11. Recursion Method: Concepts

Transcription

1 University of Rhode Island Nonequilibrium Statistical Physics Physics Course Materials Recursion Method: Concepts Gerhard Müller University of Rhode Island, Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4. License. Follow this and additional works at: nonequilibrium_statistical_physics Abstract Part eleven of course materials for Nonequilibrium Statistical Physics (Physics 66), 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. Recommended Citation Müller, Gerhard, "11. Recursion Method: Concepts" (16). Nonequilibrium Statistical Physics. Paper 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 [ntc1] 11. Recursion Method: Concepts Stage for recursion method [nln79] Modules of recursion method [nln8] Representations of recursion method [nln81] Orthogonal expansion of dynamical variables [nln8] Gram-Schmidt orthogonalization I [nln83] Relaxation function and spectral density [nln84] Moment expansion vs continued fraction I [nln85] Link to generalized Langevin equation [nln86] Orthogonal expansion of wave functions [nln9] Gram-Schmidt orthogonalization II [nln91] Structure function [nln9] Moment expansion vs continued fraction II [nln96] Genetic code of spectral densities [nln98] Spectral Lines from finite sequences of continued-fraction coefficients [nln99] Spectral densities with bounded support [nln1] Bandwidth and gap in spectral density [nln11] Spectral densities with unbounded support [nln1] Unbounded support and gap [nln13]

3 Stage for Recursion Method [nln79] Recursion method as applied to many-body dynamics: backdrop, props, protagonists. [from Viswanath and Müller 1994]

4 Modules of Recursion Method [nln8] Recursion method as applied to many-body dynamics: main lines of formal development. [from Viswanath and Müller 1994]

5 Representations of Recursion Method [nln81] Object of interest: time-dependent correlation function A(t)A. Recursive part of method: orthogonal expansion of carrier of time evolution. Heisenberg picture: time evolution prescribed by Heisenberg (or Hamilton) equation, time evolution carried by dynamical variable, Liouvillian operator generates new directions. Schrödinger picture: time evolution prescribed by Schrödinger equation, time evolution carried by wave function, Hamiltonian operator generates new directions. Liouvillian representation Hamiltonian representation d dt A(t) = ıl qua(t) = ı [H, A(t)] ı d dt A(t) = ıl cla(t) = {H, A(t)} d ψ(t) = H ψ(t) dt A(t) = e ılt A() = C k (t) f k ψ(t) = e ıht/ ψ() = k= D k (t) f k k= f = A(), f k+1 = ıl f k... f = ψ() = A φ, f k+1 = H f k... Φ(t) = [A(t), A()] + A S(t) = A(t)A() A Tr [ e βh e} ıht/ Ae {{ ıht/ } A ] e ıe t/ φ A e ıht/ A φ }{{} A(t) ψ(t)

6 Orthogonal Expansion of Dynamical Variables [nln8] A(t) = C k (t) f k. (1) k= Step #1: [M.H. Lee] Orthogonal basis, f, f 1,..., with initial condition, f = A(). Quantum statistics: f k form orthogonal set of operators. Classical statistics: f k form orthogonal set of phase-space functions. Generation of orthogonal directions: f k ıl f k =. Recurrence relations for basis vectors f k : f k+1 = ıl f k + k f k 1, k+1 = f k+1 f k+1, k =, 1,,... () f k f k Conditions: f 1. =, f. = A,. =. First three iterations spelled out in [nln83]. Step #: [M.H. Lee] Time-dependent coefficients of basis vectors: C k (t). Substitute (1) into equation of motion from [nln81]: da/dt = ıla. d/dt acts on C k (t) and L acts on f k. Comparison of coefficients in Ċ k (t) f k = k= k= C k (t) [ ] f k+1 k f k1 }{{} ıl f k yields set of coupled, linear, first-order ODEs for functions C k (t): Ċ k (t) = C k 1 (t) k+1 C k+1 (t), k =, 1,,... (4) Conditions: C 1 (t), C k () = δ k,, k =, 1,,... (3) Normalized fluctuation function (see [nln39]): C (t) = A(t) A() A() A() = Φ(t) Φ(). (5)

7 Gram-Schmidt Orthogonalization I [nln83] First three iterations in step #1 of [nln8]. Initial condition: f = A. First iteration: ıl f = f 1 f f 1 = f ıl f =. Second iteration: ıl f 1 = f 1 f f 1 f = f 1 ıl f 1 }{{} f f = f ıl f 1 Third iteration: ıl f = f 3 f 1 Γ f } {{ } f 1 f 1 if 1 = f 1 f 1 f f. + 1 f 1 f =, }{{} + 1 f f = f f 3 = f ıl f + }{{} f f 1 }{{} f 1 f 3 = f 1 ıl f }{{} f f +Γ f f =, }{{} = + f 1 f 1 + Γ f 1 f }{{} if = f f f 1 f 1, f f 3 = f ıl f + }{{} f f 1 +Γ }{{} f f = f 1 f = if Γ =.

8 Relaxation Function and Spectral Density [nln84] Relaxation function via Laplace transform: c k (z). = dt e zt C k (t). (1) Coupled ODEs for C k (t) become coupled algebraic equations for c k (z): 1 zc k (z) δ k, = c k 1 (z) k+1 c k+1 (z), k =, 1,,... () Condition: c 1 (z). Recursive construction of continued fraction representation for c (z): k = : zc (z) 1 = 1 c 1 (z) c (z) = 1 z + 1 c 1 (z) c (z), k = 1 : zc 1 (z) = c (z) c (z) c 1(z) c (z) = 1, c (z) z + c 1 (z) c (z) = z z + z + (3) Spectral density Φ(ω) from relaxation function c (z): combine Fourier transform with inverse Laplace transform. Φ (ω). = + C (t) = 1 πı C dt e ıωt C (t), dz e zt c (z) Φ (ω) = lim ɛ Re[c (ɛ ıω)]. (4) 1 Use dt e zt Ċ k (t) = [ ] e zt C k (t) dt( z)e zt C k (t) = zc k (z) C k ().

9 Moment Expansion vs Continued Fraction I [nln85] Moment expansions of fluctuation function (see [nln78]): C (t) = k= ( 1) k (k)! M kt k, M = 1. (1) Asymptotic expansion and continued-fraction representation of relaxation function: c (z). = dt e zt C (t) = M k z (k+1) = k= z z + z +. () Transformation relations between {M k } and { n } extracted from inspecting the two representations of c (z), using the algebraic equations in [nln84]. Forward direction: {M k } { n } Set values: M () k Then evaluate. = M k, M ( 1) k. =, k = 1,,..., K; 1 =. = 1. M (n) k = M (n 1) k n 1 M (n ) k, n = M (n) n (3) n sequentially for k = n, n + 1,..., K and n = 1,,..., K. Reverse direction: { n } {M k } Set values: M (k) k Then evaluate M (n 1) k. = k, M ( 1) k = n 1 M (n) k. =, k = 1,,..., K; 1 =. = 1. + n 1 n M (n ) k, M k = M () k (4) sequentially for n = k, k 1,..., 1 and k = 1,,..., K.

10 Forward direction: {M k } { n } k n M M 4 M 6 M 8 1 M M 4 M 6 M 8 M 4 M M M 6 M M 4 M 8 M M 6 3 M 6 M 4 M M 4 M 4 M M M M 8 M 6 M M 6 M 4 M M M Reverse direction: { n } {M k } n k ( 1 + ) 1 [ 3 + ( 1 + ) ] ( 1 + ) 1 [ 3 + ( 1 + ) ] ( ) 3 3 First two steps in forward directions spelled out: c (z) = z 1 M z 3 + M 4 z 5 M 6 z c 1 (z) = 1 zc (z) = M }{{} 1 z M 4 z 4 + M 6 z 6... c 1 (z) = z M 4 z 4 + M 6 z 6 M 8 z M M M ( ) M4 c (z) = c (z) zc 1 (z) = M z 3 M } {{} ( ) M6 M 4 z M

11 Link to Generalized Langevin Equation [nln86] Define two functions from the c k (z) introduced in [nln84]: Σ(z). = 1 c 1 (z) c (z), b k(z). = c k(z) c (z). (1) Rewrite algebraic equations () for k = of [nln84] using Σ(z) and b k (z): zc (z) + Σ(z)c (z) = 1, (a) zc k (z) + Σ(z)c k (z) = b k (z), k = 1,,... (b) Inverse Laplace transforms of these functions then satisfy Ċ (t) + Ċ k (t) + t t dt Σ(t t )C (t ) =, dt Σ(t t )C k (t ) = B k (t), k = 1,,... (3a) (3b) Recall orthogonal expansion (1) in [nln8] of dynamical variable: A(t) = C k (t) f k. (4) k= From (3) and (4) follows generalized Langevin equation [M. H. Lee 1983]: 1 A(t) + Orthogonal exapnsion of random force: dt Σ(t t )A(t ) = F (t). (5) F (t) = B k (t) f k, B k () = δ k,1. (6) k=1 Absence of correlations between dynamical variable and random force: F (t) A() = B k (t) f k f =. (7) k=1 Fluctuation-dissipation relation between Σ(t) (memory function) and F (t): F (t) F () = B 1 (t) f 1 f 1 = 1 B 1 (t) f f = Σ(t) f f. (8) 1 Lower integration boundary is specific to initial-value problem under consideration.

12 Orthogonal Expansion of Wave Functions [nln9] ψ(t) = D k (t) f k. (1) k= Step #1: Hamiltonian: H. = H E (generator of new directions). Ground state: φ. Dynamical variable of interest: A. Orthogonal basis, f, f 1,..., with initial condition, f = A φ. Recurrence relations for basis vectors: f k+1 = H f k a k f k b k f k 1, k =, 1,,... (a) a k = f k H f k f k f k, b k = f k f k f k 1 f k 1. (b) Conditions: f 1. =, f. = A φ, b. =. First three iterations spelled out in [nln91]. Step #: (setting = 1) Time-dependent coefficients of basis vectors: D k (t). Substitute (1) into eq. of motion from [nln81]: ı d ψ(t) = H ψ(t). dt d/dt acts on D k (t) and H on f k. Comparison of coefficients in ı Ḋ k (t) f k = D k (t) [ ] f k+1 + a k f k + b k f k 1 }{{} k= k= H f k yields set of coupled, linear, first-order ODEs for functions D k (t): ıḋk(t) = D k 1 (t) + a k D k (t) + b kd k+1 (t), k =, 1,,... (4) Conditions: D 1 (t), D k () = δ k,. Normalized correlation function: D (t) = f ψ(t) f f = φ A()A( t) φ φ A()A() φ = S(t) S() (3). = S (t). (5)

13 Gram-Schmidt Orthogonalization II [nln91] First three iterations in step #1 of [nln9]. Initial condition: f = A φ. First iteration: f 1 = H f a f Second iteration: f f 1 = f H f a f f = f = H f 1 a 1 f 1 b 1 f Third iteration: if a = f H f f f. f f = f H f 1 a }{{} 1 f f 1 b }{{} 1 f f = f 1 f 1 if b 1 = f 1 f 1 f f, f 1 f = f 1 H f 1 a 1 f 1 f 1 b 1 f 1 f = }{{} f 3 = H f a f 1 b f c f f f 3 = = if c =, f 1 f 3 = = if b = f f f 1 f 1, if a 1 = f 1 H f 1 f 1 f 1. f f 3 = = if a = f H f f f.

14 Structure Function [nln9] Laplace transform (with ıζ = z): d k (ζ). = dt e ıζt D k (t). (1) Coupled ODEs for D k (t) become coupled algebraic equations for d k (ζ): (ζ a k )d k (ζ) ıδ k, = d k 1 (ζ) + b k+1d k+1 (ζ), k =, 1,,... () Condition: d 1 (ζ). Recursive construction of continued fraction representation for d (ζ): k = : (ζ a )d (ζ) ı = b ı 1d 1 (ζ) d (ζ) =, ζ a b d 1 (ζ) 1 d (ζ) k = 1 : (ζ a 1 )d 1 (ζ) = d (ζ) + b d (ζ) d 1(ζ) d (ζ) = 1, ζ a 1 b d (ζ) d 1 (ζ) d (ζ) = ı b 1 ζ a b ζ a 1 ζ a (3) Structure function S(ω) from d (ζ): combine Fourier transform with inverse Laplace transform. S (ω). = + D (t) = 1 π C dt e ıωt D (t), dζ e ıζt d (ζ) S (ω) = lim ɛ Re[d (ω + ıɛ)]. (4)

15 Moment Expansion vs Continued Fraction II [nln96] Moment expansion of correlation function (see [nln78]): D (t) = n= ( ıt) n M n, M k as in [nln85]. (1) n! Asymptotic expansion and continued-fraction representation: d (ζ). = dt e ıζt D (t) = ı n= M n ζ (n+1) = ı b 1 ζ a ζ a 1 () Transformation relations between {M n } and {a n, b n} extracted from inspection of the two representation in (). Conversion {a n, b n} { k } in two steps. First step: {a n, b n} {M n } here. Second step: {M k } { k } in [nln85]. Calculating {a n, b n} first is most practical in many applications. Key features of dynamical quantities (bandwidth, gap, singularity structure) are most effectively extracted from { k }. Forward direction: {M n } {a n, b n} Initialize auxiliary quantities: M () k = ( 1) k M k, L () k = ( 1) k+1 M k+1, k =,..., K. (3) Evaluate sequentially for k = n,..., K n+1 (in two successive inner loops) and n = 1,..., K (outer loop): M (n) k = L (n 1) k L (n 1) n 1 M (n 1) k M (n 1) n 1, L (n) k = M (n) k+1 M (n) n M (n 1) k M (n 1) n 1 Identify continued-fraction coefficients among auxiliary quantities:. (4) b n = M (n) n, a n = L (n) n, n =,..., K. (5)

16 Reverse direction: {a n, b n} {M n } Initialize auxiliary quantities, setting b = b 1. = 1: M (n) n = b n, L (n) n = a n, n =,..., K; (6a) M ( 1) k =, k =,..., K + 1. (6b) Evaluate sequentially for n =,..., min(k, K j) (inner loop) and j =,..., K + 1 (outer loop): M (n) n+j+1 = b nl (n) n+j + b n b n 1 M (n 1) n+j, L (n) n+j+1 = M (n+1) Identify moments among auxiliary quantities: n+j+1 a n b n M (n) n+j+1. (7) M n = ( 1) n M () n, n =,..., K + 1. (8) Results of a few iterations in the forward sequence (left) and in the reverse sequence (right): a = M 1 M 1 = a b 1 = M M 1 M = a + b 1 a 1 = M 3 M1 3 M M M1 1 M 3 = (a 3 + a b 1 + b 1a 1 ) b = M 4 M 1 M 3 ] M M M1 M 4 = b1[ a + a 1 + a a 1 + b 1 + b [ ] [ ] M 3 1 M 3 M 3 M 1 M 3 [ ] M M1 1 + M M M1 1 +a a 3 + a b 1 + b 1a 1

17 Genetic Code of Spectral Densities [nln98] Justification of biological term used for analogy: Spectral densities, structure functions, dissipation functions, and Green s functions for any given classical or quantum many-body system and any choice of dynamical variable are related to each other by rigorous relations (see [nln39] and nln88]). The (symmetric) spectral density is fully characterized by a k -sequence of continued-fraction coefficients [nln84]. The recursion method presents a user-friendly and systematic way to calculate coefficients k sequentially, either directly (Liouvillian representation [nln83]) or indirectly (Hamiltonian representation [nln91]). Hence the k -sequence is a genetic code of sorts: (i) it is a code of retrievable information about key features of spectral densities as listed below; (ii) it is generative in nature in the sense that it can be used to produce spectral densities with these very features in conjunction with specific termination schems of continued fractions. Features of spectral densities that can be identified in k -sequences (incomplete list): Position and intensity of individual spectral lines [nln99]. Bandwidth of spectral densities with compact support [nln1]. Band-edge singularity of spectral densities with compact support [nln1]. Infrared singularity of spectral density with compact support [nln1]. Bandwidth and gap size of spectral densities with bounded support [nln11]. Infrared singularity of spectral densities with unbounded support [nln1]. Large-ω asymptotics of spectral densities with unbounded support [nln1]. Gap size of spectral densites with unbounded support [nln13].

18 Spectral Lines from Finite k -Sequences [nln99] Orthogonal expansion in [nln83] comes natural stop: f K+1 =. Relaxation function has K + 1 poles on imaginary z-axis. c (z) = z + 1 z + z K z = p K(z) q K+1 (z). Spectral density: Φ (ω) = π L [ u l δ(ω ωl ) + δ(ω + ω l ) ]. l=1 K: number of nonzero continued-fraction coefficients k. L: number of spectral lines (with frequencies ω l ). { L 1 (odd) if all ωl >, Relation: K = L (even) if ω l = occurs. Special case u 1 = u for L = : 1 = 1 (ω 1 + ω), = 1 3, 3 = ω 1ω. ω1 + ω Limit ω 1 implies 3 and 1.

19 Spectral Densities with Bounded Support [nln1] Consider spectral densities with convergent k -sequences. Bandwidth: If lim k k = 1 4 ω then Φ (ω) has compact support on the interval ω ω. Band edge singularity: Model spectral density with singularities only at the band edges: 1 Φ (ω) = πω β+1 ( ω B(1/, 1 + β) ω ) β, ω < ω, β > 1. (1) Associated k -sequence [Magnus 1985] and its asymptotic expansion: ω k = k(k + β) (k + β 1)(k + β + 1) = 1 4 ω [1 + 1 ] 4β +. () 4k Graphical representations for two cases [Viswanath and Müller 1994]: Analysis of limiting cases in [nex69]: β = 1 : 1 = =... = 1 4 ω Φ (ω) = 4 ω ω ω. β = 1 : 1 = 1 ω, = 3... = 1 4 ω Φ (ω) = ω ω. 1 B(x, y). = Γ(x)Γ(y)/Γ(x + y).

20 Infrared singularity: Model spectral density with infrared singularity added: Φ (ω) = πω (α+β+1) B ( (1 + α)/, 1 + β ) ω α (ω ω ) β, ω < ω, α, β > 1. (3) Associated k -sequence [Magnus 1985]: k = 4ω k(k + β) (4k + β + α 1)(4k + β + α + 1), k+1 = ω (k + α + 1)(k + β + α + 1) (4k + β + α + 1)(4k + β + α + 3). (4) Asymptotic expansion: k = 1 [ ω 1 ( 1) k α k + 1 ] 4β + ( 1) k α(β + α) +. (5) 8k Graphical representations for two cases [Viswanath and Müller 1994]: Signature of divergent infrared singularity (α < ): the k+1 converge from below and the k from above toward the same limit.

21 Bandwith and Gap in Spectral Density [nln11] Consider a model k -sequence that is periodic with period two: k 1 = o, k = e, k = 1,,... (1) Relaxation function with this genetic code: c (z) = 1 o z + z + e c (z) [ = 1 ( o + e ) + z + ( o e ) e z z o e z ]. () Spectral density has bounded support and gap: Φ (ω) = 1 ( o + e ) ω ( o e ) θ ( ) ( ω ω e ω min θ ωmax ω ) + π ] [ o e ( o e ) δ(ω), (3) e ω min = o e, ωmax = o + e. Graphical representation for two cases [Viswanath and Müller 1994]: o > e : o < e : continuum only (solid lines). continuum plus central δ-peak (dashed lines).

22 Spectral Densities with Unbounded Support [nln1] Spectral densities with unbounded support are encoded by k -sequences that grow to infinity as k. The growth law of the k -sequence determines the high-frequency decay law of the spectral density [Magnus 1985]: k k λ Φ (ω) exp ( ω /λ). (1) Model spectral density with Gaussian decay and infrared singularity: α π Φ (ω) = ω Γ ( (α + 1)/ ) ω e (ω/ω ). () Associated k -sequence has linear growth law. The intercept of the k 1 is governed by the exponent of the infrared singularity. ω k 1 = 1 ω (k 1 + α), k = 1 ω (k). (3) Graphical representations for two cases [Viswanath and Müller 1994]: Model spectral density with k -sequence of different growth laws: Φ (ω) = π/(λω α ) Γ ( λ(α + 1)/ ) ω e ω/ω /λ, M k = ω Γ ( λ(1 + α + k)/ ) Γ ( λ(1 + α)/ ), ω where the k must be determined numerically from the moment M k as described in [nln85].

23 Unbounded Support and Gap [nln13] The presence of a gap in spectral densities with unbounded support is encoded in sequences of k 1 and k that have the same growth-law exponent λ but grow with different (asymptotic) amplitudes. Model spectral density with Gaussian decay and a gap: Φ (ω) = πaδ(ω) + π ω (1 A)θ ( ω Ω ) e ( ω Ω) /ω. (1) Frequency moments: k ( k M k = π(1 A) m m= + k 1 π(1 A) m= ( k m + 1 ) ω m Ω (k m) m (m 1)!! ) Ω (k m) 1 ω m+1 m!, k = 1,,... () with the k to be determined from the M k as described in [nln85]. Graphical representations for two cases [Viswanath and Müller 1994]: In the cases shown the asymptotics set in early. A = : k+1 grow more steeply: spectral density consists of continuum split by gap alone. A = 1 : k+1 grow less steeply: spectral density consists of continuum split by gap and a central spectral line.