, an Bauer [23], an Martyna, Klein, an Tuckerman [24]. A review of these an various other constant temperature molecular ynamics methos can be foun in [25]. Only recently [26], the theory of these methos was put on a firm theoretical groun, proviing a valuable eeper view that we present in section In the original metho of Nosé, temperature control in a molecular ynamics simulation is achieve by the introuction of an aitional egree of freeom s that is use to scale time an thereby the particle velocities. This is reasonable since temperature is relate to the average of the kinetic energy. However, Nosé s original formulation, although formally correct, features substantial problems in practice, an therefore, the metho has been moifie an refine. The so-calle classical Nosé-Hoover thermostat with the extension to chains of thermostats an the so-calle emon metho have turne out to be most successful an reliable. Simply speaking, these methos exploit the equipartition theorem to etermine the equations of motion of pseuofriction coefficients that are introuce in the equations of motion of the original system. These methos may be transferre to the quantum harmonic oscillator, which is why the functionality of these classical eterministic thermostats is the subject of the following paragraphs an will be outline in etail The Nosé-Hoover metho Consier an isolate classical N-particle system in one imension escribe by a Hamiltonian, H(q, p) = N i=1 p 2 i + V (q). (2.12) 2m As usual, the ith particle is locate at position q i with momentum p i, an q (resp. p) is the N-tuple of all positions (momenta). The motion of the system in phase space is governe by Hamilton s equations, t q i = H = p i p i m, t p i = H V (q) =. (2.13) q i q i In the Nosé-Hoover metho, the equations of motion of the momenta p i are supplemente by a term similar to a frictional force. In orer to permit energy fluctuations, the frictional coefficient is regare as time-epenent an can assume both positive an negative values. In contrast to the Langevin approach, a stochastic force is not employe. One wants to obtain canonical time averages solely from varying the frictional coefficient suitably in time. 11

15 2. Methos of isothermal classical ynamics In the original notation introuce by Hoover [22], the moifie equations of motion rea t q i = p i m, t p V (q) i = ζ(t) p i, (2.14) q i where ζ(t) is a time-epenent supplementary egree of freeom ae to the system to rive the energy fluctuations require in the canonical ensemble. Therefore, the Nosé-Hoover an relate methos are frequently referre to as extene system methos. From (2.14) it can be inferre that ζ acts as a pseuofriction coefficient. Both the value an the sign of ζ vary in time. Accoringly, the momenta of the original system either ecrease or increase, which leas to a change of the kinetic energy of the system. This mechanism is use for temperature control. The key point is to etermine the time epenence of ζ such that the energy fluctuations correspon to the canonical ensemble. More precisely, we eman that the weight with which the phase space of the original system is sample in time is the canonical istribution function, exp ( βh(q, p) ). (2.15) On the level of the phase space of the extene system we postulate the istribution function ( ( f(q, p, ζ) = exp β H(q, p) Qζ2)). (2.16) Note that the Boltzmann istribution (2.15) is a marginal of f. The choice of the istribution function for ζ is relate to the linear coupling ζp i in the equation of motion (2.14). A ifferent coupling woul entail a ifferent istribution function, as will become evient in the iscussion of the emon metho. In orer to make sure that f is sample uring time evolution, the time epenence of ζ is etermine by the conition that f is the stationary solution of a generalise Liouville equation which we erive now. To fix the notation, let x enote a point in a (possibly enlarge) phase space. A istribution function f(x) satisfies a continuity equation that expresses conservation of probability, f t + iv x(fẋ) = 0. (2.17) This equation is obtaine by equating the local change of a conserve quantity insie a given volume an the flow of this quantity through the surface of this volume. On the other han, the total time erivative of f along a phase space trajectory is efine by With this ientity, (2.17) can be reexpresse as t f = f t + ẋ f x. (2.18) t f = f ( x ẋ ). (2.19) Note the moifie notation iv x ẋ x ẋ. Now, if x = (q, p) is an element of the phase space of a Hamiltonian system an the time evolution of the system is etermine by Hamilton s equations of motion (2.13), the right han sie of this equation vanishes ientically an Liouville s theorem f/t = 0 hols. Its meaning 12

16 2.2. Deterministic temperature control methos is that given an initial istribution in phase space, the local ensity of representative points oes not change if we follow the solution of Hamilton s equations. In the case of the Nosé-Hoover equations (2.14) on the enlarge phase space, i. e. with x = (q, p, ζ), we use this equation to etermine an equation of motion for ζ so as to reprouce the postulate thermal istribution (2.16). In orer to obtain an explicit equation, we impose the constraint ζ/ ζ = 0, an along with (2.14) we easily get for the right han sie of (2.19) f ( ) ẋ = f x ( N i=1 ( q i + ) ) ṗ i + q i p i ζ ζ = fnζ. (2.20) Now, we calculate the left han sie of (2.19), employing the equations of motion (2.14), t f = f p ṗ + f f q + q ζ ζ (2.21) ( N ( pi = f β + V ) ) q i βqζ i=1 mṗi q ζ i ( N ) p 2 i = f β ζ m Q ζ. i=1 Equating (2.20) an (2.21) yiels the following equation of motion for ζ: ( t ζ = 1 N ) p 2 i Q m Nk BT. (2.22) i=1 It is interesting to notice that the time evolution of ζ is etermine by the eviation of the momentary value of the kinetic energy i p2 i /2m from its canonical average value N/2 k BT. As a result, temperature control is achieve by a feeback mechanism: In case the momentary kinetic energy is larger than N/2 k B T, the time erivative of ζ is positive an ζ increases, so that the frictional force ζp i reuces the momenta, thereby cooling the system. In the opposite case, the feeback mechanism heats the system up by accelerating the particles. Besies, it is noteworthy that the influence of a heat bath may be imitate by aing a single supplementary egree of freeom to the original system. This is in striking contrast to the usual attributes of a heat bath, namely, that it is larger in comparison to the system of interest, which is usually expose as having much more egrees of freeom [21, ch. 3.6]. Furthermore, we remark that in the Nosé-Hoover metho, the canonical istribution in phase space is reprouce from a single thermoynamic average, the average of the kinetic energy. For reasons of completeness, the following equation is also solve in a simulation, since it contributes to the quantity t Θ = ζ (2.23) H = H(q, p) Qζ2 + Nk B T Θ (2.24) that is conserve by the set of equations of motion (2.14) an (2.22). We stress that the quantity (2.24) is not a Hamiltonian for the extene system; the equations of motion (2.14), (2.22) along with (2.23) o not have a Hamiltonian form. This is the reason why we consiere 13

17 2. Methos of isothermal classical ynamics the non-hamiltonian phase space (q, p, ζ) of o imensionality. We have abanone the strict Hamiltonian structure in the present context. The question of a soun generalisation of the Hamiltonian phase space notions to a non-hamiltonian system will be aresse more closely in section By construction, f is the static probability istribution generate by the ynamics (2.14) an (2.22). This conition is necessary, but not sufficient for the equivalence of a trajectory average an an ensemble average. It oes not guarantee that the correct limiting istribution will be generate by the ynamics, since it is not clear whether the system actually runs through all phase space points with the correct weight, inepenent of the initial conitions. In fact, it might be possible that some regions of phase space are unreachable for the ynamics an therefore not sample. Loosely speaking, the many boy system nees to be sufficiently complex such that the ynamics will cover the entire phase space. The analysis by Tuckerman et al. presente in section allows to expose this point very clearly. This aitional property, the equivalence of time average an ensemble average, is generally referre to as ergoicity. A strict proof of ergoic behaviour for a given system can rarely be given, however, it is observe that the more complex the ynamics of a system gets, the more likely ergoic behaviour is observe. This is intuitively unerstanable since it is clear that the number of unwante conserve quantities ecreases with increasing complexity. When using eterministic methos such as the Nosé-Hoover metho, one usually checks the marginals of the aitional egrees of freeom an hopes that if these marginals are sample correctly, the phase space of the entire system is also sample correctly [6]. There is an important example of apparent non-ergoic behaviour in the case of the Nosé- Hoover metho. The classical harmonic oscillator cannot be thermalise by the simple scheme outline above [22]. The shape of the Poincaré-sections in phase space strongly epens on the choice of the numerical value of Q, an the istributions sample o in no case correspon to a canonical istribution. For other systems, among them classical spin systems, it was foun that the simple Nosé-Hoover scheme may be ergoic for one temperature, but not ergoic for a ifferent value of T [23, 6]. In aition, in all cases of non-ergoic behaviour, a strong epenence of the initial conitions is observe, which is unacceptable. In summary, the Nosé- Hoover metho is not capable to reliably create canonical istributions. However, more general schemes like the emon metho or the metho of chain thermostats basically resolve this problem by generating a more complex ynamics. A eeper stuy using the notions of section shes light on the problem unerlying the non-ergoicity of the harmonic oscillator The emon metho In an effort to cure the problem of unpreictable non-ergoic behaviour in the Nosé-Hoover metho, Kusnezov, Bulgac, an Bauer [23] have evise a generalise coupling scheme. Two aitional egrees of freeom are use to replicate the interaction of the original system with a heat bath. The first one is couple to the equations of motion of the positions, the secon one to the momenta of the particles. This approach appears sensible since it takes into account the equality of positions an momenta in Hamiltonian mechanics. Another avantage of this metho is that the Hamilton function of the envisage system oes not have to contain a kinetic energy term for temperature control; instea, the time erivative of the pseuofriction coefficients turns out to be proportional to the ifference of two quantities whose ratio of canonical averages is k B T. This extens the range of applicability of the metho to, e. g., classical spin systems [27, 6]. 14

18 2.2. Deterministic temperature control methos Explicitly, the equations of motion of the KBB-scheme rea t q i = H g 2 p (ξ) F i(q, p), i t p i = H g 1 q (ζ) G i(q, p). (2.25) i The aitional egrees of freeom ζ an ξ are frequently referre to as emons which are couple to the equations of motion with the functions g 1 (ζ) an g 2 (ξ). F i(q, p) an G i (q, p) are arbitrary functions of all coorinates an momenta. Note that the original Nosé-Hoover equations are obtaine from (2.25) by the specific choice G i = p i, g 1 = ζ, an F i = g 2 = 0. The phase space istribution function in the (2N+2)-imensional extene phase space is chosen to be ( ( f(q, p, ζ, ξ) = C exp β H(q, p) + 1 g 1 (ζ) + 1 g 2 (ξ)) ), (2.26) κ 1 κ 2 where C is again a normalisation constant, an κ 1 an κ 2 are, at the moment, free parameters. The functions g 1 an g 2 that etermine the thermal istribution of the emons nee to be chosen such that the integral of f with respect to ζ an ξ converges. Note that their respective erivatives g 1, g 2 appear in the equations of motion (2.25). This has been the reason for choosing a Gaussian istribution function for ζ in (2.16). In orer to erive equations of motion for ζ an ξ so as to reprouce the postulate thermal istribution (2.26), we substitute the equations of motion (2.25) an the istribution function f into the generalise Liouville equation (2.19) erive in the preceing section. In aition, we have the freeom to impose the constraints ζ ζ = 0, ξ ξ = 0. (2.27) By comparing the coefficients of the functions g 1, g 2 in the generalise Liouville equation, one obtains the following equations of motion for the emons: t ζ = κ 1 t ξ = κ 2 N i=1 N i=1 ( H p i G i 1 β ( H q i F i 1 β ) G i p i ) F i q i The equations of motion (2.25) an (2.28) conserve the quantity H = H(q, p) + 1 κ 1 g 1 (ζ) + 1 κ 2 g 2 (ξ) + 1 β t t By partial integration, one easily shows that i., (2.28) [ g 1 (ζ(t )) G i + g 2 p (ξ(t )) F ] i i q i. (2.29) k B T G i p i = H p i G i, (2.30) an likewise k B T F i q i = H q i F i. (2.31) 15

19 2. Methos of isothermal classical ynamics In fact, the time epenence of the emons is etermine by the ifference of two quantities whose ratio of canonical averages is k B T. The control of the kinetic energy in the original Nosé-Hoover thermostat is only a special case. This paves a way for the generalisation of this metho to systems whose Hamiltonian oes not contain a kinetic energy term. As an example, we mention classical spin systems, where this metho has been employe successfully for extensive stuies [27, 6]. In principle, since the choice of the functions F, G, g 1, g 2 is arbitrary, this metho offers a lot of freeom. Kusnezov, Bulgac, an Bauer have investigate ifferent choices of the various functions an were able to show that the emon metho frequently resolves problems of nonergoic behaviour. The most prominent choice of functions is the following so-calle cubic coupling scheme: g 1 = 1 4 ζ4, g 2 = 1 2 ξ2, F i = q 3 i, G i = p i, (2.32) resulting in the set of equations of motion t q i = p i m ξq3 i, t ζ = κ 1 t ξ = κ 2 ( N i=1 ( N i=1 p 2 i m Nk BT t p i = V ζ 3 p i, (2.33) q i ) V q i q 3 i 3k BT These equations provie ergoic behaviour in all examples given in [23], see also section The problems of the simple Nosé-Hoover scheme epenence of the choices of Q, T, an the initial conitions are reliably resolve. The choice of the numerical values of κ 1 an κ 2 may still influence ergoicity, but rules of thumb have been foun empirically that will be iscusse in chapter The chain thermostat Another variation of the Nosé-Hoover metho has been propose by Martyna, Klein, an Tuckerman [24]. The main iea of this technique is to impose on the first thermalising pseuofriction coefficient a secon one which may be couple to yet a thir one, an so on, thereby forming a chain of thermostats. This approach of recursive thermalisation increases the size of the phase space an thus makes the ynamical evolution of the system more complex, thereby leaing to ergoicity. In a moifie notation (ζ p η /Q), the set of ynamical equations t q i = p i m, t p η = N i=1 t η = p η Q,, N i=1 q 2 i ). t p V (q) p η i = p i q i Q, (2.34) p 2 i m Nk BT, efines Nosé-Hoover ynamics. The fact that the temporal evolution of p η is governe by the eviation of the kinetic energy of the system from its canonical average value is isplaye very 16

20 2.2. Deterministic temperature control methos obviously. The variable η which is not couple to the ynamics is again inclue for reasons of completeness. The stationary istribution function reas an the conserve quantity is f(q, p, p η ) = exp ( β ( H(q, p) + p2 ) ) η, (2.35) 2Q H (q, p, p η, η) = H(q, p) + p2 η 2Q + Nk BT η. (2.36) The esire istribution (2.35) has a Gaussian epenence on the particle momenta as well as on the thermostat momentum p η. While the Gaussian fluctuations of the particle momenta are riven by p η, there is nothing to equilibrate p η itself. Therefore, it appears sensible to couple another pseuofriction coefficient to the first one, an so on. As a result, one obtains the equations of motion of the Nosé-Hoover chain metho, t q i = p i m, t p η 1 = t p η j = t p η M ( N i=1 t p V (q) i = q i ) p 2 i m Nk BT ( p 2 ηj 1 Q j 1 k B T ) = p2 η M 1 Q M 1 k B T, t η i = p η i Q i, p η1 p η2 Q 2, p ηj p ηj+1 Q j+1, p i p η1 Q 1, (2.37) where a chain of M thermostats has been implemente. These equations have the stationary phase space istribution an the conserve quantity ( f(q, p, p η ) = C exp β H(q, p) + H (q, p, η j, p ηj ) = H(q, p) + M j=1 M p 2 ) η j 2Q j=1 j p 2 η j 2Q j + Nk B T η 1 + k B T (2.38) M η j. (2.39) Although the number of egrees of freeom ae in this approach is usually larger than in the emon metho, the aition of the successive thermostats is numerically inexpensive as they form a simple one-imensional chain. Only the first thermostat interacts with all N particles. A thorough analysis of the metho has shown that it reliably leas to ergoicity, see section 2.2.5, an that it is competitive with the emon metho with regar to the spee of convergence. j=2 17

21 2. Methos of isothermal classical ynamics Non-Hamiltonian phase space as a manifol Tuckerman, Muny, an Martyna have pointe out that all erivations outline above o not properly take into account that the equations of motion such as (2.14) along with (2.22), (2.33), or (2.37) escribe non-hamiltonian systems. In [26], a consistent classical statistical mechanical theory for such systems is presente. It is base on the concepts of ifferential geometry as applie to ynamical systems an provies a soun generalisation of the usual Hamiltonian base statistical mechanical phase space principles to non-hamiltonian systems. Using these notions, a proceure is evelope that leas to the phase space counterpart of the time averages generate by a non-hamiltonian system. Besies, this approach reveals an surmounts a number of weaknesses of the original formulation, e. g., it permits to explore the reason for the apparent non-ergoicity of certain systems, notably the classical harmonic oscillator [28]. We shall briefly outline the basic ieas presente in [26]. Traitional classical statistical mechanics is base on a Hamiltonian function. A point in the phase space of a system is given by the coorinates an momenta x = (q, p). Given a time-epenent phase space istribution function f(x, t), the average of an observable B(x) in the ensemble escribe by f is given by B (t) = n x B(x)f(x, t) n x f(x, t). (2.40) The measure q p n x, which is use for the calculation of phase space averages, is preserve by Hamiltonian ynamics. This means that a subset of systems with initial conitions containe in a phase space volume element n x 0 will at a later time occupy a volume element of the same size 2 : n x 0 = n x t. This property of Hamiltonian ynamics is frequently referre to as the incompressibility of phase space flow. It is tantamount to the statement that the coorinate transformation specifie by the solution of Hamilton s equations of motion x t (t; x 0 ) has a Jacobian of absolute value 1. The existance of this time-invariant measure implies that (2.40) can be compute with respect to the phase space variables x at any time t. In the case of a general non-hamiltonian ynamical system, ẋ = ξ(x, t), (2.41) the situation becomes more complicate. The time evolution generate by the set of ifferential equations (2.41) is in general compressible an the usual phase space measure n x is no longer invariant uner the ynamical evolution. Therefore, in a more refine analysis of the situation, one must treat the phase space of the system as a general Riemannian manifol. The metric on this manifol has to be taken into account in the formulation of a continuity equation for the istribution function an in the expression of a phase space average which is given in terms of an integral over the manifol. Moreover, if one wants to relate a phase space average to a time average, the question of the integration measure nees to be consiere carefully. Analogous to the Hamiltonian case, an expression for an ensemble average is neee that uses a measure on phase space that is invariant uner time evolution. If such an invariant measure is foun, the phase space average of some property (expresse in terms of an integral over the manifol with the preserve measure) correspons to the time average of the same property over the trajectories of the system uner the usual assumption of ergoicity. 2 Together with the statement that trajectories of ientical systems o not intersect (since the solution of Hamilton s equations of motion is unique), one easily proves the theorem of Liouville, f = 0. t 18

22 2.2. Deterministic temperature control methos In orer to erive an invariant measure on this manifol, consier the solution of (2.41), x i t = xi t (t; x1 0,... xn 0 ), i = 1,... n, (2.42) as a coorinate transformation from the initial coorinates at time t = 0 to the coorinates at time t. It can be shown that the Jacobian J(x t ; x 0 ) of this transformation, obeys the equation of motion J(x t ; x 0 ) = et (x1 t,... xn t ) (x 1 0,... xn 0 ) (2.43) t J = Jκ(x t), (2.44) where the quantity κ(x t ) = n i=1 ẋi / x i is calle the phase space compressibility of the ynamical system 3. Since equation (2.44) may be rewritten as t ln J = κ, (2.45) it can be integrate easily. If we introuce the variable w(x) relate to κ by ẇ = κ, the solution reas J(x t ; x 0 ) = exp ( w(x t ) w(x 0 ) ). (2.46) The infinitesimal volume element transforms uner a coorinate transformation accoring to n x t = J(x t ; x 0 ) n x 0. (2.47) Rearranging this equation such that quantities at time t appear on one sie an quantities at t = 0 appear on the other, we get e w(xt) n x t = e w(x 0) n x 0. (2.48) This equation shows that the measure e w(xt) n x t is conserve by the ynamics. The metric eterminant of the transformation can be ientifie as g = exp( w(x)). Due to the general transformation law of the metric tensor, it is clear that the values of g at times 0 an t are relate by the Jacobian, g0 = g t J(x t ; x 0 ). (2.49) Consequently, g satisfies the following ifferential equation, g = g κ. (2.50) t Note that the same equation is fulfille by the inverse J 1 of the Jacobian J as can easily be seen from equation (2.44) an the relation J J 1 = 1. Next, if an ensemble of systems on phase space is consiere, a continuity equation that accounts for number conservation must be expresse. It may be regare as a generalisation of the theorem of Liouville. The change of local ensity is balance by a flux through the 3 This same quantity alreay occure on the right han sie of (2.19) 19

23 2. Methos of isothermal classical ynamics bounary surface taking into account the geometry of the space. As a result, the following equation is obtaine [26]: (f g) t + (f g ẋ) = 0. (2.51) This equation can be put into the form of equation (2.17) by efining a new function f = gf, but such an ientification is not general, since it entails problems with a coherent notion of the entropy of a system [26]. Finally, note that equations (2.50) an (2.51) together imply that even in the case of a non-hamiltonian system, f(x, t) is conserve, t f = 0. (2.52) The notion of ergoicity may now be expose more clearly. Suppose that the set of ynamical equations (2.41) possesses a set of n c conserve quantities Λ k (x), k = 1,... n c, satisfying t Λ k = 0. (2.53) As a consequence, the trajectories generate by equation (2.41) will only sample the intersection of the hypersurfaces {Λ k (x) = C k }, where C k is a set of constants. If all points on a given hypersurface have the same probability of being visite by a trajectory, then the system is sai to be ergoic. In this case, a time average correspons to a microcanonical average of the extene system which is a phase space average with the istribution function f(x) = n c k=1 δ(λ k C k ), (2.54) which is a prouct of δ-functions expressing the conservation laws. It is evient that f(x) satisfies the generalise Liouville equation (2.51), which is inferre most easily from its moifie form (2.52). It is worth noting that a istribution constructe from a subset of the conservation laws, f(x) = n c k=1 δ(λ k C k ), (2.55) with n c < n c, also satisfies equation (2.52). However, this solution oes not escribe the correct microcanonical istribution function. Therefore, satisfying the generalise Liouville equation (2.51) is a necessary but not sufficient conition for a ynamical system to generate a particular phase space istribution. It is essential to etermine all the conservation laws satisfie by the equations of motion. This clarifies the limitations of relying solely on the generalise Liouville equation to etermine the istribution function as in the preceent approaches presente in sections 2.2.1, 2.2.2, an In those erivations, the esire istribution function f is use to euce equations of motion for the pseuofriction coefficients on the basis of the generalise Liouville equation. Now, following Tuckerman s analysis outline above, one can start the other way roun from the Nosé-Hoover equations of motion (2.34) an show that the microcanonical average in the extene system correspons to a canonical average in the original physical system. 20

24 2.2. Deterministic temperature control methos Assuming that the quantity H = H(q, p) + p2 η 2Q + Nk BT η (2.56) is the only quantity conserve by the ynamics (2.34), the microcanonical istribution function that is sample in case of ergoicity reas δ(h C 1 ). (2.57) We want to investigate the microcanonical partition function 4 of the enlarge system, Ω T = qpp η η g δ ( H(q, p) + p2 η 2Q + Nk BT η C 1 ), (2.58) an show that the marginal of the istribution function (2.57) in the subspace of the physical variables correspons to a canonical istribution function. In the next step, the compressibility of the equations of motion (2.34) must be calculate (cf. (2.44)): κ(x) = n i=1 ẋ i = N η. (2.59) xi Thus, we obtain g = exp(nη). By integration over the non-physical variables in (2.58) the istribution function in the physical subspace may be etermine. Using the δ-function to perform the integration over η requires that Substituting this result into equation (2.58), one obtains η = 1 ( C 1 H(q, p) p2 ) η. (2.60) Nk B T 2Q Ω T = eβc 1 Nk B T p η e β p 2 η 2Q qp e βh(q,p). (2.61) The integration over p η is separate, an the istribution function in the physical subspace is, as esire, the canonical one. This illustrates that the Nosé-Hoover metho works since the microcanonical istribution function (2.57) is esigne such that its marginal in the original physical subspace is the canonical istribution function. Furthermore, the proof implies that the Nosé-Hoover equations generate a canonical istribution in the subspace of the physical variables provie that H is the only conserve quantity. In the case of the harmonic oscillator, however, Tuckerman et al. [28] have investigate the slightly moifie equations (all constants have been set equal to 1) ẋ = p p η x, ṗ = x p η p, η = p η, ṗ η = x 2 + p 2 2. (2.62) These equations create Poincaré sections in phase space that bear great resemblance to the case of the usual Nosé-Hoover ynamics for the harmonic oscillator that features apparent ergoicity problems. The well-known first conserve quantity reas H = 1 2 (x2 + p 2 + p 2 η) + 2η. (2.63) 4 The subscript T inicates that the microcanonical partition function epens parametrically on temperature. 21

25 2. Methos of isothermal classical ynamics In [28], the authors have emonstrate that the istribution function resulting from a time average over the solution of (2.62) agrees with a microcanonical istribution, that contains a secon conserve quantity, namely δ(h C 1 ) δ(k C 2 ), (2.64) K = 1 2 (x2 + p 2 )e 2η. (2.65) To show this, the compressibility associate with (2.62) nees to be evaluate in orer to calculate the metric in phase space. Subsequently, a numerical analysis reveals that the time evolution (2.62) samples marginals of (2.64). This clearly shows that one nees to inclue all conservation laws in orer to etermine the precise istribution that is sample by the trajectories of a non-hamiltonian system. In a sense, the Nosé-Hoover metho is ergoic even in the case of the harmonic oscillator; however, the term ergoicity is now use in view of two conservation laws, (2.63) an (2.65), an means that the accessible portion of phase space is correctly sample. The statement that the Nosé-Hoover metho is not ergoic for the harmonic oscillator is right if only the first conservation law is taken into account, or if the term is use loosely in the sense of the equivalence of time averages an (canonical) ensemble averages 5. The extensions of the Nosé-Hoover metho, the emon metho an the chain thermostat, create the esire canonical istribution in phase space since they violate the secon conservation law. It is a merit of the phase space analysis that the issue of apparent non-ergoicity of the Nosé-Hoover metho for the classical harmonic oscillator may be illuminate more eeply Note on eterministic chaos an ergoicity Generally speaking, for a given eterministic system it is har to etermine whether it is ergoic or not; however, it is possible to fin inications of when a system may or may not behave well in this respect. Consier a point in phase space with V (q) q i = 0, p i = 0. (2.66) At such a point, the Nosé-Hoover chain ynamics (2.37) effectively stops in the original system since all temporal erivatives of the physical variables vanish. These so-calle Hoover holes are fixe points of the ynamics. If a fixe point is stable, then in its vicinity, the equations will rive the system into the point, which is unacceptable if an ergoic system is require. The stability of a fixe point can be examine by investigating the linearise equations of motion about the point. It has been shown that neither the Nosé-Hoover chain ynamics [24] nor the emon metho with the cubic coupling scheme [23] have stable fixe points. The preceing analysis cannot etermine whether the system is ergoic. Therefore, in orer to obtain numerical evience for ergoic behaviour, brute force methos have also been applie to the chain thermostat an the cubic coupling scheme. In more etail, a necessary conition for ergoicity is that the system be chaotic [29], by which we mean that the time evelopment of the system sensitively epens on the initial conitions. In a chaotic system, ajacent trajectories in phase space iverge exponentially. 5 This is the meaning usually implie in this work, notably in chapter 5. 22

26 2.2. Deterministic temperature control methos This is characterise by a positive so-calle Liapunov exponent, whereas a negative Liapunov exponent inicates that the trajectories converge to a fixe point. The Liapunov exponent vanishes in case the system is recurrent an moves perioically on a stable orbit. Both for the chain thermostat an the emon metho, extensive investigations of the Liapunov exponents have been performe, yieling positive values in all cases. Another property encountere in this context is mixing [29, 30]. If a ynamics is mixing, it istorts any volume element of phase space so strongly that it is eventually sprea over the entire phase space, just as a rop of ink (which is suppose to correspon to a given phase space volume element) is homogeneously istribute within a glass of water after stirring. Mixing guarantees the spontaneous evolution of a non-equilibrium istribution function to equilibrium an is therefore a sufficient conition for ergoicity. This property has numerically been verfie for the cubic coupling scheme [23]. 23

27 3. Coherent states In the preceing chapter, we have escribe two ifferent techniques to incorporate the influence of a heat bath on a classical system on the level of the ynamical equations. The resulting equations of motion have the property that uner certain conitions a time average is equivalent to a canonical ensemble average. With the original Langevin equation an the Nosé-Hoover metho being techniques of classical molecular ynamics, it seems ifficult to irectly translate them to quantum mechanics. Position an momentum of a particle in quantum mechanics are escribe by operators r an p with a non-vanishing commutator, an consequently, there is no common eigenbasis of the operators. The time evolution of a quantum state is governe by Schröinger s equation which escribes a unitary transformation in the Hilbert space of state vectors. So there is no immeiate corresponence in quantum mechanics to Hamilton s classical equations of motion on phase space. However, it appears natural to start an attempt of translation to quantum ynamics with a system that features properties that allow a irect linking to classical mechanics. In this chapter, we will show that for the quantum harmonic oscillator, it is possible to construct state vectors with very special properties, the so-calle coherent states. For these states, the time evolution of the expectation values r = r, p = p (which always obey quasi-classical equations of motion accoring to Ehrenfest s theorem) escribes the full quantum mechanical time evolution. This will allow the esire access to isothermal quantum ynamics. Quantum mechanical operators are enote by an unerlying tile, e. g., we write 1 for the unit operator. While quantum mechanical expectation values are enote by., quantum canonical ensemble averages are enote by., just as classical phase space averages (1.1). There shoul be no confusion since the average quantity inicates which kin of average is implie Introuction In one imension, the quantum Hamiltonian of one particle in an external harmonic oscillator reas H (1) = 1 2 2m p mω2 r 2 = 1 2 ω( ˆp 2 + ˆr 2 ) (3.1) ( = ω a a + 1 ), 2 with ˆr = mω r an ˆp = 1 m ω p being efine as imensionless operators of position an momentum, an a = 1 2 (ˆr + i ˆp ), a = 1 2 (ˆr i ˆp ) being the so-calle annihilation an creation operators. a a is calle the number operator an has eigenstates n, a a n = n n, (3.2) 24

28 3.1. Introuction n being a non-negative integer. a an a act upon the eigenstates (3.1) in the well-known way n of the Hamiltonian a n = n + 1 n + 1, (3.3) a n = n n 1. a an a are non-hermitian operators that satisfy the basic commutation relation [a, a ] = 1. The term coherent states was initially use by R. Glauber (see, e. g., [31]) in the fiel of quantum optics. Following him, we efine coherent states as eigenstates of the annihilation operator a, a α = α α. (3.4) Since a is not hermitian, we cannot expect that its eigenvalues are real an that the appenant eigenstates α are mutually orthogonal. α can take on any complex value, an we write mω α = 2 r + i p, (3.5) 2m ω with r, p R. Note that ω α 2 = p 2 /2m + mω 2 r 2 /2. The expansion of α in terms of the number states n, α = w n n, (3.6) n=0 is foun by inserting the expression (3.6) into the eigenvalue equation (3.4). This yiels the recurrence relation w n+1 = for the expansion coefficients. From this we can infer The coefficient w 0 is etermine from the normalisation conition, which means α α = 1 = α n + 1 w n (3.7) w n = αn n! w 0. (3.8) w n 2 = n=0 α 2n w 0 2 = e α 2 w 0 2, (3.9) n! n=0 w 0 = e 1 2 α 2. (3.10) In the stanar efinition of coherent states, w 0 is arbitrarily chosen to be real. Hence, the expansion of α in eigenstates of the harmonic oscillator n reas α = e 1 2 α 2 n=0 α n n! n. (3.11) 25

29 3. Coherent states Note that the probability P n ( α ) of fining the nth eigenstate n in a coherent state, which is a measure of the energy istribution, is given by a Poisson istribution, P n ( α ) = α 2n e α 2. (3.12) n! Alternatively, coherent states are frequently introuce using the so-calle isplacement 1 or Weyl operator, by the relation D (α) = e αa α a, (3.13) α = D (α) 0, (3.14) where 0 is the harmonic oscillater groun state. The equivalence of this efinition an equation (3.11) can easily be inferre with the help of the equation [32, p.40] e A +B = e 1 2 [A,B ] e A e B, (3.15) vali whenever [A, B ] commutes with both A an B. Since we have [αa, α a ] = α 2, which is a c-number, this requirement is fulfille, an we fin from (3.14) α = e 1 2 α 2 e αa e α a 0. (3.16) The relation a 0 = 0 implies e α a 0 = 0, which allows the simplification α = e 1 2 α 2 e αa 0. (3.17) By expaning the operator e αa, we easily arrive at the esire form (3.11). The expansion (3.11) allows the calculation of the scalar prouct of two coherent states. One obtains α 2 α 1 = e 1 2 α 2 2 e 1 2 α 1 2 e α 2 α 1. (3.18) This expression oes not vanish for any pair (α 1, α 2 ), i. e., as anticipate earlier, there is no pair of mutually orthogonal coherent states. The overlap between two coherent states, α 1 α 2 2 = e α 1 α 2 2, (3.19) is given by a Gaussian. Inserting the efinitions of a, a an α, α, the following equation is obtaine from (3.14), α = e i (pr rp) 0, (3.20) which transforms into α = e i 1 2 pr e i pr i rp e 0 (3.21) 1 The isplacement operator inee performs a isplacement of the wavefunction both in coorinate an momentum space, as will become clear later on. 26

30 3.2. General properties of coherent states using equation (3.15). This expression for the state vector α has the avantage that its coorinate representation x α may be inferre easily. Starting from the coorinate representation of the groun state 0, x 0 = ( mω ) 1/4 e 1 mω x 2 2, (3.22) π which is a Gaussian situate at the origin of the x-axis, the effect of the operators in (3.21) is straightforwar. The operator e i rp shifts the argument x of the wavefunction to (x r) since p is the generator of spatial translations. Subsequently, the operator e i pr escribes a translation in momentum space by the multiplication with the phase factor e i px. The factor e i 1 2 pr is an aitional overall phase factor. On the whole we obtain x α = ( mω ) ( 1/4 exp 1 mω π 2 (x r)2 + i p(x 1 ) 2 r). (3.23) That means that a coherent state, in coorinate representation, correspons to a isplace Gaussian wavepacket with mean position α r α = r an mean momentum α p α = p. Therefore, another common notation for a coherent state is r, p α. (3.24) In this work, we will make use of both notations, epening on the respective context. It is known from stanar quantum mechanics that Gaussian wavepackets minimise Heisenberg s uncertainty relation for the operators of position an momentum. Therefore, coherent states are also frequently referre to as minimum uncertainty states General properties of coherent states The expansion (3.11) of coherent states in the basis of eigenstates of the harmonic oscillator entails very avantageous consequences. The action of the time evolution operator upon a coherent state an the matrix elements of the statistical operator may be calculate analytically. Both operators are exponentials of the Hamiltonian. Time evolution Perhaps the most prominent property of coherent states is their stability uner time evolution in a harmonic oscillator potential. The elementary calculation e iωta a α = e 1 α n 2 α 2 e inωt n (3.25) n! n=0 = e iωt α implies that a coherent state remains a coherent state for the exact quantum mechanical time evolution generate by the Hamiltonian (3.1). In other wors, the time-evolve version of a coherent state is obtaine solely by a uniform rotation of its label, α, in a clockwise manner 27

31 3. Coherent states on a circle of raius α in the complex plane. Using the obvious notation α(t) = mω 2 r(t) + i 2m ω p(t), equation (3.25) may be rephrase in terms of r(t) an p(t), r(t) = r 0 cos(ωt) + 1 mω p 0 sin(ωt), (3.26) p(t) = p 0 cos(ωt) mωr 0 sin(ωt), with α 0 = α(t = 0) = mω 2 r 0 + i 2m ω p 0. Consequently, the time evolution of a coherent state in a harmonic oscillator potential can be cast into a form that is familiar from the classical harmonic oscillator, namely t r = p m, t p = mω2 r. (3.27) This fact will play a ecisive role for the translation of the Nosé-Hoover technique to this quantum system. It has been iscovere implicitly by Schröinger in 1926 [33] who in his funamental work was the first to stuy the time evolution of a spatially isplace harmonic oscillator groun state. Accoring to (3.23), the parameters r an p correspon to mean position an mean momentum of the coherent state. Yet we stress that the statement of (3.27) goes far beyon the much weaker theorem of Ehrenfest. This theorem says that the time epenent expectation values of the operators of position an momentum (enote by r (t) = ψ(t) r ψ(t) an p (t) = ψ(t) p ψ(t) ) satisfy the equations of motion t r (t) = 1 m p (t), (3.28) t p (t) = V (t), as a result of the Schröinger time evolution of a quantum mechanical state ψ(t). These equations again strongly resemble the classical equations of motion. However, contrary to (3.27), this theorem oes not contain a statement about the complete time evolution of a quantum mechanical state, since a general quantum state ψ(t) is not fully parametrise by only two expectation values. Matrix element of the statistical operator The foregoing calculation (3.25) is particularly simple since the phase factor e iωt oes not change the absolute value of α. This is ifferent in the case of the statistical operator, e βh, for which we fin e β ωa a α = e β ωa a e 1 α n 2 α 2 n (3.29) n! = e 1 2 α 2 n=0 n=0 (e β ω α) n n! n = e 1 2 α 2 e 1 2 α 2 e 2β ω e 1 2 α 2 e 2β ω n=0 = e 1 2 α 2 (1 e 2β ω) e β ω α. (e β ω α) n n! n 28

32 3.2. General properties of coherent states We have inserte the fraction 1 = 1 e 2 α 2 e 2β ω to simplify the separation of the ket e β ω α. e 2 1 α 2 e 2β ω From this we etermine the matrix element of the statistical operator: α e βh α = e 1 2 β ω α e β ω a a α (3.30) = e 1 2 β ω e α 2 (1 e β ω). Resolution of unity an over-completeness In orer to prove the following resolution of unity, 2 1 = α π α α, 2 α = (Re α) (Im α), (3.31) one inserts again the expansion (3.11) on the right han sie. Introucing subsequently polar coorinates, α = α e iθ, one obtains = n,m=0 n,m=0 1 2 α n!m! π e α 2 α n α m m n (3.32) 1 α α m+n+1 e 1 α 2 n!m! π 0 π π θe i(m n)θ n m. With the help of the relation 1 π 2π π θeimθ = δ m,0 the summation over m can be carrie out. It then proves useful to change to the new integration variable ζ = α 2, an with the integral formula 0 ζ ζ n e ζ = Γ(n + 1) = n! the right han sie finally takes the form n=0 n n, (3.33) which is the completeness relation for the eigenstates of the harmonic oscillator. Therefore, (3.31) is in fact a vali representation of the ientity operator in terms of coherent states. In the alternative notation of (3.24), we may write r p 1 = 2π r, p r, p. (3.34) The set of all coherent states, { α, α C }, is overcomplete, which implies on the one han that the coherent states are not mutually orthogonal, on the other han that certain (even countable) subsets still span the whole Hilbert space [34]. This question has been iscusse in great etail in the literature. Moreover, the non-orthogonality leas to a number of remarkable expansion properties, e. g., that a given coherent state may be expane into a set of coherent states [35]. In Appenix C, we make use of an expansion of an eigenstate n into a circle of coherent states in orer to erive the time evolution of an arbitrary isplace eigenfunction of the harmonic oscillator. 29

33 3. Coherent states 3.3. Determination of ensemble averages with coherent states Since (3.31) is a correct resolution of unity, coherent states are a vali basis for the calculation of traces of operators an, more specifically, for the calculation of statistical ensemble averages. It will be shown that in the case of the quantum harmonic oscillator, it is possible to cast the expression for a canonical average into the form of an integral over the parameter space of coherent states. This is ue to the simple form of the matrix element of the canonical statistical operator in the basis of coherent states (3.30). The resulting special form strongly ressembles a classical phase space average, with a moifie thermal weight function for the quantum system. The subject of the following section is to etermine the quantum thermal weight functions for a single an two inistinguishable particles in an external harmonic oscillator potential, an for an arbitrary number N of fermions. This will pave the way for a translation of the classical thermostats to the quantum harmonic oscillator One particle To start with, we stuy the case of a single particle in a harmonic oscillator potential. In a manner following Schnack [36], we calculate the trace involve in a quantum canonical ensemble average, B = 1 Z (1) (β) Tr (B (1) e βh ), Z (1) (β) = Tr (e βh (1) ), (3.35) using the basis of coherent states: 1 2 α B = Z (1) (β) π α B (1) e βh α (3.36) 1 2 α = Z (1) (β) π α 1 e 2 βh (1) B e 1 2 βh (1) α 1 2 α = Z (1) (β) π e 1 2 β ω e α 2 (1 e β ω ) e 1 2 β ω α B e 1 2 β ω α 1 2 α = Z (1) (β) π e 1 2 β ω e α 2 (e β ω 1) α B α, where we have use the cyclic invariance of the trace in the first step an equation (3.29) in the secon step. A substitution α e 1 2 β ω α has been carrie out in the final step. We now efine the function w (1) (α) = e α 2 (e β ω 1) = e ( p 2 2m mω2 r 2 ) (e β ω 1)/( ω), (3.37) which specifies the weight with which a coherent state α = r, p contributes to a canonical average. In aition, we efine Z (1) (β) = e 1 2 β ω Z (1) 2 α (β) = π w(1) (α) (3.38) an = 1 e β ω 1, B(α) = α B α, (3.39) 30

34 3.3. Determination of ensemble averages with coherent states which enables us to express the canonical average as follows, B = 1 2 α Z (1) (β) π w(1) (α) B(α), (3.40) or, in (r, p) notation, B = 1 r p Z (1) (β) 2π w(1) (r, p) B(r, p). (3.41) This representation of a quantum canonical ensemble average has obviously the form of a classical phase space average. Therefore, we are le to the following interpretation: The space of the continuous parameters r an p is a phase space, an the quantum canonical ensemble average for a single particle in a harmonic oscillator may be rewritten as a phase space integral with the thermal weight function w (1) (r, p). The term B(r, p) = r, p B r, p, (3.42) is the corresponing representation of the observable B as a phase space function. The istribution function w (1) contains all quantum statistical properties of the system. From (3.37) it can be inferre that formally, it iffers from the Boltzmann istribution function of the classical harmonic oscillator by the factor ( e β ω 1 ) /(β ω). Note that this factor tens to 1 both in the classical ( 0) an in the high-temperature (β 0) limit. We note for the reaer familiar with Wigner functions that this approach to the etermination of a quantum canonical ensemble average, resulting in the expression (3.41), is entirely base on the properties of coherent states. In particular, the thermal weight function w (1) (r, p) is not the Wigner function of the thermal state ρ = 1 Z (1) (β) exp( βe n ) n n. (3.43) n=0 The matrix element x ρ x can be calculate elegantly with the ai of (3.41), see appenix B. The Wigner function is now efine by the Fourier transform of this matrix element with respect to the quantum jump from x 1 2 ξ to x ξ, W (x, p) = 1 ξ e i pξ x + 1 2π 2 ξ ρ x 1 2 ξ, (3.44) with the result W (x, p) = 1 ( 1 ) ( π tanh 2 β ω exp 2 ( 1 ) ( p 2 ω tanh 2 β ω 2m mω2 x 2)) If we compare W (x, p) to the normalise thermal weight function, 1 2π 1 Z (1) (β) w(1) (r, p) = 1 ( 2π (eβ ω 1) exp 1 ( p ω (eβ ω 2 1) 2m mω2 r 2)). (3.45), (3.46) we fin that they may be converte into one another by the formal substitution of the expression 2 tanh( 1 2 β ω) by eβ ω 1. These terms, corresponing to the inverse with (i. e., the 31

35 3. Coherent states narrowness ) of W (x, p) an w (1) (r, p), respectively, eviate consierably at low temperatures (β ), since we have ( ( 1 ) ) lim 2 tanh β 2 β ω = 2, while e β ω 1 grows exponentially with β. In the high temperature limit (β 0) or equivalently in the classical limit ( 0), both terms ten towars the same limiting expression, namely β ω. In these limits both resulting weight functions correspon to the classical Boltzmann istribution. Nevertheless, for any finite temperature, (3.45) an (3.46) are Gaussian istributions with unequal withs, w (1) being always narrower than W (x, p). Therefore there is no equivalence of w (1) to the Wigner function W (x, p). Eventually, we mention that the thermal weight function for a system of istinguishable quantum particle is simply the prouct of one-particle thermal weight functions. The case of inistinguishable quantum particles is of course much more interesting Two ientical particles In precise analogy to the case of a single particle, we are able to etermine a thermal weight function w ε (2) (α 1, α 2 ) for the system of two ientical quantum particles. The case of two ientical particles, i. e. two fermions or bosons, contains alreay the basic principle of quantum mechanical inistinguishability 2. We efine the operators of symmetrisation an antisymmetrisation in the two-particle Hilbert space, S + = 1 2 (1 + P 12 ), S = 1 2 (1 P 12 ). (3.47) In a general notation, we will use S ε, implicating ε = + or. The operators S ε are efine such that they are projection operators, an therefore they are hermitian an iempotent. Recall that in a system of ientical particles, all vali observables commute with S ε. In orer to have a common notation for both fermions an bosons, we use S α 1, α 2 if ε = A ε = (3.48) S + α 1, α 2 if ε = + for the (anti-)symmetrise two-particle state vector. The norms of these vectors may easily be calculate, an we get A ε A ε = 1 2 (1 + εe α 1 α 2 2 ). (3.49) The completeness relation for the two-fermion (two-boson) Hilbert space is obtaine by an (anti-)symmetrisation of a tensor prouct of two ientity operators, 2 α 1 2 α 2 1 ε = π π S ε α 1, α 2 α 1, α 2 S ε. (3.50) 2 Note that throughout this work, we o not consier spin egrees of freeom. 32

36 3.3. Determination of ensemble averages with coherent states In orer to get an expression with normalise state vectors, we multiply an ivie the integran by A ε A ε : 2 α 1 1 ε = π 2 α 2 π A ε A ε S ε α 1, α 2 α 1, α 2 S ε A ε A ε (3.51) This resolution of unity may now be use for the calculation of a trace an, more specifically, for the evaluation of a thermal expectation value. We write own the Hamilton operator of two non-interacting quantum particles in an external harmonic oscillator potential, H (2) = H (1) H (1). (3.52) The starting point for the calculation of w ε (2) is the following expression for the calculation of a thermal expectation value B as a phase space integral, B = = 1 Z (2) ε 1 (β) Tr Z ε (2) (β) ( B e βh (2) ) (3.53) 2 α 1 π 2 α 2 π α 1, α 2 S ε B e βh (2) S ε α 1, α 2, with Z ε (2) (β) being the respective two-particle partition function. Note that we may rop the projector S ε acting upon the ket using the cyclic invariance of the trace an the iempotency of the projector S ε. Decomposition of e βh (2) = e βh(2) /2 βh e (2) /2 an a cyclic shift of the operators uner the trace enables further simplification, taking avantage of the specific properties of coherent states in a harmonic oscillator potential, (3.29). After the following calculation, 1 2 α 1 2 α 2 B = Z ε (2) (β) π π α 1, α 2 e 1 2 βh (2) S ε B S εe 1 2 βh (2) α1, α 2 (3.54) = = = 1 Z ε (2) (β) 1 Z ε (2) (β) 1 Z ε (2) (β) 2 α 1 π 2 α 1 π 2 α 1 π 2 α 2 π e 1 2 β ω α 1 2 (1 e β ω) e 1 2 β ω α 2 2 (1 e β ω ) e 1 2 β ω α 1, e 1 2 β ω α 2 S ε B S ε e 1 2 β ω α 1, e 1 2 β ω α 2 2 α 2 π eβ ω e α 1 2 (e β ω 1) e α 2 2 (e β ω 1) α 1, α 2 S ε B S ε α 1, α 2 2 α 2 A π eβ ω e α 1 2 (e β ω 1) e α 2 2 (e β ω 1) ε B A ε A ε A ε } {{ } A ε A ε w ε (2) (α 1, α 2 ) we ientify w ε (2) (α 1, α 2 ) as the thermal weight of the expectation value Aε B Aε A ε A ε. With the (2) efinition Z ε (β) = e β ω Z ε (2) (β), we may finally write B = 1 2 α 1 Z ε (2) (β) π 2 α 2 π w(2) ε (α 1, α 2 ) A ε B A ε A ε A ε. (3.55), 33

37 3. Coherent states w ε (2) (α 1, α 2 ) is not merely the prouct of two one-particle thermal weight functions (as it woul be the case for two istinguishable particles), but contains in aition the factor A ε A ε that accounts for the quantum effects of inistinguishability. Moreover, w ε (2) (α 1, α 2 ) cannot be written as a prouct of two functions epening only on α 1 an α 2, respectively, since the term A ε A ε oes not have this separation property. This is a manifest consequence of the quantum mechanical principle of inistinguishability. In more etail, the state A ε escribing two ientical particles is entangle, which means that A ε oes not have the form of a tensor prouct α 1 α 2 α 1, α 2. On the level of the istribution function, this is reflecte by the non-separability (or entanglement) of w (2) ε. For fermions, we have A A = 1 2 (1 e α 1 α 2 2 ) (see equation (3.49)). In case α 1 = α 2, this expression vanishes along with the thermal weight function w (2), inepenent of temperature. This is unerstanable since a quantum state with two ientical fermions in the same one-particle state is forbien by the Pauli exclusion principle an therefore oes not contribute to a thermal average. In contrast, for bosons we have A + A + = 1 2 (1 + e α 1 α 2 2 ), which contains a ifferent sign that enhances the thermal weight of the quantum state with two bosons in the same one-particle state. (α 1, α 2 ) yiels the correct partition function The integral over the thermal weight function w (2) ε (2) Z ε (β). To show this, we calculate Z (2) ε (β) = 1 (3.56) 2 α 1 2 α 2 = π π e α 1 2 (e β ω 1) e α 2 2 (e β ω 1) 1 2 (1 + εe α 1 α 2 2 ) = 1 ( 2 ) 2 α 1 2 π e α 1 2 (e β ω 1) + ε 1 ( 2 α 1 2 ) α 2 2 π π e α 1 2 (e β ω 1) e α 2 2 (e β ω 1) e α 1 α 2 2 = 1 ( ) e β ω + ε 1 ( ) e 2β ω 1 = 1 2 ( Z (1) (β) 2 + ε Z (1) (2β)). This is a correct, well-known recursion relation [37] for the two-particle partition function. The secon integral is solve most easily by a change of variables from α 1, α 2 to α + = 1 2 (α 1 + α 2 ), α = 1 2 (α 1 α 2 ), which leas to a separation of the ouble integral. To set up the equations of motion of the isothermal ynamics, we stress that in orer to replace the phase space average by a time average, the function w ε (2) has to be sample uring time evolution. The expression B = 1 2 α 1 Z ε (2) (β) π 2 α 2 π e α 1 2 (e βω 1) e α 2 2 (e βω 1) A ε B A ε (3.57) that follows from (3.55) by a cancellation of A ε A ε seemingly inicates that for the calculation of B it woul suffice to sample the prouct of two one-particle thermal weight functions uring time evelopment an take the temporal average of the phase space function A ε B A ε. However, this conclusion is wrong, since the role of the normalisation (i. e., the partition function) is not properly taken into account. 34

38 3.3. Determination of ensemble averages with coherent states This can be outline more explicitly as follows. In a numerical simulation, a time average is calculate as an algebraic mean of values of the respective phase space function at points that are situate on the trajectory an equiistant in time. Let M be the number of phase space points contributing to the algebraic mean. The normalisation of the algebraic mean with the factor 1/M implies a normalisation with the histogram of the contributing phase space points, i. e. the integral of the thermal weight function that has been sample. The value of this integral is precisely the partition function, see (3.56). Consequently, if the thermal weight function sample uring a simulation is simply the function for istinguishable particles, e α 1 2 (e βω 1) e α 2 2 (e βω 1), the normalisation will correspon to the Maxwell-Boltzmann partition function for two istinguishable particles, Z(2) MB Z (β) = (1) (β) 2, an the average thus obtaine will correspon to the quantity B MB = 1 2 α 1 2 α 2 Z (2) MB (β) π π e α 1 2 (e βω 1) e α 2 2 (e βω 1) A ε B A ε, (3.58) which is not equal to (3.57). With the notation. ε for an average accoring to (3.57), we may write B ε = (2) Z MB Z ε (2) S ε B MB. (3.59) This reveals that averages of the type. MB (which coul be etermine with a simpler thermal weight function) are useless for the calculation of ensemble averages of the type. ε, since the latter can be obtaine from the former only if the complete partition function is known Case of N fermions A calculation analogous to (3.54) in the case of an arbitrary number N of fermions results in B = = 1 Z ε (N) (β) 2 α 1 2 π... α N e N 2 β ω π A e α j 2 (e β ω 1) ε B A ε A ε A ε A j=1 ε A ε } {{ } w ε (N) (α 1,... α N ) 1 2 α 1 2 Z ε (N) (β) π... α N w ε (N) (α 1,... α N ) A ε B A ε π A ε A ε N. (3.60) Note we have move the factor e N 2 β ω connecte to the one-particle groun state energy to the partition function, Z(N) ε (β) = e N 2 β ω Z ε (N) (β). In orer to calculate the explicit form of A ε A ε in the case of fermions, consier the general N-particle operator of antisymmetrisation, S (N) = 1 N! sgn (π)p π. (3.61) π π enotes a permutation of the N-tuple (1,... N), an P π correspons to the permutation operator which acts upon a N-particle prouct state as P π α 1, α 2,... α N = α π(1), α π(2),... α π(n). (3.62) 35

39 3. Coherent states (N) Using the properties of hermiticity an iempotency of S, we can easily calculate (N) A A = α 1, α 2,... α N S α 1, α 2,... α N (3.63) = 1 sgn (π) α 1 α N! π(1)... α N α π(n) π = 1 N! et ( α k α l ). This expression in terms of a eterminant will turn out to be extremely avantageous, since it will permit a far-reaching analytical evelopment of the N-fermion Nosé-Hoover metho. In the case of bosons, in contrast, it is not possible to obtain an analogous compact an useful expression for A + A +. Accoringly, it will be impossible to evise the Nosé-Hoover metho in the N-boson case. 36

40 4. Isothermal quantum ynamics for the harmonic oscillator After the preparations of the preceing chapter that provie suitable phase-space integral representations of quantum canonical average values, we are now reay to esign equations of motion which will be solve by isothermal trajectories in the parameter space of coherent states A quantum Langevin equation Inspecting expression (3.41) for a canonical average of an arbitrary observable of the quantum harmonic oscillator an expression (3.37) for the quantum thermal weight function, one sees that the time evolution of the parameters of coherent states nees to sample Gaussian istributions in orer that time averages equal canonical ensemble averages. This is in complete analogy to the classical case. Therefore, a translation of the classical Langevin equation to an equation of motion for the parameters of coherent states is straightforwar, since the istribution functions are Gaussians in both cases an iffer only with regar to the respective withs. But it is known that the with of the Gaussians is relate to the amplitue of the fluctuating force, see equation (2.4). Therefore, in orer to obtain soun equations of motion of first orer in time that may be interprete as equations of motion for the parameters of coherent states, we replace q in the classical Langevin equation (2.2) by the parameter r while keeping the letter p, resulting in r t = p m, p t = V q γp + F (t). (4.1) We interpret (4.1) as equations of motion for the parameters r an p of coherent states. Moreover, we postulate a moifie fluctuation-issipation theorem for the fluctuating force, ω F (t 1 )F (t 2 ) = 2mγ e β ω 1 δ(t 1 t 2 ). (4.2) From the analysis of the case of the classical harmonic oscillator, it is clear that in the long-time limit, a time average over the solution of (4.1) will correspon to an average over the parameter space with Gaussian istribution functions. While the classical version (2.4) of the fluctuationissipation theorem leas to a sampling of the Maxwell-Boltzmann istribution (2.11), the replacement of the term k B T by ω/(e β ω 1) will lea to a sampling of the quantum thermal weight function w (1). To summarise this section, we saw that for a single particle, it is very easy to moify the classical Langevin equation along with the corresponing classical fluctuation-issipation relation such that equations of motion for the parameters of coherent states are obtaine an the time evolution samples the istribution function w (1). Formally, the result iffers from the classical case only with regar to the temperature epenence of the amplitue of the fluctuating force. This proceure is possible since both the classical canonical istribution 37

41 4. Isothermal quantum ynamics for the harmonic oscillator function on phase space an the quantum thermal weight function on the parameter space of coherent states are Gaussians which iffer only with respect to the temperature epenence of their withs. Hence, we recognise that this stochastic approach is limite to the cases of Gaussian istribution functions, corresponing to a single or istinguishable quantum particles in an external harmonic oscillator potential. The istribution function w ε (2) of two unistinguishable bosons or fermions is not a prouct of Gaussians. Beyon, w ε (2) is not even separable into a prouct of two functions each of which only epens on the parameter of one particle. If it were possible to fin expressions V ε1 (r 1 ), V ε2 (r 2 ) such that w (2) ε ( p 2 ) ( = C exp 1 p 2 ) 2m + V ε1(r 1 ) exp 2 2m + V ε2(r 2 ), (4.3) a use of a moifie version of the classical Langevin equation woul be conceivable. But the quantum mechanical entanglement of w ε (2) prohibits a transfer of the classical Langevin equation to the case of inistinguishable quantum particles The quantum Nosé-Hoover thermostat One particle 1 In the present section, we will show that for a single quantum particle in an external harmonic oscillator potential, it is also possible to evelop sets of eterministic ynamical equations for the parameters of coherent states that ressemble closely the classical temperature control schemes that have been the subject of chapter 2. Accoring to the iea of these approaches, we will introuce pseuofriction terms into the equations of motion (3.27) of coherent states. The time epenence of the pseuofriction coefficients will be etermine in such a way that the istribution function w (1) for one particle is sample provie the time evolution is ergoic. We will start with the simple Nosé-Hoover scheme, extening it to Nosé-Hoover chains in the following. The emon metho is also sketche The Nosé-Hoover thermostat an the Nosé-Hoover chain Aopting the notation of Martyna et al. [24] that is most suitable for the generalisation to chain thermostats, we investigate the following analogue of the classical Nosé-Hoover ynamics for the quantum ynamics of coherent states: t r = p m, t p = mω2 r p p η Q. (4.4) The equation of motion of the parameter p is supplemente by a term similar to a frictional force, p p η /Q. This moification obviously touches the time evolution of the overall phase factor e i 1 2 pr of a coherent state (see equation (3.23)), but the phase factor oes not play a role in the present context of quantum statistical averages where a phase inepenent expectation value B(r, p) = r, p B r, p is average (see equation (3.41)). 1 This section, along with section 5.1, is the main content of the publication [38]. 38

42 4.2. The quantum Nosé-Hoover thermostat The key point is the time evolution the pseuofriction coefficient p η. It is etermine by the conition that the esire istribution function, ( ) f(r, p, p η ) = w (1) (r, p) exp β p2 η 2Q ( ( p 2 = exp 2m + 1 ) e 2 mω2 r 2 β ω 1 ω β p2 η 2Q is a stationary solution of the following generalise Liouville equation (cf. (2.19)) in the mixe quantum-classical phase space with elements x = (r, p, p η ): ( ) t f = f x ẋ (4.6) ( = f r ṙ + pṗ + ) p η. p η Note that constant overall prefactors of f cancel in this equation, which means that normalisations of f may be neglecte in stuies of this equation. We calculate the left han sie of (4.6), employing the equations of motion of r an p, (4.4): t f = f p ṗ + f r ṙ + f ( ( p ) e β ω = f mṗ + 1 mω2 rṙ ( p 2 p η e β ω 1 = f m Q ω ), (4.5) ṗ η (4.7) p η β p ) η Q ṗη β p η Q p η ω ). On the right han sie of (4.7), we have the freeom to impose the constraint p η / p η = 0 that is common in this context (cf. equation (2.27)). We obtain f ( x ẋ ) = f p η Q. (4.8) Equating (4.7) an (4.8) yiels the following equation of motion for p η : t p η = 1 ( p 2 e β ω ) 1 1. (4.9) β m ω A comparison of this result with (2.34) shows that as in the case of the quantum istribution function w (1), the only ifference between this equation an its classical counterpart is given by the factor (e β ω 1)/(β ω). Moreover, (4.9) retains the property of its classical counterpart that the time evolution of the pseuofriction coefficient is governe by the eviation of the actual value of a quantity relate to the kinetic energy from its canonical average value. This can be inferre by evaluating the canonical average e β ω 1 ω p2 m using (3.41). In aition, one introuces the equation of motion t η = p η Q = 1 (4.10) (4.11) 39

43 4. Isothermal quantum ynamics for the harmonic oscillator of the variable η that contributes to the conserve quantity ( p H 2 = 2m + 1 ) e 2 mω2 r 2 β ω 1 β ω + p2 η 2Q + k BT η. (4.12) The set of ynamical equations (4.4), (4.9) (along with (4.11)) forms a quantum Nosé-Hoover thermostat for a single particle in an external harmonic oscillator potential. It is expresse in terms of equations of motion for the parameters of coherent states. Since the istribution function that nees to be sample on the parameter space uring time evolution is a twoimensional Gaussian as in the classical case, we anticipate ergoicity problems for the simple Nosé-Hoover scheme. Therefore, in practice, we will employ a chain thermostat that has been escribe for the classical case in section The application of the iea of a chain of thermostats to the quantum case oes not infer anything new compare to the classical case, since only the first pseuofriction coefficient of the chain interacts with the quantum phase space variables The emon metho In section 2.2.2, we have introuce an iscusse another generalisation of the Nosé-Hoover thermostat that is frequently use in classical molecular ynamics, namely, the so-calle emon metho propose by Kusnezov, Bulgac, an Bauer [23]. Two pseuofriction coefficients, socalle emons, are introuce for temperature control. The classical equations of motion both of the positions an the momenta are supplemente by aitional terms. Now, we introuce emons into the quantum equations of motion of the parameters of coherent states: t r = p m g 2(ξ)F (r, p), t p = mω2 r g 1(ζ)G(r, p). (4.13) F (r, p), G(r, p) are arbitrary functions of the quantum phase space variables. g 1 (ζ), g 2 (ξ) are functions of the emon variables which have to be chosen such that the integration of the istribution function converges, an g 1, g 2 are the respective erivatives. The istribution function f on the phase space (r, p, ζ, ξ) = x reas ( ( p 2 f(r, p, ξ, ζ) = exp 2m mω2 r 2) e β ω 1 ( g1 (ζ) β + g 2(ξ) )) ω κ 1 κ 2, (4.14) an the time evolution of the emons is, as above, euce from the requirement that f is a solution of the generalise Liouville equation (2.19). The aitional constraints (cf. equation (2.27)), ζ ζ = 0, ξ ξ = 0, are also employe. From a comparison of coefficients of the functions g 1, g 2, we finally obtain t ζ = κ ( 1 p β t ξ = κ 2 β ω 1 m Geβ ω G ) p ( mω 2 rf eβ ω 1 F ω r, (4.15) ). (4.16) 40

44 4.2. The quantum Nosé-Hoover thermostat It is interesting to notice that G p = eβ ω 1 p G, (4.17) ω m i. e., the ratio of the canonical averages of the quantities that etermine the time erivative of the emons is eβ ω 1 ω. A comparison with equation (2.30) shows that here again, a replacement of β by eβ ω 1 ω leas from the classical to the quantum case (cf. equation (3.37)). The quantity ( p H 2 = 2m + 1 ) e 2 mω2 r 2 β ω 1 β ω + g 1(ζ) κ 1 + g 2(ξ) κ β t ( G t p g 1 + F ) r g 2 (4.18) is conserve uring the time evolution efine by (4.13), (4.15), (4.16). The cubic coupling scheme (2.32) leas in the quantum case to the special set of equations of motion t r = p m ξr3, t ζ = κ ( 1 p 2 β m t ξ = κ 2 β e β ω 1 ω t p = mω2 r ζ 3 p, (4.19) ) 1, ( mω 2 r 4 eβ ω ) 1 3r 2 ω that will be investigate in chapter 5. Finally, we note that the equations (4.15), (4.16) may easily be linke to the equations of motion propose by Kusnezov [19]. w (1) plays the role of Kusnezov s ρ(q, P ). However, while Kusnezov s scheme is limite to quantum systems of finite imensionality, the present metho works for this system with a Hilbert space of infinite imensionality because our approach takes avantage of the properties of coherent states Two ientical particles Quantum mechanical systems of ientical, i. e. inistinguishable particles feature a large number of phenomena which are unknown in classical mechanics. The consequences of inistinguishability reach very far an are extremely important. To name but two, we mention the Pauli exclusion principle for fermions that explains the perioic table of elements, an boson statistics which leas to the recently observe phase transition of a boson gas to a Bose-Einstein conensate [11, 12]. Therefore, the extension of the temperature control methos to a system of more than one quantum particle is highly esirable. We stress that a priori, it is not clear whether the techniques of the foregoing section may be applie to a many-particle quantum system at all. The structure of the two-particle istribution functions w ε (2) oes not allow a separation (4.3) into a prouct of two functions each of which epens on the parameters of only one particle. We have alreay argue that this entanglement prohibits an approach using an equation of the Langevin type (see 4.1). Consequently, it may appear unlikely that the many-particle quantum thermal weight function can be sample by a time evolution of the Nosé-Hoover type. In view of this, the following investigations give remarkable results. 41

45 4. Isothermal quantum ynamics for the harmonic oscillator The simple Nosé-Hoover thermostat We stuy the following equations of motion of the coherent states parameters (r 1, p 1, r 2, p 2 ) of two ientical quantum particles: t r 1 = p 1 m, t r 2 = p 2 m, (4.20) t p 1 = mω 2 p η1 r 1 p 1, Q 1 t p 2 = mω 2 p η2 r 2 p 2. Q 2 The equations of motion of the pseuofriction coefficients p η1, p η2 have to be etermine in a proceure analogue to the case of a single particle, i. e. we require that the esire istribution function ( ( p f ε (2) (α 1, α 2, p η1, p η2 ) w ε (2) 2 ) ) η1 (α 1, α 2 ) exp β + p2 η 2 (4.21) 2Q 1 2Q 2 ( ( = e ( α α 2 2 )(e β ω 1) (1 + εe α 1 α 2 p 2 ) ) 2 η1 ) exp β + p2 η 2 2Q 1 2Q 2 ef = e U (1 + εe V ) is a stationary solution of the generalise Liouville equation (4.6) on the phase space with elements (α 1, α 2, p η1, p η2 ) = (r 1, p 1, r 2, p 2, p η1, p η2 ) = x. The abbreviations U an V are efine as ( p U = ( α α 2 2 )(e β ω 2 ) η1 1) + β + p2 η 2 (4.22) 2Q 1 2Q 2 ( 1 = 2 mω2 (r1 2 + r2) ) e 2m (p2 1 + p 2 β ω ( 1 p 2 ) η1 2) + β + p2 η 2, ω 2Q 1 2Q 2 V = α 1 α 2 2 (4.23) = mω 2 (r 1 r 2 ) m ω (p 1 p 2 ) 2. For left han sie of the the Liouville equation, we get an for the right han sie, we calculate ( ) ( x ẋ = ṙ 1 + r 1 t f (2) ε (α 1, α 2, p η1, p η2 ) = U ε V e U e V (4.24) = p η 1 Q 1 + p η 2 Q 2, r 2 ṙ 2 + = ( U + V )f (2) ε p 1 ṗ 1 + p 2 ṗ V e U p η1 ṗ η1 + ) ṗ η2 p η2 (4.25) using the equations of motion (4.20) an imposing, in analogy to the case of a single particle, the constraints ṗ ηi p ηi = 0, i = 1, 2. (4.26) 42

46 4.2. The quantum Nosé-Hoover thermostat We obtain the following form of the Liouville equation: t f ε (2) = f ε (2) ( ( U + V )f ε (2) + V e U = f ε (2) pη1 ( ) x ẋ ) + p η 2 Q 1 Q 2 ( pη1 U = + p η 2 Q 1 Q 2 ) ε V 1 e V + ε1. (4.27) The explicit expressions for U an V are given by ( p 2 U = 1 p η1 m V = p 1 p 2 m ω ) p η2 e β ω 1 Q 2 ω ) + p2 2 Q 1 m ( p η2 p η1 p 2 p 1 Q 2 Q 1. ( pη1 + β ṗ η1 + p ) η 2 ṗ η2 Q 1 Q 2, (4.28) Hence, in the Liouville equation, only U contains the temporal erivatives of the pseuofriction coefficients. This permits to isolate ṗ η1 an ṗ η2 on one sie. A comparison of the coefficients of the terms p η1 /Q 1 an p η2 /Q 2 on both sies yiels the following equations of motion for the pseuofriction coefficients: ṗ η1 = 1 ( p 2 1 β m ṗ η2 = 1 β e β ω 1 ω 1 + εp 1 p 1 p 2 m ω ( p 2 2 e β ω 1 p 1 p 2 1 εp 2 m ω m ω ) 1 e V + ε1, (4.29) ) 1 e V + ε1. In fact, these equations of motion for the pseuofriction coefficients fulfill the requirement (4.26). Recall that the value + of ε applies for bosons, the value for fermions. The set of equations of motion (4.20), (4.29) conserves the quantity H ε = 1 β ln w(2) ε (α 1, α 2 ) + p2 η 1 2Q 1 + p2 η 2 2Q β t ( t pη1 (t ) + p η 2 (t ) ). (4.30) Q 1 Q 2 Consiering the equations of motion (4.29), we fin that the part p2 i e β ω 1 m ω 1, i = 1, 2, is familiar from the ynamics of a single thermalise particle, see equation (4.9). However, in the present case of two ientical particles we fin aitional terms that are irect consequences of the principle of inistiguishability in quantum mechanics. The ynamics of the pseuofriction coefficient of each particle epens on the parameters of both particles, which is the result of the entanglement prominent in w ε (2). In the following chapter 5, we will extensively stuy the set of equations (4.20), (4.29), an we will emonstrate how the effects of Bose-attraction an Pauli-blocking are reflecte in the thermostate ynamics. We stress that although the thermal istribution function of two ientical quantum particles w ε (2) in an external harmonic oscillator potential is not separable in the sense of equation (4.3), it has turne out to be possible to esign equations of motion for the pseuofriction coefficients p ηi such that w ε (2) is a stationary solution of the generalise Liouville equation. Surprisingly, the entanglement involve in the istribution function oes not lea to a failure of the ansatz (4.20). Therefore, the set of ynamical equations (4.20) along with (4.29) constitutes a substantial 43

47 4. Isothermal quantum ynamics for the harmonic oscillator enhancement of the applicability of the classical Nosé-Hoover scheme. A etaile analysis of the general question whether the equations of motion (4.20), (4.29) actually lea to an ergoic ynamics will be given later on, see section 5.2. In comparison to the case of a single particle, where the transition from classical to quantum mechanics essentially reuces to an alternate interpretation of the symbols r an p along with suitable moifications of the withs of Gaussian istribution functions, the present section strikingly emonstrates the flexibility an power of the eterministic Nosé-Hoover scheme. Surprisingly, its range of applicability consierably excees that of the stochastic Langevin approach iscusse in section (4.1). An aitional term in the ynamical equations of the pseuofriction coefficients is sufficient to account for the effects of the quantum mechanical principle of inistinguishability The emon metho The analogous approach using a KBB-scheme starts with the set of equations t r 1 = p 1 m g r 1 (ξ 1 )F r1 (r 1, p 1 ), (4.31) t p 1 = mω 2 r 1 g p 1 (ζ 1 )G p1 (r 1, p 1 ), t r 2 = p 2 m g r 2 (ξ 2 )F r2 (r 2, p 2 ), t p 2 = mω 2 r 2 g p 2 (ζ 2 )G p2 (r 2, p 2 ). This scheme has the obvious avantage that positions an momenta are treate symmetrically, i. e. pseuofriction coefficients are present in all equations of motion. The functions F r1, F r2, G p1, G p2 are arbitrary. In contrast to the original equations publishe in [23], we have restricte the set of variables the respective functions epen on. However, our specific choice will turn out to be sufficient to ensure ergoicity so that this simplification is acceptable. The esire istribution function reas f (2) ε (α 1, α 2, ξ 1, ξ 2, ζ 1, ζ 2 ) = w ε (2) (α 1, α 2 ) exp ( ( gr1 (ξ 1 ) β + g r 2 (ξ 2 ) + g p 1 (ζ 1 ) + g p 2 (ζ 2 ) )) κ r1 κ r2 κ p1 κ p2. (4.32) The functions g r1, g r2, g p1, g p2 are chosen such that they provie a normalisable istribution function. The time epenence of the pseuofriction coefficients is erive from a comparison of coefficients of the functions g r 1, g r 2, g p 1, g p 2 in the generalise Liouville equation (4.6) on the phase space with elements (r 1, p 1, r 2, p 2, ξ 1, ξ 2, ζ 1, ζ 2 ) = x. We obtain t ζ 1 = κ ( p 1 p1 β m G e β ω 1 p 1 G ) p 1 p 1 p εg p1 ω p 1 m ω e V, (4.33) + ε1 t ζ 2 = κ ( p 2 p2 β m G e β ω 1 p 2 G ) p 2 p 1 p 2 1 εg p2 ω p 2 m ω e V, + ε1 t ξ 1 = κ r 1 β t ξ 2 = κ r 2 β ( mω 2 e β ω 1 r 1 F r1 ω ( mω 2 e β ω 1 r 2 F r2 F r 2 ω F ) r εf r1 m ω(r 1 r 2 ) r 1 e V, + ε1 ) 1 εf r2 m ω(r 1 r 2 ) r 2 e V. + ε1 44

48 4.2. The quantum Nosé-Hoover thermostat A comparison of this result to the ynamical equations (4.15) an (4.16) reveals that the first two terms in the brackets are familiar from the one-particle case. Again, aitional terms that account for the effects of Pauli-blocking an Bose attraction are now present in the pseuofriction coefficients of all parameters. The conserve quantity reas H ε = 1 β ln w(2) ε (α 1, α 2 ) + g r 1 (ξ 1 ) + g r 2 (ξ 2 ) + g p 1 (ζ 1 ) + g p 2 (ζ 2 ) κ r1 κ r2 κ p1 κ p2 + 1 t ( t Fr1 g r β r 1 + F r 2 g r 1 r 2 + G p 1 g p 2 p 1 + G ) p 2 g p 1 p 2 2. (4.34) Case of N fermions In section 3.3.3, we have seen that the thermal istribution function of N fermions may be written in the compact form w (N) (α 1,... α N ) = 1 N! N e α j 2 (e β ω 1) et ( α k α l ). (4.35) j=1 We use the notation of the Nosé-Hoover chain (cf. equation (4.20)) in the following ansatz for the equations of motion of the coherent states parameters. However, for calculational reasons, it turns out most favourable to use a form in terms of the complex parameters α m, α m = iωα m α m α m 2 α m = iωα m + α m α m 2 p ηm Q m, (4.36) p ηm Q m. Formally, we can treat α m an α m as inepenent parameters. The thermal istribution function that we will eman to be a stationary solution of the generalise Liouville equation reas f (N) N (α, α, p ηl ) w (N) (α, p ηj α ) exp β (4.37) 2Q j=1 j N N = e α j 2 (e β ω 1) p ηj et ( α k α l ) exp β, 2Q j=1 j=1 j } {{ } } {{ } } {{ } X Y Z where we have roppe the constant overall prefactor 1/N! that cancels in consierations of the generalise Liouville equation (4.6). For this equation, we nee the total time erivative of f (N), t f (N) = ( ) ( ) ( ) t (X Y Z) = t X Y Z + X t Y Z + XY t Z. (4.38) Therefore, we calculate successively the time erivatives of X, Y, an Z. 45

49 4. Isothermal quantum ynamics for the harmonic oscillator For the first erivative, we nee an We obtain Next, we consier which requires the expression Y α m = t X = N m=1 X α m = α m (eβ ω 1) ( X α m + X ) α m α α m m N j=1 = α m (eβ ω 1) X, X α m t X = X (eβ ω 1) = X eβ ω 1 2 t Y = N m=1 e α j 2 (e β ω 1), (4.39) (4.40) = α m (e β ω 1) X. (4.41) N (α m α m + α m α m) (4.42) m=1 N m=1 α m et ( α k α l ) = p ηm Q m (α m α m )2. ( Y α m + Y ) α m α α m m N k,l=1, (4.43) ( α k α l ) α m A kl, (4.44) where the so-calle cofactor A kl of the element a kl = α k α l of the original matrix A = (a kl ) is given by A kl = ( 1) k+l M kl. (4.45) Here M kl enotes the (N 1)th orer eterminant that is erive from the original matrix A by eletion of the kth row an the lth column 2. Note that in the present case, since A is a hermitian matrix, the cofactors satisfy the relation A kl = A lk. (4.46) This is most easily seen from the following representation of the inverse matrix A 1 in terms of cofactors, (A 1 ) kl = 1 et A A lk, (4.47) 2 Equation (4.44) is an application of the general theorem: If the elements a ij of a matrix A are functions of x, then the following equation hols: x et A = i,j a ij Aij (cf. [39, ch. 14]). x 46

50 4.2. The quantum Nosé-Hoover thermostat taking into account that A 1 is also hermitian. Since we have α m α m = 1, it is clear that k l, we fin ( α k α l ) = δ α m km k l α m ( α k α l ) vanishes in case k = l = m. If ( 1 ) ( 2 α k α k α l + δ lm 1 ) 2 α l + α k α k α l. (4.48) With this, we can get to the expression N ( α k α l ) A kl = (( 1 ) ( α m 2 α m α m α k A mk + 1 ) ) 2 α m + α k α k α m A km. k,l=1 k m (4.49) We insert this expression as well as the equations of motion (4.36) on the right han sie of (4.43). Using the abbreviations we fin t Y = m C km = α k α k α m A km (4.50) D mk = 1 ) 2 α m ( α m α k A mk + α k α m A km + C km, ( iω( α m C km + α m C km ) + p η m α m α ) m ( D mk + Dmk Q m 2 ). (4.51) k m Fortunately, this expression simplifies consierably, because the first aen in the sum vanishes, m C km + α m k m( α mckm ) = α m C km + α mckm (4.52) m k m m k m = α m C km + α k C mk = 0, m k m m k m since we have α mc km = α kc mk, see the efinition of C km, equation (4.50). The final expression we obtain after a reorganisation of terms reas t Y = p ηm α m α m Q m 2 m Lastly, we nee k m {( α ( ) ) } m α m α k A mk + c.c. + α k α m α k A mk c.c. 2 t Z = βz For the right han sie of the generalise Liouville equation, we fin ( ) N ( f x ẋ αm = X Y Z + ) α m α m α m = X Y Z N j=1 m=1 N m=1. (4.53) p ηj Q j ṗ ηj. (4.54) p ηm Q m. (4.55) 47

51 4. Isothermal quantum ynamics for the harmonic oscillator Here again, the constraint ṗ ηm / p ηm = 0 has been impose. We insert equations (4.42), (4.53) an (4.54) into (4.38) an ivie both sies of the generalise Liouville equation by f (N) = X Y Z. In the resulting equation, we compare the coefficients of the term p ηm /Q m, an the following equations of motion for the pseuofriction coefficients are obtaine: t p η m = (4.56) ( 1 eβ ω 1 (α m α m β 2 ) Y α m α m 2 k m {( α ( ) ) m α m α k A mk + c.c. + α k α m α k A mk c.c.} ). 2 This set of equations of motion gives the analytical generalisation of (4.29) to an arbitrary number N of fermions. Since the coupling of these equations to a sufficient number of chain thermostats to ensure ergoicity is straightforwar, there is no oubt that these equations of motion will permit to sample the istribution function w (N). It the case N = 2, m = 1, (4.56) reuces to t p η 1 = 1 ( eβ ω 1 (α 1 α 1 β 2 )2 1 + α 1 α 1 2 (α 1 α 1 α 2 + α 2 ) ) e α 1 α 2 2 1, (4.57) which can clearly be ientifie as the known result (4.29) (with ε = ) written in terms of the complex coherent states parameters α m. For reasons of completeness, we give also the final results for the emon metho. The equations of motion rea t α m = iωα m g 2m(ξ m )F m (α m, α m) i 1 mω g 1m(ζ m )G m (α m, α m), (4.58) t α m = iωα m g 2m F m + i 1 mω g 1m G m, where we assume the F m an G m are real value functions. The thermal istribution function is given by f (N) (α, α, ζ, ξ) = w (N) ( β exp N m=1 ( g1m + g ) ) 2m, (4.59) κ 1m κ 2m an from a comparison of coefficients of g 1m we get the equation of motion of ζ m, ( t ζ m = κ 1m i eβ ω 1 β + i G m mωy k m mω (α m α m )G m (4.60) {( α ( ) ) } m α m α k A mk + c.c. + α k α m α k A mk c.c. 2 + i ( Gm mω α G ) ) m, m α m 48

52 4.2. The quantum Nosé-Hoover thermostat whereas the equation of motion of ξ m is obtaine from a comparison of coefficients of g 2m, ( t ξ m = κ 2m (e β ω 1)(α m + α m β )F m (4.61) F m Y k m {( α ( ) ) } m α m α k A mk + c.c. + α k α m α k A mk c.c. 2 ( Fm α + F ) ) m. m α m The present subsection impressingly emonstrates the possibility to come to analytical results for fermions. In contrast, a similar treatment for bosons is not possible, since the permanent involve in w (N) + oes not feature a comparable richness of calculational properties as the eterminant. However, the practical calculation of thermoynamic properties of manyparticle quantum systems, e. g. using Path Integral Monte Carlo methos, is in general easier for bosons, since the calculations for fermionic systems are frequently prevente by the so-calle sign problem, see chapter 1. The sign problem in our metho occurs in the calculation of the eterminants. 49

53 5. Results In the preceing chapter, we have evelope schemes that are suppose to prouce an isothermal time evolution that gives time averages corresponing to quantum canonical ensemble averages. In the case of a single particle, the quantum methos are closely relate to their classical counterparts such that we o not expect new an rich insights; this is why we o not iscuss results using the quantum Langevin equation propose in 2.1. However, since we are able to apply the Nosé-Hoover an relate schemes to the unpreceente case of ientical particles, we will also present finings for a single particle in orer to become familiar with the quantum Nosé-Hoover metho. The key question that nees to be iscusse in the following is whether the equations of motion propose in section 4.2 reliably lea to ergoic behaviour. From the analysis of the classical techniques in section it is clear that ergoicity is estroye as soon as there are conserve quantities other that the pseuoenergy H. Unfortunately, the analytical etermination of such conserve quantities from the equations of motion is extremely ifficult, so that one has to resort to numerical stuies. As a rule of thumb, it is wiely accepte that the more complex a system is, the more probable becomes ergoic behaviour. The effects of Bose-attraction an Pauli-blocking in the case of ientical quantum particles effectively act like a many-boy-interaction an obviously make the overall ynamics more complex. Hence, we have teste whether it is sufficient to couple only one particle to a thermostat; however, this metho fails to prouce ergoic behaviour. Throughout this chapter, we have set the numerical values of the constants m, ω,, an k B equal to 1. This implies that in our units the energy spacing ω of the eigenvalues of the harmonic oscillator Hamiltonian is equal to 1, as well as the temperature T corresponing to the energy k B T = ω One particle Nosé-Hoover metho We start with a iscussion of the quantum Nosé-Hoover metho for a single particle in a harmonic oscillator potential. Formally, the transition from the classical Boltzmann istribution, equation (2.11), to the quantum phase space ensity w (1), equation (3.37), results from replacing the inverse temperature β by the expression ( e β ω 1 ) /( ω). This applies analogously on the level of the respective isothermal equations of motion. Since this term is only a c-number that epens on temperature, but not on the coherent states parameters r an p, it is clear that it oes not influence the overall characteristics of the ynamics. Therefore, we expect that ergoicity problems in the quantum case will arise uner the same circumstances as in the classical case. We show results for the set of parameters chosen in [24] to enable a irect comparison. We chose Q = 1 an the initial conitions r(0) = 1, p(0) = 1. The numerical integration of the equations of motion (4.4), (4.9) was carrie out with a fourth-orer Runge-Kutte algorithm using a step size that ensure conservation of the pseuoenergy H (see eq. (4.12)) to at least 50

54 5.1. One particle Figure 5.1.: Results of time averaging with a simple Nosé-Hoover scheme, T = 1. Left panel: p η (0) = 1, right panel: p η (0) = 10. From above: (r, p) ensity plot, position istribution, momentum istribution. The soli line shows the respective normalise marginal of w (1), e. g. f(r) = (1/ Z (1) ) (p/(2π ))w (1) (r, p). The istributions sample by time averaging are presente as histograms. six significant figures. All runs were mae over a total integration time of τ = 2000 T, where the time unit is given by T = 2π/ω. In figure 5.1, we present results for the case T = 1, with the initial conition p η (0) = 1 on the left, p η (0) = 10 on the right panel. On the left panel, the system performs rotational motions in the (r, p) parameter space, resulting in a very regularly shape ensity plot. On the right panel, the motion looks more chaotic. However, in both cases, the histograms eviate consierably from the respective marginal of w (1). It is also evient that the change of the initial conitions substantially affects the istributions that are sample. This is unacceptable as an invariant probability istribution is esire. Figure 5.2 presents results obtaine with the initial conition p η (0) = 10, but at temperatures T = 0.1 on the left an T = 10 on the right panel. At the low temperature, the 51

55 5. Results Figure 5.2.: Results of time averaging with a simple Nosé-Hoover scheme at a low temperature (T = 0.1, left panel), an at a high temperature (T = 10, right panel). histograms o not at all agree with the respective marginals of w (1). Aitional investigations for moifie initial conitions an parameter values (e. g., Q) show ifferent patterns in the (r, p) ensity plots, but the theoretical istribution function w (1) is not matche. This also applies to the marginal istribution of the pseuofriction coefficient p η. Altogether, strong non-ergoic behaviour is observe in the low temperature regime. Although the overall agreement is better in the high temperature regime, the corresponing (r, p) plot reveals that the histograms o not correspon to Gaussians, since the istribution that is sample has a eplete probability ensity at the origin of the harmonic potential which is also prominent in the marginal f(r). From more extensive investigations, it can be inferre that generally speaking, the agreement with the esire istribution improves with temperature. Therefore, it will be necessary to carefully investigate in particular the lowtemperature behaviour of all ynamical schemes. The examples given here suffice to illustrate that the simple quantum Nosé-Hoover scheme oes not prouce ergoic motion for the harmonic oscillator. Hence, this ynamical scheme 52

56 5.1. One particle will not reprouce all canonical ensemble averages correctly. The sample histograms strongly epen on temperature an on the initial conitions as well as on the particular parameter values. This general ifficulty was to be expecte from the classical harmonic oscillator where the simple Nosé-Hoover scheme also fails to sample Gaussian istribution functions. Anticipating the improvements of ergoic behaviour that will be attaine in the following for the more refine schemes (e. g., by a temperature epenent aaptation of thermostat masses or coupling constants), we stress that these measures have turne out to be insufficient in the present case. Therefore, alternate methos nee to be stuie. 53

57 5. Results Nosé-Hoover chain metho The situation changes raically with the successive introuction of further thermostating variables acting upon the first pseuofriction coefficient p η, see figure 5.3. We consier chain Figure 5.3.: Results of time averaging at T = 0.1 using Nosé-Hoover chains of length M = 2 (left panel), an M = 4 (right panel). 54

58 5.1. One particle thermostats of length M = 2 an M = 4 (for the meaning of M, see equation (2.37)) at T = 0.1, using in the beginning equal thermostat masses Q i = 1. The istribution functions sample by the scheme using a chain thermostat of length M = 2 (figure 5.3, left panel) are much closer to the theoretical Gaussian istributions than in the case of the simple Nosé-Hoover scheme, cf. figure 5.2, left panel. Nevertheless, the histograms still eviate noticeably from the theoretical Gaussians, since the feature of a eplete probability ensity at the origin of the harmonic potential is still visible. This problem is completely resolve in the case of a chain of length M = 4. The histograms reprouce the respective marginals of w (1) extremely well. More etaile investigations show that changes of the initial conitions o not have observable effects on the results. Likewise, the values of the initial conitions may be moifie, an further thermostating variables may be ae. In all cases, we foun that the ynamics generate in this way is ergoic, even at low temperatures. The following figure 5.4 illustrates the short-time evolution resulting from an ergoic Nosé- Hoover chain ynamics (M = 4) in the (r, p) plane. Instea of rotating clockwise on a circle of constant raius, the parameter trajectory approaches the origin of the harmonic potential on a spiral. This correspons to cooling the particle. At a later time, the raius increases again. Figure 5.4.: Short-time behaviour of an ergoic Nosé-Hoover chain ynamics. The initial conitions are given by r(0) = 0.5, p(0) = 0.5, p ηi = 1, an we have T = 0.2. The open circles represent the values of the coherent states parameters at time istances 0.05 T, the total integration time is 6.4 T. The connecting line is rawn to guie the eye. An alternate way of improving the ergoicity of the scheme consists of an aaptation of the numerical values of the thermostat masses. This iea has first been iscusse in the context of the emon metho by Kusnezov, Bulgac, an Bauer [23]. They foun a rule of thumb for the values of the coupling constants that consierably improves ergoicity (see next section). Aopting this rule, we take Q i T 2, an fin that even the chain of length M = 2 is ergoic own to a temperature value of T = Note that in orer to avoi numerical ifficulties at such low temperatures, the initial state nees to be chosen very close to the groun state. 55

59 5. Results The emon metho We have also investigate the cubic coupling scheme given by the set of functions (2.32), choosing the initial values r(0) = 1, p(0) = 1, ξ(0) = 1, ζ(0) = 1. The coupling constants were initially chosen to be κ 1 = κ 2 = 1. At temperatures above T = 0.3, no ergoicity problems are encountere. The ynamics provies ergoic behaviour in all examples investigate, an the histograms obtaine by time averages converge rapily against w (1). However, using ientical parameter values an initial conitions, but lowering the temperature to T = 0.2, the ynamics samples wrong istributions (see figure 5.5, left panel). At even lower temperatures, the agreement becomes progressively worse. This is a striking example that even if a system is ergoic at some temperature, it is Figure 5.5.: Results of time averaging using the cubic coupling scheme. At T = 0.3, we obtain an ergoic ynamics (right panel). The ashe line represents the marginal of the corresponing classical canonical istribution function. The left panel illustrates that at a lower temperature (T = 0.2), non-ergoic motion occurs. The trajectories appear to be regular an recurrent an thus o not reprouce w (1). 56

60 5.1. One particle not necessarily ergoic at a ifferent temperature. Kusnezov, Bulgac, an Bauer have foun this feature also in the classical case [23]. From their own results, they propose to aapt the choice of the coupling constants accoring to T, κ 1 1/T an κ 2 1/T 2, which increases the numerical values of the time erivatives of the emons at low T, cf. (2.28). This increase mobility of the emons resolves the problem in the classical case. In the quantum case, we fin that the aaptation of the coupling constants is not as successful; it ensures ergoicity only own to a temperature value of T 0.2 (not shown). In comparing the chain thermostat an the emon metho in conjuction with the cubic coupling scheme, it appears as an avantage of the chain thermostat that it can easily be mae more complex if require. The aition of more pseuofriction terms in the chain is simple to perform an numerically inexpensive. Moreover, a convenient choice of the thermostat masses leas to ergoic behaviour at temperature values well below the regime in which the cubic coupling scheme is ergoic. A combination of the emon metho an the chain metho is of course thinkable an sensible, since the emon metho has the avantage of treating the parameters r an p equally. Finally, we point out that the statistics obtaine by time averaging over the moifie quantum time evolution is the quantum statistics of the harmonic oscillator. To make this evient, 1 we present on the right han panel of figure 5.5 plots of w(1) (r, p)/ Z (1) (β) along with plots of its classical limit 1 lim e β ω 1 β ω 1 2π 1 Z (1) (β) w(1) (r, p) = βω ( ( p 2 2π exp β 2m mω2 r 2)) 2π, (5.1) which is precisely the normalise classical canonical Boltzmann istribution function. Since ( e β ω 1 ) /(β ω) > 1 for all β, the quantum istribution function is narrower compare to its classical limit Convergence spee In orer to check the convergence spee of the eterministic time averages, we consier the eviation of the histograms from the theoretical marginals of w (1). As an example, we investigate the emon metho using the cubic coupling scheme at T = 1. We efine the eviation of an r histogram at the sampling time τ (enote by f hist (r, τ)) to the exact theoretical marginal istribution (in percent) as r (τ) = 100 r f(r) f hist (r, τ), (5.2) an analogously we have p, ξ, ζ for the parameter p an the emons. The evolution of with time is an inication of the convergence spee of the coupling scheme. It is well known that the error of the average value of a series of N statistically inepenent measurements ecreases like N 1/2. This result is erive by the application of the law of error propagation to the algebraic average. Therefore, in figure 5.6, the bol straight line correspons to τ 1/2, accoring to the case of optimal sampling, i. e. statistically inepenent samples. An inspection of figure 5.6 shows that in this case the convergence spee is very close to its theoretical optimum. 57

61 5. Results p r 10 ξ ζ (τ) τ Figure 5.6.: log log representation of the time evolution of the eviation of the sample histograms from the respective theoretical istribution using the cubic coupling scheme, T = 1. The bol straight line correspons to (τ) τ 1/ Mean values of selecte observables It is a priori clear that if the relevant istribution function is correctly sample uring time evolution, the time average of any observable will match its ensemble average. To illustrate this, we investigate two typical observables, the internal energy an its variance. The analytical expressions for the canonical ensemble averages are given by U(β) = H = ω 2 + ω e β ω 1 = ω ( ) 1 2 coth 2 β ω, (5.3) ( ) 2 var(h ) = H 2 H 2 ω = 2 sinh( 1 2 β ω). (5.4) Interestingly, in the expression for the internal energy, we fin again that apart from the non-zero groun state energy the replacement of β by (e β ω 1)/ ω leas from the classical to the quantum statistical mechanics result. In orer to calculate the respective time averages, the following matrix elements are neee: ( α H α = ω α ) 12 p2 = ω + 2 2m mω2 r 2, (5.5) ( ( α H 2 α = 2 ω 2 α ) 2 + α 2). (5.6) 2 These matrix elements have to be regare as phase space functions in the sense of equation (3.39). Figure 5.7 shows results obtaine by time averaging using the cubic coupling scheme with a total sampling time τ = 1000 T. The agreement with the exact results is excellent. We observe only small eviations that are of statistical origin. 58

62 5.1. One particle Figure 5.7.: Values of the internal energy an its variance for the harmonic oscillator obtaine from time averaging with a KBB ynamics (crosses) compare to the exact quantum canonical ensemble result (soli line). In closing this section, we remark that the question whether a given scheme of isothermal ynamics prouces ergoic motion or not cannot be answere in close form. The numerical values of the temperature an the free parameters (like coupling constants or thermostat masses), an the choice of functions in the KBB-scheme affect this question in an unpreictable manner. However, the problems occur only if the value of T lies an orer of magnitue below the typical energy unit of the consiere system; an even in this case, simple moifications of the ynamics like an extension of the chain or aapte values of the coupling constants resolve the problem reliably. Moreover, in all examples we stuie, non-ergoic motion coul easily be etecte by monitoring the marginals of the pseuofriction coefficients. We foun that if an only if the marginals of the pseuofriction coefficients were sample correctly, the marginals f(r) an f(p) were sample correctly. Although the conition of ergoic sampling in the subspace of the aitional egrees of freeom is only necessary for ergoicity on the whole space, we i not encounter a counterexample an therefore consier such a case ergoicity of the pseuofriction coefficients, non-ergoicity of the original system as highly exceptional. 59

63 5. Results 5.2. Two particles In the preceing section, we have presente results of the quantum Nosé-Hoover an relate methos for a single particle. We briefly stuie questions of ergoicity an convergence, stressing the close relation to the classical case. Now, consiering two ientical quantum particles, we enter an entirely new fiel that oes not have a classical counterpart. The quantum istribution functions w ε (2) are entangle an o not have the form of a prouct of two functions epening separately on the parameters of one particle. Therefore, the isothermal equations of motion erive in section 4.2 postulating that w ε (2) be a stationary solution of the generalise Liouville equation contain aitional terms. The impact of these terms on the ynamics will be stuie in the following sections Bose-attraction an Pauli-blocking The ifferent signs for bosons an fermions in the equations of motion of the pseuofriction coefficients stem from the factor A ε A ε inclue in w ε (2). We investigate the consequences for the movements of the particles. Obviously, the effects of Bose-attraction an Pauli-blocking will be most pronounce at low temperatures, when both particles ten to occupy the one-particle groun state an thereby get close to one another in phase space. As a simple example, we examine the Nosé-Hoover scheme, see equation (4.20) an (4.29). When two fermions approach in phase space, V = α 1 α 2 2 becomes very small an the factor 1/(e V 1) grows like 1/V, overcompensating the linear ecrease of the prefactor p 1 p 2. This causes a strong acceleration of p η1 an p η2, an thereby of p 1 an p 2. The irections of the accelerations of the momenta are oppose ue to ifferent signs: In case p 1 an p 2 approach the origin of the harmonic potential with the same sign, the signs in equation (4.29) in front of the aitional term are opposite; in case p 1 an p 2 approach the origin with ifferent signs, the opposite irections of the acceleration stem from the equations of motion (4.20) where p 1 an p 2 appear in front of p η1 an p η2. Effectively, a close approach of the particles in phase space, corresponing to V 0, is avoie by the ynamics. This may be regare as the ynamical consequence of the exclusion principle. In the case of fermions, the aitional terms in the equations of motion (4.29) act like a repulsive force. Hence, we will refer to these terms as statistical interaction forces. In the case of bosons, the opposite signs in the equations (4.29) cause an acceleration of the parameters into the irection of one another, favouring a meeting of the particles in phase space. Moreover, if V = 0, the factor (p 1 p 2 )/(e V + 1) vanishes, an the statistical interaction forces vanish at this point. So the case V = 0 is not exclue at all; on the contrary, it is aime at by the boson version of the ynamics. Figure 5.8 illustrates the simple Nosé-Hoover ynamics of two fermions at low temperature. Initially, both particles are coole an move to the origin of the potential, i. e. to states of lower energy. But immeiately after approaching closely, the fermions are strongly riven away from each other. Figure 5.9 illustrates the short-time behaviour of two bosons at low temperature. After cooling, the bosons are locate close to each other an stay at the origin of the potential for a consierable perio, an only after t 8 T, they are again riven away from the origin. These ifferent short-time appearances will obviously lea to ifferent thermal istribution functions. We remark that the repulsive or attractive statistical interaction forces occur although we are treating a system of non-interacting particles. The interaction is purely a consequence of the quantum statistics of a system of ientical particles an is meiate by the influence of the 60

64 5.2. Two particles Figure 5.8.: Isothermal Nosé-Hoover ynamics of two ientical fermions. The initial values of {r 1, p 1, r 2, p 2, p η1, p η2 } are {0, 0.1, 0, 0.1, 0, 0}, the value of the temperature is T = 0.1, an Q 1 = Q 2 = 0.5. The time istance between the symbols is T, an the total integration time is 6.5 T. The two fermions reach their closest position after t 5.5 T an are riven away from each other immeiately. Figure 5.9.: Isothermal Nosé-Hoover ynamics of two ientical bosons. All values are chosen as in figure 5.8. However, the total integration time is 8.1 T, since the bosons o not separate earlier. pseuofrictional forces. As we will see in section 5.2.2, the simple Nosé-Hoover scheme that has been use here to visualise the effects of the statistical interaction forces fails to prouce ergoic motion in the low temperature regime. Nevertheless, we have chosen this scheme for the present illustrative purpose since the effects of Bose-attraction an Pauli-blocking are most luci in this elementary version of the equations of motion. The schemes that lea to ergoic behaviour have a slightly ifferent short-time appearance with more pronounce rotational movements aroun the origin that are reminiscent of figure

65 5. Results Ergoicity investigations for two fermions The impact of quantum mechanical inistiguishability on the thermostate ynamics of ientical particles effectively looks like an attractive or repulsive interaction. One may anticipate that this interaction which is most pronounce at low temperature values influences the complexity an thereby the ergoicity of the ynamics. One may hope that this leas to improvements, especially in the low temperature range where the emon metho shows non-ergoic behaviour, cf. section Furthermore, one coul think of linking only one particle to a thermostat an hope that the secon one thermalises sympathetically ue to the statistical interaction. This will be iscusse in section In orer to check the ergoicity of the ifferent methos, we will stuy marginal istributions of the thermal weight function w ε (2). Algebraically, it is more convenient an elegant to etermine the marginal istributions written in terms of the relative an the center-of-mass coorinate (i. e., switch to α + = 1 2 (α 1 + α 2 ), α = 1 2 (α 1 α 2 ), as inicate for the calculation of the two-particle partition function, equation (3.56)). However, to avoi a reformulation of the equations of motion in terms of new coorinates an to be able to compare the histograms of the thermalise variables irectly to the correct respective istributions, we prefer to stick to the representation in terms of r 1, p 1, r 2, p 2. As an example, we give the analytical expressions obtaine for the normalise marginal istributions of r 1 an p 1 : f ε (r 1 ) = 1 Z (2) ε = 1 Z (2) ε f ε (p 1 ) = 1 Z (2) ε = 1 Z (2) ε p1 r 2 p 2 2π 2π ( 1 mω 8π Nosé-Hoover metho 1 8πm ω w (2) ε (r 1, r 2, p 1, p 2 ) (5.7) mω (e β ω e 1) ε e 2β ω 1 r1 r 2 p 2 2π 2π ( 1 (e β ω 1) 3 e 1 + ε e 2β ω 1 2 (e β ω 1)r 2 1 ) 1 e β ω e mω sinh(β ω)r1 2, w (2) ε (r 1, r 2, p 1, p 2 ) (5.8) p 2 1 2m (eβ ω 1)/( ω) ) 1 e β ω e 1 m ω sinh(β ω)p2 1. To start with, we present results obtaine by averaging over trajectories generate by the equations of motion (4.20) along with (4.29). At low temperature values T 1.0, an extensive stuy shows that the scheme exhibits non-ergoic behaviour, just as in the one-particle case. An example is given in figure 5.10, left panel. For the given set of initial values an parameters, we obtain histograms that strongly resemble the left panel of figure 5.1. An aaptation of the thermostat masses accoring to the known rule Q i T 2 oes not resolve the problem. However, if the scheme is investigate at higher temperature values, the agreement between the sample histograms an the theoretical marginals becomes graually better. In the temperature range T 1.2, we fin very goo agreement; the right han panel of figure 5.10 gives an example. This shoul be contraste with the one-particle case, where non-ergoicity was foun in the simple Nosé-Hoover scheme even for very high temperatures, cf. figure 5.2, right 62

66 5.2. Two particles Figure 5.10.: Results of time averaging with the simple Nosé-Hoover scheme for two fermions, left panel: T = 0.1, right panel: T = 1.2. In both cases, we use ientical initial conitions, r 1 (0) = r 2 (0) = 0.01, p 1 (0) = p 2 (0) = 0.01, p η1 (0) = p η2 (0) = 0.01, Q 1 = Q 2 = 1.0. We present ensity plots an histograms for r 1, p 1, an p η1 ; the respective results for r 2, p 2, an p η2 look similar. 63

67 5. Results Figure 5.11.: Density plots an marginal istributions at T = 0.2 for a Nosé-Hoover scheme in which only particle 1 is couple to a chain thermostat of length M = 2. The equations of motion of particle 2 have not been moifie. Note that on the right panel, ifferent scales have been use on all axes. panel. We attribute this improve ergoicity to the increase complexity of the two-particle ynamics ue to the statistical interaction force. The problem of non-ergoic behaviour at low temperatures is resolve by a scheme involving the coupling of a Nosé-Hoover chain of length M = 2 to both p 1 an p 2. If in aition the values of the parameters Q 1 an Q 2 are aapte accoring to the rule of thumb Q i T 2, ergoic behaviour is foun over the entire temperature range investigate that reaches own to T = A stuy of even lower temperatures was prohibite by numerical ifficulties. We note that it is not sufficient to couple a secon thermostating pseuofriction coefficient to only one parameter, say p η1. In this case, the marginals of p η2 are not sample correctly at temperature values T 0.3. This illustrates the limits of the improve ergoicity ue to the statistical interaction. 64

68 5.2. Two particles Figure 5.12.: Short-time behaviour of the scheme in which only one particle is couple to a chain thermostat, T = 0.2. The initial conitions are r 1 (0) = r 2 (0) = 1.0, p 1 (0) = p 2 (0) = 0, an the total integration time is 0.9 T. It is nicely visible that particle 1 is coole by the Nosé-Hoover chain, while particle 2 performs an unisturbe oscillatory motion in the harmonic potential. More rastically, if only p 1 is couple to a Nosé-Hoover chain of length M = 2, the ynamics of the parameters r 2 an p 2 is not affecte at all. Hence, the energy of particle 2 is an aitional conserve quantity. Although the parameters of particle 2 influence the motion of particle 1 via the value of V, there is no feeback mechanism in the opposite irection. Consequently, only particle 1 experiences the statistical interaction (which remains very weak, since the two fermions stay a consierable istance apart in phase space), whereas particle 2 rotates uniformly on a circle of constant raius, corresponing to the unperturbe motion of a particle in a harmonic oscillator potential. The resulting marginal istributions are shown in figure Although the marginal istributions of r 1 an p 1 are not sample exactly, the eviation is fairly small, an vanishes completely at higher temperature values (not shown). In contrast, the uniform rotation of the secon particle leas to a microcanonical istribution. Figure 5.12 illustrates the short-time behaviour of this ynamical scheme The emon metho We have also applie the cubic coupling scheme (2.32) to the case of two fermions, using the rules of thumb for the ajustment of the coupling constants that have been presente in section These aaptation rules turn out to be very efficient in proviing ergoic motion in the present two-particle case, in contrast to what has been observe for a single particle. Yet, at temperature values below T 0.1, we fin the following feature unpreceente in the classical applications of the metho: The pseuofriction coefficients ξ 1 an ξ 2 freeze, i. e., they remain very close to their initial value, see figure Although the marginals of the original system are still sample with goo precision, the istributions of ξ 1 an ξ 2 are narrowly peake aroun their initial value an therefore completely wrong. This eficiency is not cure by choosing moifie values for the coupling constants κ r1 an κ r2. However, since we fin that the marginals of the coefficients ζ 1 an ζ 2 are well matche, we attribute it to the linear coupling of ξ 1 an ξ 2 in the two-particle version of the equations of motion (4.19). 65

69 5. Results Therefore, we have also investigate the following set of equations of motion, t r 1 = p 1 m ξ3 1r 1, (5.9) t p 1 = mω 2 r 1 ζ1 3 p 1, t r 2 = p 2 m ξ3 2 r 2, t p 2 = mω 2 r 2 ζ2 3 p 2, t ξ 1 = κ ( r 1 mω 2 r 2 e β ω ) mω 2 r 1 r 2 1 r 1 β ω ω e V, 1 t ξ 2 = κ ( r 2 mω 2 r 2 e β ω ) mω 2 r 1 r 2 1 r 2 β ω ω e V, 1 t ζ 1 = κ ( p 1 p 2 1 e β ω 1 1 p ) 1 p 1 p 2 1 β m ω m ω e V, 1 t ζ 2 = κ ( p 2 p 2 2 e β ω p ) 2 p 1 p 2 1 β m ω m ω e V, 1 in which ξ 1 an ξ 2 are cubically couple to the equations of motion, just as ζ 1 an ζ 2. Choosing all coupling constants to be of the orer of magnitue 1/T, we foun ergoic behaviour own to T = 0.05 for this scheme, see figure 5.13, right panel. 66

70 5.2. Two particles Figure 5.13.: Marginal istributions obtaine at T = 0.05 with a cubic coupling scheme (left panel), an with the scheme escribe by the set of equations of motion (5.9). The initial value of ξ 1 on the left panel is ξ 1 (0) = 5.1, an the freezing is striking. Surprisingly, all other marginals are well reprouce. The right panel illustrates that the problem is resolve by the moifie scheme (5.9). 67

71 5. Results Ergoicity investigations for two bosons The extensive stuy of the preceing section may be repeate for the two-boson system. Basically, it turns out that the main features are preserve, which is why we give only a brief survey of the results Nosé-Hoover metho Like in the fermionic case, the simple Nosé-Hoover scheme is not ergoic in the temperature range T 1.0. Above that value, with the known aaptation of the thermostat masses, the agreement between the sample histograms an the theoretical marginals becomes satisfying. The scheme with two Nosé-Hoover chains of length M = 2 is foun to be ergoic over the entire temperature range. The smallest temperature value that has been investigate is T = The emon metho Employing the cubic coupling scheme, we fin again that the pseuofriction coefficients ξ 1 an ξ 2 that are linearly couple to the equations of motion of r 1 an r 2, respectively, freeze at temperatures below T 0.1. As can be inferre from figure 5.14, this problem is cure by coupling ξ 1 an ξ 2 cubically, in analogy to the scheme (5.9). 68

72 5.2. Two particles Figure 5.14.: Same figure as 5.13, but for a two-boson system. The figure is given to exemplify the statement that the key features of the ynamics are preserve in the boson case. The ashe lines in the upper four figures correspon to the respective fermionic marginals. 69

73 5. Results Convergence spee Figure 5.15 shows in analogy to figure 5.6 that the convergence spee of the histograms to the exact theoretical marginal istributions in the two-fermion case is again close to its theoretical optimum. The result in the bosonic case is similar. p1 r1 ξ1 (τ) 10 ζ τ Figure 5.15.: log log representation of the time evolution of the eviation of the sample histograms from the respective theoretical istribution using the scheme (5.9) at T = The bol straight line correspons to (τ) τ 1/ Mean values of typical observables As in the case of a single particle, we compare time averages obtaine with our metho to the analytically accessible canonical ensemble averages. The matrix element that is neee for the calculation of the internal energy for fermions reas A H (2) A A A ( 1 = ω 2 α 1 + α ( ) 1 2 coth 2 α 1 α 2 2) α 1 α , (5.10) where H (2) enotes the Hamiltonian of two non-interacting particles in an external harmonic oscillator potential, (3.52). For two bosons, we obtain A + H (2) A + A + A + ( 1 = ω 2 α 1 + α ( ) 1 2 tanh 2 α 1 α 2 2) α 1 α (5.11) For the variance of the internal energy, var(h (2) ) = H (2) 2 H (2) 2, one also nees the matrix element of H (2) 2, A ε H (2) 2 A ε A ε A ε. (5.12) 70

74 5.2. Two particles The result for the numerator reas α 1,α 2 S ε H (2) 2 α 1, α 2 = (5.13) ( 1 ( 2 2 ω 2 α ) 2 ( + α1 2 + α ( + α2 2 2) α )( α ) 2 2 ( + ε (α 2 α 1) 2 α 1 α 2 + (α 1 α 2) 2 α 2 α 1 + e α 1 α 2 2 ( 2 α1 2 α (α 1 α 2 + α 2 α 1) + 1 ))). The analytical expressions for the ensemble averages of the internal energy an its variance are easily erive from the respective partition function Z ε (2) (cf. equation (3.56)) via the relations U ε (2) (T ) = H (2) (T ) = k B T 2 T var(h (2) ) ε (T ) = k 2 BT 4 ln Z(2) ε (T ), (5.14) 2 ln Z(2) T 2 ε (T ). (5.15) Note that if we use the moifie partition function woul obtain a function Ũ ε (2) (T ) satisfying Z (2) ε = e β ω Z (2) (T ) in relation (5.14), we ε Ũ (2) ε (T ) = U (2) ε (T ) + ω. The result for var(h (2) ) ε is not affecte. We restrict ourselves to the presentation of results for the fermionic case, see figure 5.16, since the resulting curves for the two-boson system are ientical (variance of the internal energy) or very similar (the internal energy of two bosons iffers only by a total shift ue to a ifferent groun state energy) 1. Another interesting an more complex observable that we investigate in the following is the two-particle ensity enote by ρ (2) ε (x 1, x 2 ). It gives the conitional probability to fin one particle within the interval [x 2, x 2 + x 2 ] on the x-axis provie that the other particle is locate within [x 1, x 1 + x 1 ]. Of course, ρ (2) ε (x 1, x 2 ) is given by the thermal average of the absolute square of the normalise two-particle wavefunction, ρ (2) ε (x 1, x 2 ) = x 1, x 2 A ε 2 A ε A ε which can be calculate analytically by the evaluation of the integral, (5.16) ρ (2) ε (x 1, x 2 ) = 1 r1 p 1 r 2 p 2 Z ε (2) 2π 2π w(2) ε (r 1, p 1, r 2, p 2 ) x 1, x 2 A ε 2 A ε A ε. (5.17) Since the thermal weight function w ε (2) contains the factor A ε A ε in the numerator, this term cancels an all integrations can be carrie out. We give the exact result of the thermal average 1 Given the phase space functions (5.10) an (5.11) an the respective thermal weight functions w ε (2), both insights are somewhat noteworthy. Note that they imply that ieal Fermi an Bose gases containe in a one-imensional harmonic oscillator potential have the same specific heat [40]. 71

75 5. Results Figure 5.16.: Results of time averages for the internal energy an its variance for two fermions using the cubic coupling scheme (crosses). The soli lines correspon to the respective exact analytical result. The values of the coupling constants have been aapte to temperature accoring to the rule of thumb κ i 1/T, an the sampling time was τ = 2000 T for every temperature value. of the two-particle ensity for bosons an fermions in terms of the relative an center-of-mass variables x = 1 2 (x 1 x 2 ), x + = 1 2 (x 1 + x 2 ), ρ (2) ε (x, x + ) = 1 ( mω 1 Z ε (2) 2π e 2β ω 1 exp mω ( 1 ) ) tanh 2 β ω (x x2 ) ( ( 1 + ε exp 4 mω e β ω ) ) e 2β ω 1 x2. (5.18) As a consequence of the Pauli exclusion principle, the fermionic two-particle ensity vanishes for x = 0, whereas for bosons, small values of the relative coorinate have an increase probability ensity. In orer to give a useful representation of our finings, we present results of time averages along with the respective analytical results for fixe values of x + that have been chosen arbitrarily. Figure 5.17 shows an excellent agreement. Finally, we compare the mean occupation numbers n (T ) of the nth oscillator eigenstate in the canonical ensemble at temperature T for the two-fermion system. For finite Fermi systems, the gran canonical limit is not applicable, an the mean occupation numbers iffer from the Fermi istribution function. Therefore, in orer to isplay the exact results, we have use a recursion relation ue to Schönhammer [41]. Figure 5.18 isplays again an excellent agreement between time averages an ensemble averages. 72

76 5.2. Two particles Figure 5.17.: Results of time averages for the bosonic (left figure) an fermionic (right figure) two-particle ensity. The soli curves correspon to the respective analytical results as given by equation (5.18). The crosses are obtaine from the simulations that gave the right han panels of figure 5.13 an figure While the upper line has been obtaine by setting x + = 0, the lower line correspons to the value x + = Figure 5.18.: Mean occupation numbers for a two-fermion system at various temperature values. The results obtaine by a time average employing the scheme (5.9) (total sampling time τ = 1000 T ) are isplaye by symbols, whereas the exact occupation numbers are given by crosses ( ) that are linke by straight lines to guie the eye. 73

77 6. Summary an iscussion In this work, we have presente an extension of the powerful techniques of heat bath coupling in classical MD simulations to the quantum harmonic oscillator. The particular framework provie by coherent states allows an immeiate access that is able to yiel vali isothermal equations of motion both in the elementary case of a single quantum particle an in the more involve cases of inistinguishable particles. Time averages can be performe by irect numerical integration of these equations. For the quantum Nosé-Hoover an relate techniques, we have shown that the results thus obtaine are in agreement with the respective analytical canonical ensemble averages. Problems of non-ergoic behaviour are a constant topic of concern, but we have propose an teste a number of effective remeies, like ajustment of coupling constants or aition of further egrees of freeom. These measures can be transferre irectly from the classical case, since the thermalising aitional egrees of freeom remain essentially classical variables. Moreover, in the case of inistinguishable quantum particles, we observe that ue to the statistical interaction the ergoicity of the ynamics improves (see ). This confirms the wiesprea opinion that the more involve a ynamics is, the less important ergoicity problems become. Compare to the approaches of Grilli an Tosatti [17] an Kusnezov [19], substantial progress in basic methoology has been mae. On the one han, the quantum Nosé-Hoover metho is more reliable an easier to hanle than the Grilli-Tosatti metho, an on the other han, its range of application reaches well beyon the practial scope of Kusnezov s metho which is restricte to finite-imensional quantum systems. Moreover, although euce in close analogy to classical approaches, the metho turns out to be suitable for the sampling of quantum entangle istribution functions. This unexpecte feature is a istinction of the quantum Nosé-Hoover metho over the approach using a Langevin equation. However, the present evelopment stage of the metho oes not yet permit the investigation of quantum systems other than the harmonic oscillator that are not solvable with more traitional methos. It is not clear how an isothermal ynamics scheme can be evise for systems for which the phase space istribution is unknown. Another serious rawback of the entire iea of isothermal quantum ynamics is that it requires that the time evolution of the quantum system is computable, which is just not the case for complicate many-boy quantum systems. Put ifferently, if the time evolution of a quantum system is known, the calculation of its thermoynamical properties is usually also feasible, since the operator of time evolution an the statistical ensity operator are both exponentials of the Hamiltonian operator of the system. So the solution of the Schröinger equation an the calculation of the partition function usually amount to the same problem. The most promising prospect that our metho offers lies in a combination with approximate quantum ynamics schemes. A variety of such schemes is available some of which are base on the time-epenent quantum variational principle, δ t2 t 1 t Q(t) i t H Q(t) = 0, (6.1) which allows to erive approximations to the time-epenent Schröinger equation. Typically, 74

78 the trial states Q(t) are given in terms of parameters, an (6.1) is use to etermine the equations of motion of these parameters. In the metho of Fermionic Molecular Dynamics (FMD) [10], Q(t) is a Slater eterminant of single-particle Gaussian wave packets parametrise by mean position, mean momentum, an complex with 1. FMD has been evelope an successfully employe in nuclear physics, e. g. for the escription of the nuclear liqui-gas phase transition [42]. Beyon, its statistical properties have been stuie extensively [43], an even a thermostating metho using a thermometer an a feeback mechanism has been evelope (cf. chapter 1). Given the availability of these powerful approximate quantum ynamics schemes, a new fiel of possible research opens. How can we combine the thermostating metho evelope in this work with, e. g., FMD in orer to obtain an isothermal ynamical scheme for a complex interacting fermion system? This question is very timely in view of recent experiments investigating the behaviour of trappe Fermi gases. In view of the fact that at low temperatures, the trappe fermions form an ieal gas an the trapping potential may be consiere harmonic, the isothermal quantum ynamics scheme evelope in the present work can be consiere a goo starting point for a perturbative treatment that is correct in the limit of vanishing interaction an a harmonic confining potential. Another iea of combining FMD with a thermostat is to cool the system of interest sympathetically, i. e. via an interaction between particles that are kept at a constant temperature by a quantum Nosé-Hoover-chain an the physical system uner investigation. This correspons precisely to the experimental technique of sympathetic cooling currently employe to investigate ultracol fermionic gases [13]. Again, the thermalising of the particles couple to a Nosé-Hoover chain woul correspon to a thermalising of non-interacting particles, which is only approximately correct since we nee to employ an interaction to enable the sympathetic cooling. Despite these ifficulties, an important asset of an isothermal MD scheme is that it can possibly provie temporal information, in particular time correlation functions. Although it is not clear to which egree the extene system methos realistically mimic the heat bath interaction, the unerlying equations of motion are physically reasonable. Therefore, in principle, this metho is tailor-mae to moel the particle ynamics at constant temperature in a magnetic trap. Employing a two-boy interaction an FMD, the resulting trajectory is a well-efine approximation of the exact quantum time evelopment which is etermine by forces an the heat bath interaction, proviing a goo picture of temporal correlations. This is to be contraste to MC calculations where no ynamical equations are solve an the resulting MC trajectories are unphysical. It is noteworthy that our metho offers in combination with FMD a new perspective for the calculation of fermionic systems at low temperatures, while the path-integral MC methos have been very successful for bosons. Hence, our metho may be regare as complementary to MC approaches also in this respect. In summary, the evelopments of the present work offer a variety of new approaches in the simulation of quantum systems at finite temperature. In combination with the ongoing investigations of ultracol trappe quantum gases, new an rich physical insights are to be expecte in this fiel. 1 These wave packets are frequently referre to as squeeze states an may be regare as generalisations of coherent states. Therefore, an application of the thermostats evelope in the present work to FMD appears feasible. 75

79 A. The Grilli-Tosatti thermostat This appenix gives a brief general review of the quantum thermostat metho propose by Grilli an Tosatti [17] along with a iscussion of the problems encountere when applying it to the harmonic oscillator. The metho is base on the following theorem. Consier a general quantum Hamiltonian H, H (r, p ) = N i=1 p2 i 2m i + V (r ). (A.1) The thermal average of a quantum observable B, which is suppose to be some function of the operators of position an momentum, B (r, p ) = Tr (B e βh ) Tr e βh is equal to the following microcanonical average ps s Tr ( B (sr, p/s)δ(e H ext) ), (A.2) B mc = ps s Tr ( δ(e H ext) ), (A.3) where H ext enotes the extene Hamiltonian H ext = N p2 i 2m i=1 i s 2 + V (sr ) } {{ } H s = H (sr, p/s) + p2 s 2Q + k BT ln s, (A.4) where H s is obtaine from the original Hamiltonian by a scaling of the operators r an p to sr an p /s, respectively. The proof of the theorem is base on the fact that this scaling leaves the trace of any operator B (r, p ) invariant. This is seen most easily if one consiers that the following unitary operator 1, precisely performs the scalings ( i ) U s = exp 2 ln s(r p + pr ), (A.5) U sr U s = sr, U sp U s = p /s. (A.6) 1 has been set equal to 1 in this appenix. 76

80 As a result, we have U sh U s = H s, (A.7) an any unitary transformation is obviously trace-invariant. Therefore, the numerator of (A.3) can be transforme into p s s Tr ( B (sr, p/s)δ(e H s) ) = (A.8) p s s x x B (r, p )δ(e H (r, p ) p2 s 2Q k BT ln s) x, where we have roppe the unitary scaling of H an use a representation of the ientity operator 1 = x x x (with x being the eigenstates of r, i. e., r x = x x ) to evaluate the trace. An analogous calculation hols for the enominator. Now the integrations over the variables s an p s insie the δ-function can be carrie out, an after cancelling overall prefactors, we obtain the result B mc = x x B e βh x x x e βh x, (A.9) where the right han sie correspons to the usual quantum canonical average. The theorem of Grilli an Tosatti is potentially very powerful since contrary to the quantum Nosé-Hoover metho evelope for the harmonic oscillator in this work it applies to any Hamiltonian of the general form (A.1). However, the practical application of the theorem turns out to be elicate. In orer to be able to replace the microcanonical average of equation (A.3) by a time average, a ynamical scheme is neee that samples the microcanonical istribution function δ(e H s) on the mixe quantum-classical state space consisting of quantum states Ψ an two real numbers s, p s. In other wors, the equations of motion have to conserve the quantity E = Ψ H s Ψ + p2 s 2Q + k BT ln s (A.10) an nee to be ergoic. motion, Grilli an Tosatti have propose the following set of equations of i t Ψ(t) = H s Ψ(t), t s = p s M, t p s = T ( ) s Ψ(t) s H s Ψ(t). (A.11) In the following, we show that in the case of the harmonic oscillator, this ynamical scheme oes not prouce ergoic motion, which means that the metho unfortunately fails in this elementary, but ubiquitous moel. For simplicity, we consier the motion of a single particle in a one-imensional harmonic oscillator. The eigenstates of the original H = 1 2m p mω2 2 r (A.12) 77

81 A. The Grilli-Tosatti thermostat an the scale Hamilton operator H s = 1 2ms 2 p ms2 ω 2 2 r (A.13) are, respectively, H n = E n n, H s n, s = E n n, s. (A.14) Since in the case of the harmonic oscillator the scaling correspons to a scaling of the mass, m ms 2, it is particularly evient that the eigenvalues E n = ω(n+1/2) are not affecte by the scaling, i. e., H an H s are isospectral. Moreover, the following relation for the wavefunctions of the eigenstates hols: x n, s = s sx n. (A.15) The time-epenent quantum state Ψ(t) can be represente with respect to the basis { n, s(t) }, which itself is time-epenent through the time-epenence of s: Ψ(t) = n=0 c n (t) e ient n, s(t). (A.16) Then the left han sie of the Grilli-Tosatti ynamical equation for the quantum state reas ( i t Ψ(t) = i n + n + n an for the right han sie we obtain H s Ψ(t) = n ( ie n )c n (t)e ient E n n, s ċ n (t)e ient n, s ) c n (t)e ient( ) t n, s, E n c n (t)e ient n, s. (A.17) (A.18) The equation of motion simplifies since two sums cancel. The term /t n, s can be calculate in a straightforwar manner. With the obvious notation (using the imensionless operators ˆr an ˆp of equation (3.1)) a s = 1 2 (sˆr + i ˆp /s), (A.19) we obtain t n, s = t ( ) 1 (a s )n 0, s n! = 1 n! ( ( t (a s) n) 0, s + (a s) n t 0, s ). (A.20) 78

82 Using we fin We also have t a s = ṡ ( ) 1 2 (sr s ip/s) = ṡ s a s, ( t (a s) n) 0, s = ṡ s n(n 1) (n 2)! n 2, s. 2 t 0, s = ṡ 2, s. 2 s (A.21) (A.22) (A.23) Therefore, we arrive at t n, s = ṡ ( ) n(n 1) n 2, s (n + 1)(n + 2) n + 2, s. (A.24) 2s Altogether, we obtain an equation of motion for the expansion coefficients which reas t c n = ṡ ) (c n 2 n(n 1) cn+2 (n + 1)(n + 2) 2 s. (A.25) Equation (A.24) implies that in this case, the Grilli-Tosatti time evolution conserves the inital partition of probability among states of even an o parity, since occupation probability is transferre either among basis states with even n (even parity) or o n (o parity). This is not surprising since H s in the equation of motion of the quantum state (A.11) commutes with the parity operation. Clearly, this aitional conservation law is incompatible with ergoicity. This result is corroborate by numerical investigations which show that neither the quantum system attains the esire temperature nor that the marginal of the variable p s is accurately sample. In orer to break this conservation law, consier the shifte harmonic oscillator potential (r 0 being a fixe real number), H = 1 2m p mω2 (r r 0 ) 2, (A.26) which is not symmetric with respect to the parity operation. Extensive numerical investigations have shown that the sampling of the marginal of the variable p s is improve, but still, the quantum system oes not equilibrate [44]. Nonetheless, we stress that the theorem provie by Grilli an Tosatti appears extremely powerful an eserves further reflection. It will be necessary to evise a ynamical scheme that reliably leas to ergoicity both for elementary an more involve systems. 79

83 B. Position representation of the canonical ensity operator As an example for the usefulness of the formula (3.41), B = 1 r p Z (1) (β) 2π w(1) (r, p) B(r, p), (B.1) with B(r, p) = r, p B r, p, (B.2) we present here an elementary an straightforwar calculation of the position representation of the ensity operator ρ = 1 Z (1) (β) exp( βe n ) n n, n=0 (B.3) i. e., we calculate the matrix elements ρ(x, x ) = x ρ x. (B.4) Note that the probability istribution of the position of a particle in a harmonic oscillator potential at temperature T, g(x) = 1 Z(β) exp( βe n ) x n 2, n=0 (B.5) is equal to the iagonal matrix element x ρ x of the ensity operator. Moreover, g(x) is closely relate to the Wigner function W (x, p) of the thermal state (B.3), since the marginal istribution of the variable x of the Wigner function correspons to g(x), g(x) = p W (x, p). (B.6) ρ(x, x ) may be calculate by an immeiate evaluation of the expression (B.4) using the representation (B.3) for ρ. For this irect calculation, an elaborate relation for Hermite polynomials that is accessible from an integral representation of the Hermite polynomials is inispensable [35]. Another approach which works only for the iagonal matrix element consists of solving a ifferential equation for x ρ x [45]. In aition to these known approaches, we propose here to rephrase ρ(x, x ), which is in general a complex number, as the thermal expectation 80

84 value of a non-hermitian operator, ρ(x, x ) = = 1 Z (1) (β) 1 Z (1) (β) exp( βe n ) x n n x n=0 exp( βe n ) n x x n n=0 = x x, (B.7) an to use (B.1) for the calculation of the thermal average. The evaluation of the integral ρ(x, x ) = 1 Z (1) (β) r p 2π w(1) (r, p) r, p x x r, p (B.8) is straightforwar. The integration over p is trivial, an the integration over r can be carrie out irectly by aition an subtraction of the quaratic completion in the exponent. Using the ientities 2 ( 1 ) e β ω 1 = coth 2 β ω 1, (B.9) an 2 ( 1 ) e β ω + 1 = tanh 2 β ω 1, (B.10) we finally obtain the result mω ( 1 ) ρ(x, x ) = π tanh 2 β ω ( exp mω ( 1 ) 4 tanh 2 β ω (x + x ) 2 mω ( 1 ) ) 4 coth 2 β ω (x x ) 2, (B.11) in agreement with the irect evaluation of (B.4) [35]. The spatial probability ensity of a particle on the x-axis is mω ( 1 ) ( ρ(x, x) = g(x) = π tanh 2 β ω exp mω ( 1 ) ) tanh 2 β ω x 2. (B.12) We stress that compare to the other known approaches this calculation has the peagogical avantage of being very simple an straightforwar. 81

85 C. Sinusoial oscillations of isplace harmonic oscillator eigenfunctions 1 The remarkably simple time evolution of a coherent state in a harmonic oscillator potential, equation (3.25), initially foun by Schröinger [33], is at the heart of the translation of the classical Nosé-Hoover metho to quantum ynamics. It implies that the mean position an the mean momentum of the Gaussian wavepacket follow the solutions of the equations of motion of the corresponing classical oscillator, an, moreover, that the shape of the wavepacket remains constant. This behaviour is a special case of a more general equation foun in 1954 by I. R. Senitzky [46]. He has shown that in a harmonic oscillator potential, all the various eigenfunctions u n (x) = x n, if isplace in coorinate an momentum space, oscillate sinusoially in time while maintaining a fixe shape, i. e. ψ(x, t) 2 = u n (x q 0 (t)) 2, q 0 (t) = a cos(ωt + θ), (C.1) where a an θ are arbitrary constants. The time evolution of a coherent state is a special case of this equation which is obtaine by setting n = 0. Senitzky erive this result by performing an elegant analysis of the time-epenent Schröinger equation in the coorinate representation. The purpose of this appenix is to present two alternate instructive methos for proving Senitzky s result, the first using coherent states, the secon using the isplacement operator. In section 3.2 we have expresse the time evolution of a coherent state as follows: e i H t α = e iωt(a a ) D (α) 0 (C.2) = e iωt 1 2 e iωt α = e iωt 1 2 D (e iωt α) 0. Using the notation of the isplacement operator, Senitzky s fining, (C.1), that any initially isplace eigenfunction unergoes sinusoial oscillations while maintaining constant shape, may be rephrase as e iωt(a a ) D (α) n = e iωt(n+ 1 2 ) D (e iωt α) n. (C.3) We will prove this form of the theorem first by means of an expansion of n into coherent states, thereby arriving at a very straightforwar calculation. Seconly, we will erive a general relation, (C.9), between the isplacement an the time evolution operators, from which (C.3) follows immeiately. 1 This appenix is similar to a part of the article Almost-perioic wave packets an wave packets of invariant shape that has been submitte to American Journal of Physics, in collaboration with Prof. M. Luban, Ames Laboratory, Ames, Iowa, USA. 82

86 a. Our first metho of establishing the theorem of (C.3) exploits the fact that the general energy eigenstate n may be expane as an integral over the coherent states β efine on a circle in the complex plane with raius β = β 0, where β 0 is an arbitrary positive number. In orer to fin this expansion, we start with the familiar expansion of a coherent state α in an infinite series of oscillator eigenstates m, α = e 1 2 α 2 m=0 α m m! m. (C.4) Writing β e iθ for α, multiplying both sies of this equation by e inθ an integrating on θ from π to π one obtains π n = N n ( β ) θe inθ β e iθ, π (C.5) the normalization factor N n ( β ) being N n ( β ) = 1 2π e 1 n! 2 β 2 β n. (C.6) Using the well-known formula [32, p. 40] e A +B = e A e B e [A,B ]/2 which applies if both A an B commute with [A, B ], one easily obtains the following ientity for the isplacement operator: D (α + β) = D (α)d (β)e i Im(α β). (C.7) Using equations (C.2), (C.5), an (C.7), the following calculation is straightforwar: π e iωta a D (α) n = N n ( β ) = N n ( β ) = N n ( β ) π π π π π = D (e iωt α)n n ( β ) θe inθ e iωta a D (α)d (β) 0 (C.8) θe inθ e iim(αβ ) D (e iωt (α + β)) 0 θe inθ D (e iωt α)d (e iωt β) 0 π π = e inωt D (e iωt α)n n ( β ) = e inωt D (e iωt α) n. This provies our first proof of the theorem of (C.3). θe inθ e iωt β π π θe in(θ ωt) e i(θ ωt) β b. A secon metho for proving (C.3) consists of the following. Inspecting the final result of (C.8), one is le to conjecture the operator ientity e iωta a D (α) = D (e iωt α)e iωta a, (C.9) 83

87 C. Sinusoial oscillations of isplace harmonic oscillator eigenfunctions which immeiately implies the claime ientity (C.3). To prove it, we efine the three operators A = a a, (C.10) B +(α) = αa + α a, B (α) = αa α a. These satisfy the following commutation relations: [A, B ±(α)] = B (α), (C.11) [B +(α), B (α)] = 2 α 2 1, [A, [A, B (α)]] = γb (α), with γ = 1. Because of the thir commutation relation we may use the stanar theorem, vali for any c-number λ [32], e λa B (α)e λa = B (α) cosh(λ γ) + [A, B (α)] γ sinh(λ γ). (C.12) Setting λ = iωt an using the first equation of (C.11) as well as the efinition of B (α) we obtain e iωta B (α)e iωta = B (αe iωt ). (C.13) It then follows that e iωta (B (α)) 2 e iωta = e iωta B (α)e iωta e iωta B (α)e iωta = (B (αe iωt )) 2 (C.14) an thus e iωta e B (α) e iωta = e B (αe iωt ). (C.15) But we have e B (α) = D (α), so that (C.15) is the conjecture ientity (C.9). We remark that the harmonic oscillator oes not possess any other wave packets of constant shape. This follows from the fact that if one trie to replace the eigenket n in (C.3) by a linear combination, ψ, of two or more eigenkets, the right-han sie coul not be written as a single time-epenent factor multiplying D (e iωt α) ψ. 84

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90 Acknowlegements This thesis has been prepare uring the past three years in the group Makroskopische Systeme un Quantentheorie at the physics epartment of the University of Osnabrück. I thank all group members for the open an fruitful working atmosphere. In particular, I am very grateful to PD Dr. J. Schnack who supporte the project with both powerful ieas an his unmatche enthusiasm. His ongoing encouragement was very helpful, an I enjoye working with him very much. He also raise the funs for the project Isotherme Dynamik kleiner Quantensysteme, giving me the occasion to prepare my thesis in Osnabrück. Financial support of this project by the DFG is gratefully acknowlege. Prof. K. Bärwinkel an Prof. H.-J. Schmit provie both support an freeom which helpe making my research enjoyable an successful. I also appreciate the collaboration with Dipl.-Phys. F. Homann, his extensive computer support, an his sorcerous inspiration in many respects. In aition, I woul like to express my thanks to Prof. M. Luban who agree to be an external referee for the present work. I have been working with him for several years on magnetic molecules, but our contact reaches far beyon that. I am especially grateful for the kin hospitality uring my stays at the Ames Lab, an for a large number of helpful an enjoyable iscussions. Last but not least, I thank Dr. C. Schröer for a lot of support, avice an guiance over the years, an many helpful an elightful iscussions. 87