10. Zwanzig-Mori Formalism

Transcription

1 University of Rhode Island Nonequilibrium Statistical Physics Physics Course Materials Zwanzig-Mori Formalism Gerhard Müller University of Rhode Island, Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Follow this and additional works at: nonequilibrium_statistical_physics Part of the Physics Commons Abstract Part ten of course materials for Nonequilibrium Statistical Physics (Physics 626), taught by Gerhard Müller at the University of Rhode Island. Entries listed in the table of contents, but not shown in the document, exist only in handwritten form. Documents will be updated periodically as more entries become presentable. Updated with version 2 on 5/3/206. Recommended Citation Müller, Gerhard, "0. Zwanzig-Mori Formalism" (205). Nonequilibrium Statistical Physics. Paper 0. This Course Material is brought to you for free and open access by the Physics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in Nonequilibrium Statistical Physics by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.

2 Contents of this Document [ntc0] 0. Zwanzig-Mori Formalism Introduction [nln28] Time-dependence of expectation values (quantum and classical) [nln77] Zwanzig s kinetic equation: generalized master equation [nln29] [nex68] Projection operator method (Mori formalism) [nln3] Kubo inner product [nln32] Projection operators [nln33] First and second projections [nln34] [nln35] Continued-fraction representation of relaxation function [nln36] n-pole approximation [nln87] Relaxation function with uniform continued-fraction coefficients [nex69] Link to Green s function formalism [nln88] Structure function of harmonic oscillator [nex7], [nex72], [nex73]

3 Zwanzig-Mori formalism [nln28] Beginnings: Two phenomenological approaches for the dynamics of systems close to or at thermal equilibrium: Phenomenological equations of motion for probability distributions (e.g. master equation, Fokker-Planck equation). Phenomenological equations of motion for dynamical variables (e.g. Langevin equation) In these approaches, the focus is on selected degrees of freedom. All other degrees of freedom are taken into account summarily in the form of ad-hoc randomness. Completions: Microscopic foundations for these phenomenological approaches. Zwanzig (960): Rigorous derivation of a generalized master equation from first principles, i.e. from the Liouville equation. Mori (965): Rigorous derivation of a generalized Langevin equation from first principles, i.e. from the (quantum) Heisenberg equation or the (classical) canonical equations. The focus is again on selected degrees of freedom but here the effects of the other degrees of freedom are taken into account on a basis that is exact and amenable to systematic approximation. Variants: Zwanzig s approach leads to a kinetic equation of a particular kind. There exist alternative ways to derive kinetic equations from the Liouville equations via systematic approximations (e.g. via BBGKY hierarchy). Mori s approach has been formulated in more than one rendition. The version named projection operator formalism is most illuminating regarding the physical meaning of systematic approximations. The version named recursion method is most readily amenable to computational applications.

4 Time Dependence of Expectation Values [nln77] Quantum Hamiltonian system: A t = Tr{ρ(0)A(t)} (Heisenberg representation) = Tr { ρ(0)e ıht/ A(0)e ıht/ } = Tr { e ıht/ ρ(0)e ıht/ A(0) } = Tr{ρ(t)A(0)} (Schroedinger representation) da dt = ıˆla = ı [H, A] dρ dt = ıˆlρ = [H, ρ] ı ˆL. = [H, ] (Heisenberg equation) (quantum Liouville equation) (quantum Liouville operator) [, ] (commutator) Classical Hamiltonian system: A t = = d n q d n q d n p ρ ( q,..., q n ; p,..., p n ; 0 ) A ( q,..., q n ; p,..., p n ; t ) d n p ρ ( q,..., q n ; p,..., p n ; t ) A ( q,..., q n ; p,..., p n ; 0 ) da dt = ıˆla = {H, A} (Hamilton equation) dρ dt = ıˆlρ = {H, A} (classical Liouville equation) ˆL =. n ( H {H, } = ı H ) (classical Liouville operator) q j p j p j q j {, } j= (Poisson bracket)

5 Zwanzig s method [nln29] Consider classical phase-space density ρ(t) = ρ(q, p ; q 2, p 2 ;... ; q n, p n ; t). Distinguish system (q, p ) and heat bath (q 2, p 2 ;... ; q n, p n ). Probability density of system: ρ (t) = ˆP ρ(t) via projection. Probability density of heat bath: ρ 2 (t) = ˆQρ(t), where ˆQ = ˆP. Implementation of projection: ρ = ˆP ρ(q, p ; q 2, p 2 ;... ; q n, p n ; t) = ρ eq (q 2, p 2 ;... ; q n, p n )σ(q, p ; t), where ρ eq / t = ıˆlρ eq = 0 and σ(q, p ; t) = dq 2 dp 2 dq n dp n ρ(q, p ; q 2, p 2 ;... ; q n, p n ; t). Idempotency, ˆP ρ (t) = ρ (t), satisfied by construction. Liouville equation, ρ(t)/ t = ıˆlρ(t), split into two coupled equations: ˆP t ρ(t) = t ρ (t) = ı ˆP ˆL [ ρ (t) + ρ 2 (t) ], () ˆQ t ρ(t) = t ρ 2(t) = ı ˆQˆL [ ρ (t) + ρ 2 (t) ]. (2) Formal solution of (2) [nex68] to be substituted into (): ρ 2 (t) = e ı ˆQˆLt ρ 2 (0) ı t 0 dτ e ı ˆQˆLτ ˆQˆLρ (t τ). Zwanzig s kinetic equation (generalized master equation): t ρ (t) = ı ˆP ˆLρ (t) ı ˆP ˆLe ı ˆQˆLt ρ 2 (0) t # Autonomous part of system s time evolution. 0 dτ ˆP ˆLe ı ˆQˆLτ ˆQˆLρ (t τ). #2 Autonomous part of heat bath s time evolution. Instantaneous effect of heat bath on system at time t. #3 Effect caused by system on heat bath at time t τ propagates in heat bath and feeds back into system at time t. First term often vanishes and second term (inhomogeneity) can be made zero by judicious choice of initial conditions. Zwanzig s kinetic equation can be used as the starting point for the derivation (via approximations) of a master equation or a Fokker-Planck equation.

6 [nex68] Zwanzig s kinetic equation. In the derivation of Zwanzig s kinetic equation, t t ρ (t) = i ˆP ˆLρ (t) i ˆP ˆLe i ˆQ ˆLt ρ 2 (0) dτ ˆP ˆLe i ˆQ ˆLτ ˆQˆLρ (t τ), () 0 from two projections of the Liouville equation, ˆP ρ t = ρ t = i ˆP ˆL[ρ + ρ 2 ], we postulate the formal solution ˆQ ρ t = ρ 2 t = i ˆQˆL[ρ + ρ 2 ], (2) t ρ 2 (t) = e i ˆQ ˆLt ρ 2 (0) i dτ e i ˆQ ˆLτ ˆQˆLρ (t τ). (3) 0 Verify that (3) is a solution of (2). The derivation involves one integration by parts. Solution:

7 Projection operator method [nln3] Goal: Determination of symmetrized time-correlation function (fluctuation function) for a dynamical variable A(t) of a quantum or classical many-body Hamiltonian system H in thermal equilibrium. Fluctuation function (real, symmetric, normalized): C 0 (t). = A(t) A A A = A A( t) A A = A e ılt A. A A Dirac notation symbolizes inner product of choice as explained in [nln32]. Some properties of dynamic quantities depend on choice of inner product. Relaxation function (via Laplace transform): c 0 (z) = 0 dt e zt A e ılt A A A = A A A z + ıl A. Projection operator method determines relaxation function via systematic approximation. Inverse Laplace transform, C 0 (t) = 2πı C dz e zt c 0 (z), involves integral along straight path from ɛ ı to ɛ + ı for ɛ > 0. In practical applications, the (real, symmetric) spectral density is inferred from the relaxation function as limit process, Φ 0 (ω) = 2 lim ɛ 0 R{c 0 (ɛ ıω)}, and the fluctuation function via inverse Fourier transform, C 0 (t) = + dω 2π e ıωt Φ 0 (ω). The last bracket is also known as a Green s function.

8 Kubo inner product [nln32] General properties of inner products: A B = B A, A λb = λ A B, A A = A 2 0, A B+C = A B + A C. Kubo inner product for quantum system: where A B. = β β 0 dλ e λh A e λh B, A = Z Tr{e βh A}, Z = Tr{e βh }, β = k B T. Alternative inner product for quantum systems: 2 A B. = 2 A B + BA. Both inner products have the same classical limit: 3 A B =. d n q d n p e βh(q,p) A(q, p)b(q, p). Z Inner products of [nln3] employ... quantum Liouville operator: L = [L, ], da Heisenberg equation of motion: dt = ı [H, A] = ıla. classical Liouville operator: L = ı{h, } = ı Hamilton s equation of motion: da dt n ( H j= q j = {H, A} = ıla H p j p j q j ), Designed to satisfy classical fluctuation-dissipation theorem in [nln39]. 2 Designed to satisfy quantum fluctuation-dissipation theorem in [nln39]. 3 Option for all inner products: subtract A B.

9 Projection operators [nln33] The relaxation function c 0 (z) is determined recursively by a succession of subdivisions of the many-body dynamics into components that are treated rigorously and a remainder that is treated phenomenologically. It is expected that the remainder diminishes in importance as the number of rigorous components is increased systematically. The time evolution of the dynamical variable A(t) can be conceived as a pirouette performed by the vector A(t) through the Hilbert space. The subdivisions are implemented by a sequence of projections onto onedimensional Hilbert subspaces traversed by A(t). Initial condition: f 0. = A(0) = A. Projection operators P n and Q n = P n, n = 0,,...,.. P 0 = f0 f 0 f 0 f 0, P0 2 = P 0, P 0 = P 0, P 0 Q 0 = Q 0 P 0 = 0. Orthogonal direction: f = ıl f 0, f 0 f = 0, P 0 f = 0, Q 0 f = f P 0 f = f, P = f f f f, Q = P. The systematic generation of further orthogonal direction will be discussed in the context of the recursion method. Successive projections filter out particular aspects of the many-body dynamics to be taken into account rigorously. The filters are applied in series. What passes through n filters is the remainder to be treated phenomenologically. The physical content of this process can be gleaned from the first two projections carried out in detail: First projection [nln33], Second projection [nln34]. Unitary transformation e ılt makes ıla orthogonal to A, implying A ıla = 0.

10 First projection [nln34] Rewrite relaxation function from [nln3] with projection operators from [nln33] and apply Dyson identity (X + Y ) = X X Y (X + Y ) : f 0 f 0 c 0 (z) = f 0 z + ıl f 0 = f 0 z + ılp 0 + ılq 0 f 0 = f 0 z + ılq 0 f 0 + f 0 ılp 0 z + ılq 0 z + ıl f 0. Simplify both terms: [ f 0 + ( ı) z z LQ 0 + ( ı)2 f 0 ıl z + ılq 0 f 0 f 0 f 0 c 0 (z) = [ z f 0 ıl z + ılq 0 f 0 = f 0 = f 0 ( ı)lq 0 = ] LQ z 2 0 LQ 0 + f 0 f 0 z + ıl f 0 z + z, f 0 ıl f 0 f 0 z + ılq 0 f 0 = z f 0 f 0. = f 0 ıl z + ılq 0 f 0 c 0 (z). ] z (z + ılq 0 ) + ıl z + ılq 0 z + ılq 0 f 0 ıl z + ılq 0 f 0 = f 0 ( ı)lq 0 Q 0 ıl z + ıq 0 LQ 0 f 0 z + ıl f ; f = Q 0 f = Q 0 ıl f 0, L = Q 0 LQ 0. f Relaxation function after first projection expressed via memory function: c 0 (z) = z + Σ (z), Σ (z) = f f 0 f 0 z + ıl f. Memory function Σ (z) of original problem, {L, f 0 }, can be reinterpreted as the (as yet non-normalized) relaxation function of a new dynamical problem, {L, f }. Projection operator Q 0 acts as filter on the Liouvillian L, absorbing that part of dynamics dealt with explicitly in first projection. Explicit information contained in normalization constant of Σ (z). Direct consequence of operator identity (X+Y )(X+Y ) = X(X+Y ) +Y (X+Y ) =, here with X. = z + ılq 0, Y. = ılp 0.

11 Second projection [nln35] Rewrite memory function from [nln34] with projection operators from [nln33] and apply Dyson identity: f 0 f 0 Σ (z) = f z + ıl f = f z + ıl P + ıl Q f = f z + ıl Q f f ıl P z + ıl Q z + ıl f = z f f f ıl z + ıl Q f f0 f 0 f f Σ (z), where simplifications analogous to [nln34] are carried out. Σ (z) = f f / f 0 f 0 z + z, f ıl f f z + ıl Q f z f ıl z + ıl Q f = = f ( ı)l Q Q ıl z + ıq L Q f = f 2 z + ıl 2 f 2, with f 2 = Q ıl f, L 2 = Q L Q. Memory function (first termination function) after second projection expressed via second termination function: Σ (z) = z + Σ 2 (z), Σ 2(z) = f 2 f f z + ıl 2 f 2 with continued-fraction coefficients = f f / f 0 f 0. The n th projection yields Σ n (z) = n z + Σ n (z), Σ n(z) = f n f n f n with n = f n f n / f n 2 f n 2 and f n = Q n ıl n f n, L n = Q n L n Q n. z + ıl n f n

12 Continued-fraction representation [nln36] Relaxation function after n successive projections: c 0 (z) = z + z + z n z + Σ n (z) The explicit dynamical information extracted from the original many-body system, {L, f 0 }, in the first n projections is contained in the continuedfraction coefficients:,..., n. Each projection adds a layer of projection operators around the original Liouvillian: L n = Q n Q 0 LQ 0 Q n. Expectation (somewhat naively): If n is sufficiently large, all distinctive spectral features of L will have been filtered out and incorporated explicitly into the relaxation function via the i. Whatever features of L still shine through the n filters are adequately represented by a source of white noise. The memory function associated with white noise is a constant, commonly represented by a relaxation time: Σ n (z) = τ n = const. This completion of the continued fraction is known under the name n-pole approximation. The relaxation function is characterized by n poles in the complex frequency plane. Options used in practical applications: A number of continued-fraction coefficients are determined on phenomenological grounds along with a terminating relaxation time τ n. Examples: classical relaxator ( pole), classical oscillator (two poles). A number of continued-fraction coefficients are calculated from the original many-body system via the recursion method along with a termination function Σ n (z) inferred from an extrapolation scheme.

13 n-pole Approximation [nln87] Relaxation function after n successive projections (Mori formalism [nln36]) or n iterations (recursion method [nln84]), c 0 (z) = z + z + z + 2, () + n z + τ n has termination function represented by a relaxation time τ n : c n (z) Γ n (z) = n c n (z) =. (2) τ n Consequences: Singularity structure of c 0 (z) reduced to n poles in complex z-plane. Frequency moments M 2k = ω 2k divergent for k n. Limiting cases: τ n : All poles approach ω-axis: z = ɛ ıω with ɛ 0. Spectral density Φ 0 (ω) is a sum of δ-functions. τ n 0: Transition to (n )-pole approximation: n z + /τ n τ n. Illustrations for n =, 2, 3: relaxation function: c 0 (z) = memory function: Σ(z), z + Σ(z), spectral density: Φ 0 (ω) = 2 lim ɛ 0 Re [ c 0 (ɛ ıω)].

14 -pole spectral approximation Memory function representing white noise: Σ(z) = τ = const. Relaxation function: c 0 (z) = z + = τ τ z +. τ Spectral density: Φ 0 (ω) = 2τ + τ 2 ω 2. 2

15 2-pole spectral approximation Memory function representing classical relaxator: Σ(z) = z + τ. Relaxation function: Spectral density: c 0 (z) = Φ 0 (ω) = τ 2 z + τ 2 z 2 + z + τ 2. 2 τ 2 τ 2 2 ( ω 2 ) 2 + ω 2. 3

16 3-pole spectral approximation Memory function representing damped classical oscillator: Σ(z) = z + 2 z + τ 3. Relaxation function: Spectral density: c 0 (z) = Φ 0 (ω) = τ 3 z 2 + z + 2 τ 3 τ 3 z 3 + z 2 + ( + 2 )τ 3 z τ 3 [τ 3 ω(ω 2 2 )] 2 + (ω 2 ) 2. 4

17 [nex69] Relaxation function with uniform continued-fraction coefficients. Find closed-form expressions for the relaxation function c 0 (z), the spectral density Φ 0 (ω), and the fluctuation function C 0 (t), of some physical system if we know that the (infinite) sequence of continued-fraction coefficients is (a) uniform: = 2 =... = 4 ω2 0, (b) almost uniform: = 2 ω2 0, 2 = 3... = 4 ω2 0. Solution:

18 Link to Green s Function Formalism [nln88] Retarded Green s function (two varieties): 23 G ± (t t ) =. A(t); B(t ) ± = ıθ(t t ) [A(t), B(t )] ±. () Equations of motion are hierarchical in nature: ı t A(t); B(t ) ± = δ(t t ) [A, B] ± + [A(t), H] ; B(t ) ±. (2) Frequency-dependent Green s functions via Fourier transform: G ± (ω). = A; B ± ω = + dt e ıωt G± (t), (3) ω A; B ± ω = [A, B] ± + [A, H] ; B ± ω. (4) Spectral representations relate the Green s functions G ± (ζ) to the functions S(ω), Φ(ω), and χ (ω) as defined in [nln39]: G ± (ζ) = + d ω S( ω) [ ] ± e β ω, ζ = ω + ıɛ, ɛ > 0, (5a) 2π ζ ω + d ω Φ( ω) + G + (ζ) = 2 2π ζ ω, G d ω χ ( ω) (ζ) = 2 2π ζ ω, (5b) Inverse relations: Φ(ω) = lim Im [ G + (ω + ıɛ) ] ɛ 0 χ (ω) = lim Im [ G (ω + ıɛ) ] ɛ 0 S(ω) = Φ(ω) 2( + e βω ) = χ (ω) 2( e βω ) (spectral density) (dissipation function) (structure function) Relation to relaxation function for ıζ = z (see [nln3] and [nln84]): c 0 (z) = 0 ıζt Φ(t) dt e Φ(0) = ı + dω Φ(ω) Φ(0) 2π ζ ω = ı 2 Φ(0) G +(ζ). Here we set = as is common practice. 2 The bracket [, ] stands for commutators and the bracket [, ] + for anticommutators. 3 One variety coincides with Kubo s response function from [nln27]: G (t) = χ AB (t).

19 [nex7] Structure function of harmonic oscillator I. Consider the quantum harmonic oscillator (for = ), H = p2 2m + 2 mω2 0q 2 = ω 0 ( a a + ), 2 where q = (a + a)/ 2mω 0, p = i mω 0 /2(a a) relate the position and momentum operators ([q, p] = i) to the boson creation and annihilation operators ([a, a ] = ). Use the recursion method with inner product A B = 2 ( AB + BA ) to calculate the structure function S qq (ω) for the position variable at temperature T, where a a = n B = (e βω ), β = /k B T. Solution:

20 [nex72] Structure function of harmonic oscillator II. Consider the classical harmonic oscillator, H = p2 2m + 2 mω2 0q 2. Use the recursion method with inner product A B = AB to calculate the structure function S qq (ω) for the position variable at temperature T, where p 2 /2m = 2 mω2 0 q 2 = 2 k BT, according to the equipartition theorem. Solution:

21 [nex73] Structure function of harmonic oscillator III. Consider the quantum harmonic oscillator (for = ), H = p2 2m + 2 mω2 0q 2 = ω 0 ( a a + ), 2 where q = (a + a)/ 2mω 0, p = i mω 0 /2(a a) relate the position and momentum operators ([q, p] = i) to the boson creation and annihilation operators ([a, a ] = ). Solve the equation of motion ζ A; B ± ζ = [A, B] ± + [A, H]; B ± ζ for (a) the Green s function q; q ζ and (b) the Green s function q; q + ζ. Infer from each Green s function the structure function S qq (ω) for the position variable at temperature T. Solution: