Isothermal quantum dynamics: Investigations for the harmonic oscillator


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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 slieown from this summit, as the principle is applie to various cases, or the climbup 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 NoséHoover metho The emon metho The chain thermostat NonHamiltonian 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 NoséHoover thermostat One particle The NoséHoover thermostat an the NoséHoover chain The emon metho Two ientical particles The simple NoséHoover thermostat The emon metho Case of N fermions Results One particle NoséHoover metho NoséHoover chain metho The emon metho Convergence spee Mean values of selecte observables Two particles Boseattraction an Pauliblocking Ergoicity investigations for two fermions
5 Contents NoséHoover metho The emon metho Ergoicity investigations for two bosons NoséHoover metho The emon metho Convergence spee Mean values of typical observables Summary an iscussion 74 A. The GrilliTosatti 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 NoséHoover scheme, T = Time averaging with a simple NoséHoover scheme, T = 0.25, Time averaging with NoséHoover chains, T = Shorttime behaviour of a NoséHoover chain ynamics Time averaging with the cubic coupling scheme, T = 0.2, Convergence spee of histograms Mean values of selecte observables Dynamical illustration of Pauliblocking Dynamical illustration of Boseattraction Time averaging with a simple NoséHoover scheme, T = 0.1, Density plots an marginal istributions for a NoséHoover scheme, only one particle is couple to a thermostat Shorttime behaviour of a NoséHoover 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 twoparticle ensity Results of time averages for the mean occupation numbers for a twofermion 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 grancanonical (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 highimensional 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 manyparticle 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 nonergoic 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 socalle Langevin equation, an it has been shown in 1945 that uner appropriate conitions, the longtime 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 finitetemperature simulations of quantum systems, the most successful approach is base on the pathintegral 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 socalle 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 λtransition 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 finitetemperature 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 manyboy 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 nonphysical 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 manyboy 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 nonequilibrium 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 tailormae 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 BoseEinstein conensation in weakly interacting Bose gases [11, 12]. On the fermionic sie, researchers stuy the large impact of FermiDirac statistics on the behaviour of socalle 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 swave 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 manyparticle 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 twolevel 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, farreaching methoological evelopments. For a single quantum particle, the analogy to classical physics is very close, whereas for two noninteracting 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 (socalle 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 qaxis 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 fluctuationissipation 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 WienerKhintchinetheorem 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 MaxwellBoltzmann 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 MaxwellBoltzmann istribution uses a FokkerPlanck equation associate to (2.1) [3, 21]. In general, for a oneimensional Markov process y(t), the FokkerPlanck 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 FokkerPlanck equation for a oneimensional 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 FokkerPlanck equation for an nimensional 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 FokkerPlanck 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 twoimensional Gaussian istribution in q an p with timeepenent average values an withs. In the limit t, one obtains the MaxwellBoltzmann 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 longtime 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 FokkerPlanck equation. The with of the Gaussians is etermine by the amplitue of the fluctuating force via the fluctuationissipation 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 socalle classical NoséHoover thermostat with the extension to chains of thermostats an the socalle 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 NoséHoover metho Consier an isolate classical Nparticle 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 Ntuple 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 NoséHoover 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 timeepenent 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 timeepenent supplementary egree of freeom ae to the system to rive the energy fluctuations require in the canonical ensemble. Therefore, the NoséHoover 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 NoséHoover 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 NoséHoover 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 nonhamiltonian 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 nonhamiltonian 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 NoséHoover 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 nonergoic 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 Poincarésections 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 NoséHoover scheme may be ergoic for one temperature, but not ergoic for a ifferent value of T [23, 6]. In aition, in all cases of nonergoic 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 nonergoicity of the harmonic oscillator The emon metho In an effort to cure the problem of unpreictable nonergoic behaviour in the NoséHoover 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 KBBscheme 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 NoséHoover 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 NoséHoover 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 socalle 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 NoséHoover 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 NoséHoover 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 NoséHoover 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 NoséHoover 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 oneimensional 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 NonHamiltonian 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 nonhamiltonian 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 nonhamiltonian systems. Using these notions, a proceure is evelope that leas to the phase space counterpart of the time averages generate by a nonhamiltonian 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 nonergoicity 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 timeepenent 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 timeinvariant 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 nonhamiltonian 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
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