Chap 5. Dynamics in Condensed Phases

Transcription

1 PG/KA/Chap 5-1 Chap 5. Dynamcs n Condensed Phases Generalzed Langevn Equaton Phenomenologcal ntroducton m v = m Γ(t τ)v(τ)dτ + R(t) Γ(t) : frcton kernel frcton depends on the past ( = memory effect : delayed response of the surroundng meda) R(t) : random force Later, GLE wll be derved from a model Hamltonan (and thus, GLE may be tme reversble)

2 PG/KA/Chap 5-2 Coarse granng If : Γ(t) 2 γδ(t) (no delay) m v = γmv(t) + R(t) (Langevn eq) Smlarly, f we look at the dynamcs n (macroscopc) tme scale t much larger than the (mcroscopc) decay tme of Γ(t), (.e., coarse-granng n tme) m v(t) = Γmv(t) + R(t) ( Γ t Γ(τ)dτ Γ(τ)dτ ) In ths way, the tme reversblty of the (classcal mechancal) dynamcs s lost by the coarse-granng of tme scale. (But, GLE may be tme reversble)

3 PG/KA/Chap 5-3 Laplace transform [math preparaton Defnton : Dervatves : Convoluton : Useful stuffs : L{e ωt } = L{f(t)} = e st f(t)dt = f(s) where (s > ) L{ f(t)} = s f(s) f(), L{ f(t)} = s 2 f(s) sf() f() L{f(t)} L{g(t)} = L{ f(t τ)g(τ)dτ} proof : rght hand sde = dte st t f(t τ)g(τ)dτ varable transformaton (t, τ) (τ, τ t τ) (Jacoban = 1) = = { dτ dτ e s(τ+τ ) f(τ )g(τ) e sτ f(τ )dτ }{ e s(s ω)t dt = 1 s ω e sτ g(τ)dτ} = L{f}L{g} L{cos ωt} = L{(e ωt + e ωt )/2} = 1 2 [ 1 s ω + 1 s + ω = L{cos ωt} = ω s 2 + ω 2 s s 2 + ω 2

4 PG/KA/Chap 5-4 Mcroscopc model for GLE System + Harmonc bath H = p2 s 2 + V (s) + ( p2 2 + ω2 2 x2 ) + c x s Classcal eqs of moton : System-bath couplng : c = ( V (s, x) s x )pot mn ṡ = H p s, ṗ s = H s s = V (s) s, ẋ = H p c x,, ṗ = H x ẍ = ω 2 x c s 1. (formally) solve the 2nd EOM for x 2. enter back to the 1st EOM for s

5 PG/KA/Chap 5-5 Laplace transform : λ 2 x (λ) λx () ẋ () = ω 2 x (λ) c s(λ) λ 1 1 x (λ) = λ 2 + ω 2 x () + λ 2 + ω 2 ẋ () c λ 2 + ω 2 s(λ) Back transformaton ẋ () x (t) = x () cos ω t + ω sn ω t c t ω sn ω (t τ)s(τ)dτ Partal ntegraton of the last ntegral sn ω (t τ)s(τ)dτ = [ 1 ω cos ω (t τ)s(τ) t 1 t ω cos ω (t τ)ṡ(τ)dτ Enter nto EOM for s { V (s) s = s ( c ω ) 2 t } cos ω (t τ)ṡ(τ)dτ s(t) + s() cos ω t + R(t) c ω ẋ () sn ω t R(t) c x () cos ω t Defne frcton kernel : Γ(t) ( c ω ) 2 cos ω t GLE form : s = V (s) s + Γ()s(t) Γ(t τ)ṡ(τ)dτ s()γ(t) + R(t) For harmonc V (s) = Ω2 2 s2 V (s) : s + Γ()s(t) (Ω 2 Γ() }{{} )s(t) e, frequency shft (potental softenng) due to frcton Ω 2 eff

6 PG/KA/Chap 5-6 Fluctuaton-dsspaton theorem R()R(t) = k B T Γ(t) TCF of random force = frcton kernel temperature For the harmonc bath system, (Both stems from the medum moton) x ()x j () = for ( j) : bath modes are ndependent x ()ẋ () = poston and velocty are (locally) ndependent ω2 2 x () 2 = k B T 2 : equpartton theorem Thus, R()R(t) = c 2 x () 2 cos ω t = k B T ( c ω ) 2 cos ω t = k B T Γ(t)

7 PG/KA/Chap 5-7 Matrx parttonng method Multdmensonal potental : V (x) Expand around the mnmum x : (e, ( V x V (x) = V (x ) Ω2 (x x ) 2 + ) x=x = ) [ Ω 2 ( 2 V x 2 )x Off-dagonal (Ω 2 ) j = couplng between x and x j (Note : dagonalzaton of Ω 2 normal mode analyss) Suppose : we are only nterested n small number of x ( = 1, 2,, n ) out of total N degrees of freedom. (n < N) Now we denote the rest of x ( = n + 1,, N ) by y

8 PG/KA/Chap 5-8 Matrx parttonng d 2 dt 2 [ x y = [ Ω 2 xx Ω 2 yx Ω 2 xy Ω 2 yy [ x y ẍ = Ω2 xx x Ω2 xy y ÿ = Ω 2 yxx Ω 2 yyy Smlarly as before, (1) formally solve for y, (2) enter back nto eq for ẍ s 2 2 ỹ(s) sy() + ẏ() = Ωyx x(s) Ω2yyỹ(s) ỹ(s) = (s Ω 2 yy ) 1 (sy() ẏ() Ω 2 yx x(s) y(t) = cos Ω yy t y() Ω 1 yy sn Ω yyt ẏ() Ω 1 yy sn Ω yy(t τ) Ω 2 yx x(τ)dτ Partal ntegraton, and defne random force and frcton kernel R(t) Ω 2 xy cos Ω yyt y() Ω 2 xy Ω 1 yy sn Ω yyt ẏ() Γ(t) Ω 2 xy Ω 2 yy cos Ω yyt Ω 2 yx GLE form : ẍ = Ω 2 eff x(t) Γ(t τ)ẋ(τ)dτ Γ(t)x() + R(t) (Verfy Fluctuaton-dsspaton theorem : R()R(t) = k B T Γ(t) ) Set up models for R(t) Stochastc trajectory methods

9 PG/KA/Chap 5-9 Projecton operator methods (1) The dvson nto x and y n the prevous secton s also obtaned by applyng projecton operator matrces P [ 1 n and Q 1 N P = [ 1 N n (Note : P 2 = P, Q 2 = Q projecton operator) Startng from orgnal (full N dm) : ẍ = Ω 2 x = Ω 2 (P + Q)x (P ) Pẍ = (PΩ 2 P)(Px) (PΩ 2 Q)(Qx) (Q ) (Defne Px x P etc.) Qẍ = (QΩ 2 P)(Px) (QΩ 2 Q)(Qx) { ẍ P = Ω 2 P P x P Ω 2 P Qx Q ẍ Q = Ω 2 QP x P Ω 2 QQx Q [ We may try to defne more general projecton matrces to extract physcal varables of specfc nterests.

10 PG/KA/Chap 5-1 Projecton operator methods (2) Projector onto a (fnte) target space {φ } ( = 1, 2,, n) ˆP n φ φ, ˆQ 1 ˆP = φ φ =1 =n+1 Tme-dependent Schrodnger eq : ψ = ī h Ĥψ = ī h Ĥ( ˆP + ˆQ)ψ ˆP ψ = ī h {( ˆP Ĥ ˆP ) ˆP ψ + ( ˆP Ĥ ˆQ) ˆQψ} ψ ˆQ ψ = ī h {( ˆQĤ ˆP ) ˆP ψ + ( ˆQĤ ˆQ) ˆQψ} P = ī h (H P P ψ P + H P Q ψ Q ) ψ Q = ī h (H QP ψ P + H QQ ψ Q ) Formal soluton of the 2nd lne (Laplace Tr.) s ψ Q (s) ψ Q () = ī h H QP ψ P (s) ī h H QQ ψ Q (s) 1 ψ Q (s) = s+h QQ / h {ψ Q() ī h H QP ψ P (s)} ψ Q (t) = e H QQt/ h ψ Q () ī h e H QQ(t τ)/ h H QP ψ P (τ)dτ

11 PG/KA/Chap 5-11 Usually, we assume that the ntal wavefuncton ψ() s n the target space, n other words, ψ Q () = ˆQψ() = Then the eq for ψ P becomes t ψ P (t) = ī h H P P ψ P (t) + ( ī h )2 1st term = evoluton due to H P P 2nd term = transton from and to Q-space H P Q e H QQ(t τ)/ h H QP ψ P (τ)dτ (Note : Green s functon representaton dampng theory ) We can also carry out smlar projecton for the Louvlle eq (cf Chap 7) t ρ = ˆLρ (Louvlle operator ˆLA 1 h [H, A ) Ths leads to the Master equaton formalsm of the densty matrx ρ