Chap 5. Dynamics in Condensed Phases ● Generalized Langevin Equation Phenomenological introduction
Г( t ) : friction kernel ～ friction depends on the past = memory effect : delayed response of the surrounding media) R(t) : random force Later, GLE will be derived from a model Hamiltonian (and thus, GLE may be time reversible) ● Coarse graining If : Γ(t) ~ 27δ(t)     (no delay) ⇒ mv = −γmv(t) + R(t) (Langevin eq) Similarly, if we look at the dynamics in (macroscopic) time scale ∆ t much larger than the (microscopic) decay time of Γ( t ) , (i.e., “coarse-graining” in time) mÿ(t) = −Γmv(t) + R(t)     ( Γ ≡ ƒ∀ť Γ(τ)dτ ~ ƒ∞ Γ(τ)dτ ) In this way, the time reversibility of the (classical mechanical) dynamics is lost by the coarse-graining of time scale. (But, GLE may be time reversible)
● Laplace transform
[math preparation] L{f(t)} = fo e-st f(t)dt = f(s) where (s > 0 ) Definition : L{f(t)} = sf(s) − f(0) , L{f(t)} = s2f(s) − sf(0) − f(0) Derivatives : Convolution :     L{f(t)} · L{g(t)} = L{f(t = t)g(t)dt} proof :  right hand side =  f ∞ dte-st  f ∞ f (t − τ)g(τ)dτ variable transformation (t, τ) → (τ, τ′ ≡ t − τ) (Jacobian = 1) = ſ ∞ d τ ſ ∞ d τ ′ e − s (τ+τ′) f (τ′ ) g (τ) = { f ∞ e-s τ ′ f(τ ′ )dτ ′ }{ f ∞ e-s τ g(τ)dτ} = L{f}C{g} Useful stuffs :
●  L{cosωt} = L{(eiωt + e-iωt)/2} = 2[3 - iω + ___] S s² + ω² ● L{cosωt} =  s² + ω² Microscopic model for GLE ● System + Harmonic bath
Classical eqs of motion :
1. (formally) solve the 2nd EOM for x i 2. enter back to the 1st EOM for s
ñ¿(λ) =  ∑² + ωž x i (0) +  ∑² + ωž x i (0) − c i ∑² + ωž š(λ) Back transformation x i (t) = x i (0) cos ω i t +  i (0) sin ω i t −  c i  f t sin ω i (t − τ)s(τ)dτ Partial integration of the last integral f o sin ω i (t − τ)s(τ)dτ = [ ζ i cos ω i (t − τ)s(τ)] o − ζ i f o cos ω i (t − τ)s(τ)dτ Enter into EOM for s š = − ∂V(s) − ∑ i (ニッ)² { f ζ cos ω i (t − τ) š (τ) dτ − s(t) + s(0) cos ω i t } + R(t) R(t) ≡ − − ), c i x i (0) cos ω i t − − − − − − x i (0) sin ω i tくすううDefine friction kernel : Γ( t ) ≡ ∑ , (ﾆｼﾞ)² cos ω i t GLE form : ↑
ie, frequency shift (potential softening) due to friction Fluctuation-dissipation theorem (R(0)R(t)} = kBT Γ(t) TCF of random force = friction kernel × temperature (Both stems from the medium motion) For the harmonic bath system, ●  {x i (0)x j (0)} = 0 for (i ≠ j) : bath modes are independent ● (x i (0) x i (0) = 0 position and velocity are (locally) independent ( − x i (0) 2 ) = k B T : equipartition theorem Thus,
i k B T
Matrix partitioning method Multidimensional potential : V(x) Expand around the minimum x 0 : (ie, ( ∂Y )x=x 0
Off-diagonal (Ω²) ij = coupling between x i and x j (Note : diagonalization of Ω² ⇒ normal mode analysis) Suppose : ●  we are only interested in small number of x i ( i = 1, 2, . . . , n ) out of total N degrees of freedom. (n < N) ●  Now we denote the rest of x i (i = n + 1 , …, N ) by y i ● Matrix partitioning
Similarly as before, (1) formally solve for y , (2) enter back into eq for x [s²¥(s) — sy(0) + ¥(0) = — Ω² x ¥(s) — Ω² y ¥(s) ỹ(s) = (s²1 + Ω²y)−¹(sy(0) − ÿ(0) − Ω²yxx(s) y(t) = cos Ωyyt · y(0) — Ωyy sin Ωyyt · ÿ(0) — Ωyy fo sin Ωyy(t — τ) · Ωyxx(τ)dτ Partial integration, and define random force and friction kernel R(t) ≡ Ω žy cos Ωyyt · y(0) — Ω žyΩ−1 sin Ωyyt · ÿ(0) Г(t) ≡ Ω² yΩ² cos Nyyt · Ω² GLE form : rt ↑ ÿ = −Ωঙƒƒ×(t) −   |   Γ(t − τ)ż(τ)dτ − Γ(t)x(0) + R(t) 0 J (Verify Fluctuation-dissipation theorem : 〈R(0)R(t)〉＝ k BT Γ(t) ) ● Set up models for R(t) ⇒ Stochastic trajectory methods Projection operator methods (1) The division into x and y in the previous section is also obtained by applying projection operator matrices
[We may try to define more general projection matrices to extract physical] variables of specific interests. Projection operator methods (2)
Time-dependent Schrodinger eq : ψ = − ψ Âψ = − ψ Â( ˆ P + ˆ Q)ψ
⇒ ψQ(t) = e-iHQQt/hψQ(0) - ñ ∫ t e-iHQQ(t-τ)/ñ HQpψp(τ)dτ Usually, we assume that the initial wavefunction ψ(0) is in the target space, in other words, ψ (0) = Qψ (0) = 0 Then the eq for ψρ becomes /2 HpQe-iHQQ(t-t)/k HQpψp(t)dt 0 J ●  1st term = evolution due to Hpp ●  2nd term = transition from and to Q-space (Note : Green's function representation ⇒ damping theory ) We can also carry out similar projection for the Liouville eq (cf Chap 7) ∂t ρ = −i λρ (Liouville operator ÊA ≡ њ[H, A] ) This leads to the Master equation formalism of the density matrix ρ