Nonlinear physics (solitons, chaos, discrete breathers)

Transcription

1 Nonlinear physics (solitons, chaos, discrete breathers) N. Theodorakopoulos Konstanz, June 2006

2 Contents Foreword vi 1 Background: Hamiltonian mechanics Lagrangian formulation of dynamics Hamiltonian dynamics Canonical momenta Poisson brackets Equations of motion Canonical transformations Point transformations Hamilton-Jacobi theory Hamilton-Jacobi equation Relationship to action Conservative systems Separation of variables Periodic motion. Action-angle variables Complete integrability Symmetries and conservation laws Homogeneity of time Homogeneity of space Galilei invariance Isotropy of space (rotational symmetry of Lagrangian) Continuum field theories Lagrangian field theories in 1+1 dimensions Symmetries and conservation laws Perturbations of integrable systems Background: Statistical mechanics Scope Formulation Phase space Liouville s theorem Averaging over time Ensemble averaging Equivalence of ensembles Ergodicity The FPU paradox The harmonic crystal: dynamics The harmonic crystal: thermodynamics The FPU numerical experiment i

3 Contents 4 The Korteweg - de Vries equation Shallow water waves Background: hydrodynamics Statement of the problem; boundary conditions Satisfying the bottom boundary condition Euler equation at top boundary A solitary wave Is the solitary wave a physical solution? KdV as a limiting case of anharmonic lattice dynamics KdV as a field theory KdV Lagrangian Symmetries and conserved quantities KdV as a Hamiltonian field theory Solving KdV by inverse scattering Isospectral property Lax pairs Inverse scattering transform: the idea The inverse scattering transform The direct problem Time evolution of scattering data Reconstructing the potential from scattering data (inverse scattering problem) IST summary Application of the IST: reflectionless potentials A single bound state Multiple bound states Integrals of motion Lemma: a useful representation of a(k) Asymptotic expansions of a(k) IST as a canonical transformation to action-angle variables Solitons in anharmonic lattice dynamics: the Toda lattice The model The dual lattice A pulse soliton Complete integrability Thermodynamics Chaos in low dimensional systems Visualization of simple dynamical systems Two dimensional phase space dimensional phase space dimensional phase space; nonautonomous systems with one degree of freedom Small denominators revisited: KAM theorem Chaos in area preserving maps Twist maps Local stability properties Poincaré-Birkhoff theorem Chaos diagnostics The standard map The Arnold cat map The baker map; Bernoulli shifts ii

4 Contents The circle map. Frequency locking Topology of chaos: stable and unstable manifolds, homoclinic points Solitons in scalar field theories Definitions and notation Lagrangian, continuum field equations Static localized solutions (general KG class) General properties Specific potentials Intrinsic Properties of kinks Linear stability of kinks Special properties of the SG field The Sine-Gordon breather Complete Integrability Atoms on substrates: the Frenkel-Kontorova model The Commensurate-Incommensurate transition The continuum approximation The special case ɛ = 0: kinks and antikinks The general case ɛ > 0: the soliton lattice Breaking of analyticity FK ground state as minimizing periodic orbit of the standard map Small amplitude motion Free end boundary conditions Metastable states: spatial chaos as a model of glassy structure Solitons in magnetic chains Introduction Classical spin dynamics Spin Poisson brackets An alternative representation Solitons in ferromagnetic chains The continuum approximation The classical, isotropic, ferromagnetic chain The easy-plane ferromagnetic chain in an external field Solitons in antiferromagnets Continuum dynamics The isotropic antiferromagnetic chain Easy axis anisotropy Easy plane anisotropy Easy plane anisotropy and symmetry-breaking field Solitons in conducting polymers Peierls instability Electrons decoupled from the lattice Electron-phonon coupling; dimerization Solitons and polarons in (CH) x A continuum approximation Dimerization The soliton The polaron iii

5 Contents 12 Solitons in nonlinear optics Background: Interaction of light with matter, Maxwell-Bloch equations Semiclassical theoretical framework and notation Dynamics Propagation at resonance. Self-induced transparency Slow modulation of the optical wave Further simplifications: Self-induced transparency Self-focusing off-resonance Off-resonance limit of the MB equations Nonlinear terms Space-time dependence of the modulation: the nonlinear Schrödinger equation Soliton solutions Solitons in Bose-Einstein Condensates The Gross-Pitaevskii equation Propagating solutions. Dark solitons Unbinding the double helix A nonlinear lattice dynamics approach Mesoscopic modeling of DNA Thermodynamics Nonlinear structures (domain walls) and DNA melting Local equilibria Thermodynamics of domain walls Pulse propagation in nerve cells: the Hodgkin-Huxley model Background The Hodgkin-Huxley model The axon membrane as an array of electrical circuit elements Ion transport via distinct ionic channels Voltage clamping Ionic channels controlled by gates Membrane activation is a threshold phenomenon A qualitative picture of ion transport during nerve activation Pulse propagation Localization and transport of energy in proteins: The Davydov soliton Background. Model Hamiltonian Energy storage in C=O stretching modes. Excitonic Hamiltonian Coupling to lattice vibrations. Analogy to polaron Born-Oppenheimer dynamics Quantum (excitonic) dynamics Lattice motion Coupled exciton-phonon dynamics The Davydov soliton The heavy ion limit. Static Solitons Moving solitons iv

6 Contents 17 Nonlinear localization in translationally invariant systems: discrete breathers The Sievers-Takeno conjecture Numerical evidence of localization Diagnostics of energy localization Internal dynamics Towards exact discrete breathers A Impurities, disorder and localization 164 A.1 Definitions A.1.1 Electrons A.1.2 Phonons A.2 A single impurity A.2.1 An exact result A.2.2 Numerical results A.3 Disorder A.3.1 Electrons in disordered one-dimensional media A.3.2 Vibrational spectra of one-dimensional disordered lattices Bibliography 173 v

7 Foreword The fact that most fundamental laws of physics, notably those of electrodynamics and quantum mechanics, have been formulated in mathematical language as linear partial differential equations has resulted historically in a preferred mode of thought within the physics community - a linear theoretical bias. The Fourier decomposition - an admittedly powerful procedure of describing an arbitrary function in terms of sines and cosines, but nonetheless a mathematical tool - has been firmly embedded in the conceptual framework of theoretical physics. Photons, phonons, magnons are prime examples of how successive generations of physicists have learned to describe properties of light, lattice vibrations, or the dynamics of magnetic crystals, respectively, during the last 100 years. This conceptual bias notwithstanding, engineers or physicists facing specific problems in classical mechanics, hydrodynamics or quantum mechanics were never shy of making particular approximations which led to nonlinear ordinary, or partial differential equations. Therefore, by the 1960 s, significant expertise had been accumulated in the field of nonlinear differential and/or integral equations; in addition, major breakthroughs had occurred on some fundamental issues related to chaos in classical mechanics (Poincaré, Birkhoff, KAM theorems). Due to the underlying linear bias however, this substantial progress took unusually long to find its way to the core of physical theory. This changed rapidly with the advent of electronic computation and the new possibilities of numerical visualization which accompanied it. Computer simulations became instrumental in catalyzing the birth of nonlinear science. This set of lectures does not even attempt to cover all areas where nonlinearity has proved to be of importance in modern physics. I will however try to describe some of the basic concepts mainly from the angle of condensed matter / statistical mechanics, an area which provided an impressive list of nonlinearly governed phenomena over the last fifty years - starting with the Fermi-Pasta-Ulam numerical experiment and its subsequent interpretation by Zabusky and Kruskal in terms of solitons ( paradox turned discovery, in the words of J. Ford). There is widespread agreement that both solitons and chaos have achieved the status of theoretical paradigm. The third concept introduced here, localization in the absence of disorder, is a relatively recent breakthrough related to the discovery of independent (nonlinear) localized modes (ILMs), a.k.a. discrete breathers. Since neither the development of the field nor its present state can be described in terms of a unique linear narrative, both the exact choice of topics and the digressions necessary to describe the wider context are to a large extent arbitrary. The latter are however necessary in order to provide a self-contained presentation which will be useful for the non-expert, i.e. typically the advanced undergraduate student with an elementary knowledge of quantum mechanics and statistical physics. Konstanz, June 2006 vi

8 1 Background: Hamiltonian mechanics Consider a mechanical system with s degrees of freedom. The state of the mechanical system at any instant of time is described by the coordinates {Q i (t), i = 1, 2,, s} and the corresponding velocities { Q i (t)}. In many applications that I will deal with, this may be a set of N point particles which are free to move in one spatial dimension. In that particular case s = N and the coordinates are the particle displacements. The rules for temporal evolution, i.e. for the determination of particle trajectories, are described in terms of Newton s law - or, in the more general Lagrangian and Hamiltonian formulations. The more general formulations are necessary in order to develop and/or make contact with fundamental notions of statistical and/or quantum mechanics. 1.1 Lagrangian formulation of dynamics The Lagrangian is given as the difference between kinetic and potential energies. particle system interacting by velocity-independent forces For a L({Q i, Q i }) = T V (1.1) T = 1 s m i Q 2 i 2 i=1 V = V ({Q i }, t). where an explicit dependence of the potential energy on time has been allowed. Lagrangian dynamics derives particle trajectories by determining the conditions for which the action integral t S(t, t 0 ) = dτl({q i, Q i, τ}) (1.2) t 0 has an extremum. The result is d L dt Q = L (1.3) i Q i which for Lagrangians of the type (1.2) becomes i.e. Newton s law. m i Qi = V Q i (1.4) 1.2 Hamiltonian dynamics Canonical momenta Hamiltonian mechanics, uses instead of velocities, the canonical momenta conjugate to the coordinates {Q i }, defined as P i = L Q i. (1.5) 1

9 1 Background: Hamiltonian mechanics In the case of (1.2) it is straightforward to express the Hamiltonian function (the total energy) H = T + V in terms of P s and Q s. The result is H({P i, Q i }) = s I=1 P 2 i 2m i + V ({Q i }). (1.6) Poisson brackets Hamiltonian dynamics is described in terms of Poisson brackets {A, B} = s i=1 { A B A } B Q i P i P i Q i (1.7) where A, B are any functions of the coordinates and momenta. The momenta are canonically conjugate to the coordinates because they satisfy the relationships Equations of motion According to Hamiltonian dynamics, the time evolution of any function A({P i, Q i }, t) is determined by the linear differential equations Ȧ da dt = {A, H} + A t. (1.8) where the second term denotes any explicit dependence of A on the time t. Application of (1.8) to the cases A = P i and A = Q i respectively leads to P i = {P i, H} Q i = {Q i, H} (1.9) which can be shown to be equivalent to (1.4). The time evolution of the Hamiltonian itself is governed by dh dt = H ( = V ). (1.10) t t Canonical transformations Hamiltonian formalism important because the symplectic structure of equations of motion (from Greek συµπλɛκω = crosslink - of momenta & coordinate variables -) remains invariant under a class of transformations obtained by a suitable generating function ( canonical transformations). Example, transformation from old coordinates & momenta {P, Q} to new ones {p, q}, via a generating function F 1 (Q, q, t) which depends on old and new coordinates (but not on old and new momenta - NB there are three more forms of generating functions - ): P i = F 1(q, Q, t) Q i p i = F 1(q, Q, t) q i K = H + F 1 t (1.11) 2

10 1 Background: Hamiltonian mechanics new coordinates are obtained by solving the first of the above eqs., and new momenta by introducing the solution in the second. It is straightforward to verify that the dynamics remains form-invariant in the new coordinate system, i.e. ṗ i = {p i, K} q i = {q i, K} (1.12) and dk(p, q, t) K(p, q, t) =. (1.13) dt t Note that if there is no explicit dependence of F 1 on time, the new Hamiltonian K is equal to the old H Point transformations A special case of canonical transformations are point transformations, generated by F 2 (Q, p, t) = i f i (Q, t)p i ; (1.14) New coordinates depend only on old coordinates - not on old momenta; in general new momenta depend on both old coordinates and momenta. A special case of point transformations are orthogonal transformations, generated by F 2 (Q, p) = i,k a ik Q k p i (1.15) where a is an orthogonal matrix. It follows that q i = k p i = k a ik Q k a ik P k. (1.16) Note that, in the case of orthogonal transformations, coordinates transform among themselves; so do the momenta. Normal mode expansion is an example of (1.16). 1.3 Hamilton-Jacobi theory Hamilton-Jacobi equation Hamiltonian dynamics consists of a system of 2N coupled first-order linear differential equations. In general, a complete integration would involve 2N constants (e.g. the initial values of coordinates and momenta). Canonical transformations enable us to play the following game: 1 Look for a transformation to a new set of canonical coordinates where the new Hamiltonian is zero and hence all new coordinates and momenta are constants of the motion. 2 Let (p, q) be the set of original momenta and coordinates in eqs of previous section, 1 Hamilton-Jacobi theory is not a recipe for integration of the coupled ODEs; nor does it in general lead to a more tractable mathematical problem. However, it provides fresh insight to the general problem, including important links to quantum mechanics and practical applications on how to deal with mechanical perturbations of a known, solved system. 2 Does this seem like too many constants? We will later explore what independent constants mean in mechanics, but at this stage let us just note that the original mathematical problem of integrating the 2N Hamiltonian equations does indeed involve 2N constants. 3

11 1 Background: Hamiltonian mechanics (α, β) the set of new constant momenta and coordinates generated by the generating function F 2 (q, α, t) which depends on the original coordinates and the new momenta. The choice of K 0 in (1.11) means that F 2 t + H(q 1, q s ; F 2 q 1,, F 2 q s ; t) = 0. (1.17) Suppose now that you can [miraculously] obtain a solution of the first-order -in general nonlinear-pde (1.17), F 2 = S(q, α, t). Note that the solution in general involves s constants {α i, i = 1,, s}. The s + 1st constant involved in the problem is a trivial one, because if S is a solution, so is S + A, where A is an arbitrary constant. It is now possible to use the defining equation of the generating function F 2 β i = S(q, α, t) α i (1.18) to obtain the new [constant] coordinates {β i, i = 1,, s}; finally, turning inside out (1.18) yields the trajectories q j = q j (α, β, t). (1.19) In other words, a solution of the Hamilton-Jacobi equation (1.17) provides a solution of the original dynamical problem Relationship to action It can be easily shown that the solution of the Hamilton-Jacobi equation satisfies or S(q, α, t) S(q, α, t 0 ) = ds dt = L, (1.20) t t 0 dτ L(q, q, τ) (1.21) where the r.h.s involves the actual particle trajectories; this shows that the solution of the Hamilton-Jacobi equation is indeed the extremum of the action function used in Lagrangian mechanics Conservative systems If the Hamiltonian does not depend explicitly on time, it is possible to separate out the time variable, i.e. S(q, α, t) = W (q, α) λ 0 t (1.22) where now the time-independent function W (q) (Hamilton s characteristic function) satisfies ( H q 1, q s ; W,, W ) = λ 0, (1.23) q 1 q s and involves s 1 independent constants, more precisely, the s constants α 1, α s depend on λ 0. 4

12 1.3.4 Separation of variables 1 Background: Hamiltonian mechanics The previous example separated out the time coordinate from the rest of the variables of the HJ function. Suppose q 1 and W q 1 enter the Hamiltonian only in the combination φ 1 (q 1, W q 1 ). The Ansatz W = W 1 (q 1 ) + W (q 2,, q s ) (1.24) in (1.23) yields H (q 2, q s ; W,, W ; φ 1 q 2 q s ( q 1, W ) ) 1 = λ 0 ; (1.25) q 1 since (1.25) must hold identically for all q, we have H ( φ 1 q 1, W ) 1 q 1 ) (q 2, q s ; W,, W ; λ 1 q 2 q s = λ 1 = λ 0. (1.26) The process can be applied recursively if the separation condition holds. Note that cyclic coordinates lead to a special case of separability; if q 1 is cyclic, then φ 1 = W q 1 = W 1 q 1, and hence W 1 (q 1 ) = λ 1 q 1. This is exactly how the time coordinate separates off in conservative systems (1.23). Complete separability occurs if we can write Hamilton s characteristic function - in some set of canonical variables - in the form W (q, α) = i W i (q i, α 1,, α s ). (1.27) Periodic motion. Action-angle variables Consider a completely separable system in the sense of (1.27). The equation p i = S q i = W i(q i, α 1,, α s ) q i (1.28) provides the phase space orbit in the subspace (q i, p i ). Now suppose that the motion in all subspaces {(q i, p i ), i = 1,, s} is periodic - not necessarily with the same period. Note that this may mean either a strict periodicity of p i, q i as a function of time (such as occurs in the bounded motion of a harmonic oscillator), or a motion of the freely rotating pendulum type, where the angle coordinate is physically significant only mod 2π. The action variables are defined as J i = 1 2π p i dq i = 1 2π dq i W i (q i, α 1,, α s ) q i (1.29) and therefore depend only on the integration constants, i.e. they are constants of the motion. If we can turn inside out (1.29), we can express W as a function of the J s instead of the α s. Then we can use the function W as a generating function of a canonical transformation to a new set of variables with the J s as new momenta, and new angle coordinates θ i = W J i = W i(q i, J 1,, J s ) J i. (1.30) 5

13 1 Background: Hamiltonian mechanics In the new set of canonical variables, Hamilton s equations of motion are J i = 0 θ i = H(J) J i ω i (J). (1.31) Note that the Hamiltonian cannot depend on the angle coordinates, since the action coordinates, the J s, are - by construction - all constants of the motion. In the set of action-angle coordinates, the motion is as trivial as it can get: J i = const θ i = ω i (J) t + const. (1.32) Complete integrability A system is called completely integrable in the sense of Liouville if it can be shown to have s independent conserved quantities in involution (this means that their Poisson brackets, taken in pairs, vanish identically). If this is the case, one can always perform a canonical transformation to action-angle variables. 1.4 Symmetries and conservation laws A change of coordinates, if it reflects an underlying symmetry of physical laws, will leave the form of the equations of motion invariant. Because Lagrangian dynamics is derived from an action principle, any such infinitesimal change which changes the particle coordinates and adds a total time derivative to the Lagrangian, i.e. q i q i = q i + ɛf i (q, t) q i q i = q i + ɛf i (q, t) (1.33) L = L + ɛ df dt, (1.34) will leave the equations of motion invariant. On the other hand, the transformed Lagrangian will generally be equal to and therefore the quantity will be conserved. L ({q i, q i}) = L({q i, q i}) = L({q i, q i }) + = L({q i, q i }) + = L({q i, q i }) + s i=1 s [ L ɛf i + L ] ɛf q i q i i s [ ( ) d L ɛf i + L ] ɛf dt q i q i i s ( ) d L f i dt q i i=1 i=1 i=1 Such underlying symmetries of classical mechanics are: L q i f i F (1.35) 6

14 1.4.1 Homogeneity of time 1 Background: Hamiltonian mechanics L = L(t + ɛ) = L(t) + ɛdl/dt, i.e. F = L; furthermore, q i = q i(t + ɛ) = q i + ɛ q i, i.e. f i = q i. As a result, the quantity s L H = q i L (1.36) q i (Hamiltonian) is conserved. i= Homogeneity of space The transformation q i q i + ɛ (hence f i = 1) leaves the Lagrangian invariant (F = 0). The conserved quantity is s L P = (1.37) q i (total momentum). i= Galilei invariance The transformation q i q i ɛt (hence f i = t) does not generally change the potential energy (if it depends only on relative particle positions). It adds to the kinetic energy a term ɛp, i.e. F = m i q i. The conserved quantity is (uniform motion of the center of mass). s m i q i P t (1.38) i= Isotropy of space (rotational symmetry of Lagrangian) Let the position of the ith particle in space be represented by the vector coordinate q i. Rotation around an axis parallel to the unit vector ˆn is represented by the transformation q i q i + ɛ f i where f i = ˆn q i. The change in kinetic energy is ɛ i q i f i = 0. If the potential energy is a function of the interparticle distances only, it too remains invariant under a rotation. Since the Lagrangian is invariant, the conserved quantity (1.35) is s i=1 L q i f i = s m i qi (ˆn q i ) = ˆn I, i=1 where is the total angular momentum. s I = m i ( q i q i ) (1.39) i=1 7

15 1.5 Continuum field theories 1 Background: Hamiltonian mechanics Lagrangian field theories in 1+1 dimensions Given a Lagrangian in 1+1 dimensions, L = dxl(φ, φ x, φ t ) (1.40) where the Lagrangian density L depends only on the field φ and first space and time derivatives, the equations of motion can be derived by minimizing the total action S = dtdxl (1.41) and have the form d dt ( L φ t ) + d ( ) L L dx φ x φ = 0. (1.42) Symmetries and conservation laws The form (1.42) remains invariant under a transformation which adds to the Lagrangian density a term of the form ɛ µ J µ (1.43) where the implied summation is over µ = 0, 1, because this adds only surface boundary terms to the action integral. If the transformation changes the field by δφ, and the derivatives by δφ x, δφ t, the same argument as in discrete systems leads us to conclude that L L δφ + δφ x + L ( dj0 δφ t = ɛ φ φ x φ t dt + dj ) 1 (1.44) dx which can be transformed, using the equations of motion, to ( ) d L δφ + L δφ t + d ( ) L δφ + L ( dj0 δφ x = ɛ dt φ t φ t dx φ x φ x dt + dj ) 1 dx (1.45) Examples: 1. homogeneity of space (translational invariance) x x + ɛ δφ = φ(x + ɛ) φ(x) = φ x ɛ δφ t = φ t (x + ɛ) φ t (x) = φ xt ɛ δφ x = φ x (x + ɛ) φ x (x) = φ xx ɛ δl = dl dl δx = dx dx ɛ J 1 = L, J 0 = 0. (1.46) Eq. (1.45) becomes ( ) d L φ x + L φ xt + d ( ) L φ x + L φ xx = dl dt φ t φ t dx φ x φ x dx or (1.47) ( ) d L φ x + d ( ) L φ x L = 0 ; (1.48) dt φ t dx φ x 8

16 1 Background: Hamiltonian mechanics integrating over all space, this gives dx L φ x P φ t (1.49) i.e. the total momentum is a constant. 2. homogeneity of time t t + ɛ δφ = φ(t + ɛ) φ(t) = φ t ɛ δφ t = φ t (t + ɛ) φ t (t) = φ tt ɛ δφ x = φ x (t + ɛ) φ x (t) = φ xt ɛ δl = dl dl δt = dt dt ɛ J 0 = L, J 1 = 0. (1.50) Eq. (1.45) becomes ( ) d L φ t + L φ tt + d ( ) L φ t + L φ tx = dl dt φ t φ t dx φ x φ x dt or (1.51) ( ) d L φ t L + d ( ) L φ t = 0 ; (1.52) dt φ t dx φ x integrating over all space, this gives [ ] L dx φ t L H (1.53) φ t i.e. the total energy is a constant. 3. Lorentz invariance 1.6 Perturbations of integrable systems Consider a conservative Hamiltonian system H 0 (J) which is completely integrable, i.e. it possesses s independent integrals of motion. Note that I use the action-angle coordinates, so that H 0 is a function of the (conserved) action coordinates J j. The angles θ j are cyclic variables, so they do not appear in H 0. Suppose now that the system is slightly perturbed, by a time-independent perturbation Hamiltonian µh 1 (µ 1) A sensible question to ask is: what exactly happens to the integrals of motion? We know of course that the energy of the perturbed system remains constant - since H 1 has been assumed to be time independent. But what exactly happens to the other s 1 constants of motion? The question was first addressed by Poincaré in connection with the stability of the planetary system. He succeeded in showing that there are no analytic invariants of the perturbed system, i.e. that it is not possible, starting from a constant Φ 0 of the unperturbed system, to construct quantities Φ = Φ 0 (J) + µφ 1 (J, θ) + µ 2 Φ 2 (J, θ), (1.54) where the Φ n s are analytic functions of J, θ, such that {Φ, H} = 0 (1.55) 9

17 1 Background: Hamiltonian mechanics holds, i.e. Φ is a constant of motion of the perturbed system. The proof of Poincaré s theorem is quite general. The only requirement on the unperturbed Hamiltonian is that it should have functionally independent frequencies ω j = H 0 / J j. Although the proof itself is lengthy and I will make no attempt to reproduce it, it is fairly straightforward to see where the problem with analytic invariants lies. To second order in µ, the requirement (1.55) implies {Φ 0 + µφ 1 + µ 2 Φ 2, H 0 + µh 1 } = 0 {Φ 0, H 0 } + µ ({Φ 1, H 0 } + {Φ 0, H 1 }) + µ 2 ({Φ 2, H 0 } + {Φ 1, H 1 }) = 0. The coefficients of all powers must vanish. Note that the zeroth order term vanishes by definition. The higher order terms will do so, provided {Φ 1, H 0 } = {Φ 0, H 1 } (1.56) {Φ 2, H 0 } = {Φ 1, H 1 }. The process can be continued iteratively to all orders, by requiring {Φ n, H 0 } = {Φ n+1, H 1 }. (1.57) Consider the lowest-order term generated by (1.57). Writing down the Poisson brackets gives s ( Φ1 H 0 Φ ) 1 H 0 s ( Φ0 H 1 = Φ ) 0 H 1. (1.58) θ i J i J i θ i θ i J i J i θ i j=1 The second term on the left hand side and the first term on the right-hand side vanish because the θ s are cyclic coordinates in the unperturbed system. The rest can be rewritten as s j=1 ω i (J) Φ 1 θ i = j=1 s j=1 Φ 0 J i H 1 θ i. (1.59) For notational simplicity, let me now restrict myself to the case of two degrees of freedom. The perturbed Hamiltonian can be written in a double Fourier series H 1 = n 1,n 2 A n1,n 2 (J 1, J 2 ) cos(n 1 θ 1 + n 2 θ 2 ). (1.60) Similarly, one can make a double Fourier series ansatz for Φ 1, Φ 1 = n 1,n 2 B n1,n 2 (J 1, J 2 ) cos(n 1 θ 1 + n 2 θ 2 ). (1.61) Now apply (1.59) to the case Φ 0 (J) = J 1. Using the double Fourier series I obtain B (J n 1) 1 n 1,n 2 = A n1,n n 1 ω 1 + n 2 ω 2, (1.62) 2 which in principle determines the first-order term in the µ expansion of the constant of motion J 1 which should replace J 1 in the new system. It is straightforward to show, using the same process for J 2, that the perturbed Hamiltonian can be written in terms of the new constants J 1 as H = H 0 (J 1, J 2) + O(µ 2 ). (1.63) Unfortunately, what looks like the beginning of a systematic expansion suffers from a fatal flaw. If the frequencies are functionally independent, the denominator in (1.62) will in general vanish on a denumerably infinite number of surfaces in phase space. This however means that Φ 1 cannot be an analytic function of J 1, J 2. Analytic invariants are not possible. All integrals of motion - other than the energy - are irrevocably destroyed by the perturbation. 10

18 2 Background: Statistical mechanics 2.1 Scope Classical statistical mechanics attempts to establish a systematic connection between microscopic theory which governs the dynamical motion of individual entities (atoms, molecules, local magnetic moments on a lattice) and the macroscopically observed behavior of matter. Microscopic motion is described - depending on the particular scale of the problem - either by classical or quantum mechanics. The rules of macroscopically observed behavior under conditions of thermal equilibrium have been codified in the study of thermodynamics. Thermodynamics will tell you which processes are macroscopically allowed, and can establish relationships between material properties. In principle, it can reduce everything - everything which can be observed under varying control parameters ( temperature, pressure or other external fields) to the equation of state which describes one of the relevant macroscopic observables as a function of the control parameters. Deriving the form of the equation of state is beyond thermodynamics. It needs a link to microscopic theory - i.e. to the underlying mechanics of the individual particles. This link is provided by equilibrium statistical mechanics. A more general theory of non-equilibrium statistical mechanics is necessary to establish a link between non-equilibrium macroscopic behavior (e.g. a steady state flow) and microscopic dynamics. Here I will only deal with equilibrium statistical mechanics. 2.2 Formulation A statistical description always involves some kind of averaging. Statistical mechanics is about systematically averaging over hopefully nonessential details. What are these details and how can we show that they are nonessential? In order to decide this you have to look first at a system in full detail and then decide what to throw out - and how to go about it consistently Phase space An Hamiltonian system with s degrees of freedom is fully described at any given time if we know all coordinates and momenta, i.e. a total of 2s quantities (=6N if we are dealing with point particles moving in three-dimensional space). The microscopic state of the system can be viewed as a point, a vector in 2s dimensional space. The dynamical evolution of the system in time can be viewed as a motion of this point in the 2s dimensional space (phase space). I will use the shorthand notation Γ (q i, p i, i = 1, s) to denote a point in phase space. More precisely, Γ(t) will denote a trajectory in phase space with the initial condition Γ(t 0 ) = Γ Note that trajectories in phase space do not cross. A history of a Hamiltonian system is determined by differential equations which are first-order in time, and is therefore reversible - and hence unique. 11

19 2.2.2 Liouville s theorem 2 Background: Statistical mechanics Consider an element of volume dσ 0 in phase space; the set of trajectories starting at time t 0 at some point Γ 0 dσ 0 lead, at time t to points Γ dσ. Liouville s theorem asserts that dσ = dσ 0. (invariance of phase space volume). The proof consists of showing that the Jacobi determinant (q, p) D(t, t 0 ) (q 0, p 0 (2.1) ) corresponding to the coordinate transformation (q 0, p 0 ) (q, p), is equal to unity. Using general properties of Jacobians (q, p) (q, p) (q 0, p 0 = ) (q 0, p) (q 0, p) (q 0, p 0 ) = (q) (p) (q 0 ) p=const (p 0 ) (2.2) q=const and D(t, t 0 t = t=t0 s i=1 ( qi + ṗ ) i = q i p i t=t0 s i=1 ( 2 H q i p i ) 2 H = 0, (2.3) p i q i and noting that D(t 0, t 0 ) = 1, it follows that D(t, t 0 ) = 1 at all times Averaging over time Consider a function A(Γ) of all coordinates and momenta. If you want to compute its longtime average under conditions of thermal equilibrium, you need to follow the state of the system over a long time, record it, evaluate the function A at each instant of time, and take a suitable average. Following the trajectory of the point in phase space allows us to define a long-time average 1 T Ā = lim dta[γ(t)]. (2.4) T T 0 Since the system is followed over infinite time this can then be regarded as a true equilibrium average. More on this later Ensemble averaging On the other hand, we could consider an ensemble of identically prepared systems and attempt a series of observations. One system could be in the state Γ 1, another in the state Γ 2. Then perhaps we could determine the distribution of states ρ(γ), i.e. the probability ρ(γ)δγ, that the state vector is in the neighborhood (Γ, Γ + δγ). The average of A in this case would be < A >= dγρ(γ)a(γ) (2.5) Note that since ρ is a probability distribution, its integral over all phase space should be normalized to unity: dγρ(γ) = 1 (2.6) A distribution in phase space must obey further restrictions. Liouville s theorem states that if we view the dynamics of a Hamiltonian system as a flow in phase space, elements of volume are invariant - in other words the fluid is incompressible: d dt ρ(γ, t) = {ρ, H} + ρ(γ, t) = 0. (2.7) t 12

20 2 Background: Statistical mechanics For a stationary distribution ρ(γ) - as one expects to obtain for a system at equilibrium - {ρ, H} = 0, (2.8) i.e. ρ can only depend on the energy 2. This is a very severe restriction on the forms of allowed distribution functions in phase space. Nonetheless it still allows for any functional dependence on the energy. A possible choice (Boltzmann) is to assume that any point on the phase space hypersurface defined by H(Γ) = E may occur with equal probability. This corresponds to ρ(γ) = 1 δ {H(Γ) E} (2.9) Ω(E) where Ω(E) = dγ δ {H(Γ) E} (2.10) is the volume of the hypersurface H(Γ) = E. This is the microcanonical ensemble. Other choices are possible - e.g. the canonical (Gibbs) ensemble defined as ρ(γ) = 1 Z(β) e βh(γ) (2.11) where the control parameter β can be identified with the inverse temperature and Z(β) = dγe βh(γ) (2.12) is the classical partition function Equivalence of ensembles The choice of ensemble, although it may appear arbitrary, is meant to reflect the actual experimental situation. For example, the Gibbs ensemble may be derived - in the sense that it can be shown to correspond to a small (but still macroscopic) system in contact with a much larger reservoir of energy - which in effect holds the smaller system at a fixed temperature T = 1/β. Ensembles must - and to some extent can - be shown to be equivalent, in the sense that the averages computed using two different ensembles coincide if the control parameters are appropriately chosen. For example a microcanonical average of a function A(Γ) over the energy surface H(Γ) = ɛ will be equal with the canonical average at a certain temperature T if we choose ɛ to be equal to the canonical average of the energy at that temperature, i.e. < A(Γ) > micro ɛ =< A(Γ) > canon T if ɛ =< H(Γ) > canon T. If ensembles can be shown to be equivalent to each other in this sense, we do not need to perform the actual experiment of waiting and observing the realization of a large number of identical systems as postulated in the previous section. We can simply use the most convenient ensemble for the problem at hand as a theoretical tool for calculating averages. In general one uses the canonical ensemble, which is designed for computing average quantities as functions of temperature Ergodicity The usage of ensemble averages - and therefore of the whole edifice of classical statistical mechanics - rests on the implicit assumption that they somehow coincide with the more physical time averages. Since the various ensembles can be shown to be equivalent (cf. 2 or - in principle - on other conserved quantities; in dealing with large systems it may well be necessary to account for other macroscopically conserved quantities in defining a proper distribution function. 13