1549380323-Statistical Mechanics Theory and Molecular Simulation

584 Langevin and generalized Langevin equations and averaging over a canonical distribution of the initial conditions at tempera ...

Examples 585 equation into a simple algebraic equation. Taking the Laplace transform of both sides of eqn. (15.3.12), and solvin ...

586 Langevin and generalized Langevin equations ω ̃^2 =ω^2 − 2 ∑ α g^2 α μmαω^2 α , (15.3.19) so that W(q) = 1 2 μω ̃^2 q^2. (15 ...

Examples 587 where it has been assumed thats 0 ̃γ(s 0 ) is of the same order ass 1. Using the fact that s^20 =−ω ̃^2 and solving ...

588 Langevin and generalized Langevin equations vibrational and energy relaxation phenomena as an application of the GLE in grea ...

Vibrational dephasing 589 to a bath with memory, it was shown that, when the frequency of the oscillator is high compared to the ...

590 Langevin and generalized Langevin equations will decay as exp(−ζ′( ̃ω)t/μ). This time scale corresponds toT 1 and is given s ...

Molecular dynamics 591 thatCqq(t) is an oscillatory function with an exponential decay envelope. Thus, the general approximate f ...

592 Langevin and generalized Langevin equations 15.5.1 Numerical integration of the Langevin equation Because the simple Langevi ...

Molecular dynamics 593 between stochastic processes (Kuboet al., 1985). The latter is expressed in differential form as dq(t) =v ...

594 Langevin and generalized Langevin equations is simple yet elegant. We begin by integrating eqns. (15.5.5) fromttot+ ∆tto yie ...

Path sampling 595 q(t+ ∆t) =q(t) + ∆tv(t) +A(t) v(t+ ∆t) =v(t) + ∆tf(q(t)) + 1 2 ∆t^2 v(t)f′(q(t)) +σ √ ∆tξ(t)−∆tγv(t)−γA(t), (1 ...

596 Langevin and generalized Langevin equations 0 5 10 15 20 t /T -3 0 3 q(t) 0 5 10 15 20 t /T -3 0 3 q(t) -4 0 4 q -4 0 4 p -4 ...

Path sampling 597 where the displacementδxdis purely deterministic, andδxris due to the random force. If we take the random forc ...

598 Langevin and generalized Langevin equations If xn∆t∈B, accept the trial move, and reject it otherwise. If the path is rejec ...

Mori–Zwanzig theory 599 Geometrically, recall that the projection of a vectorbalong the direction of another vectorais given by ...

600 Langevin and generalized Langevin equations and orthogonal toA(0). This is done by inserting the identity operatorIinto eqn. ...

Mori–Zwanzig theory 601 (s−iL)−^1 −(s−QiL)−^1 = (s−iL)−^1 (s−QiL−s+iL) (s−QiL)−^1 = (s−iL) − 1 (I−Q)iL(s−QiL) − 1 = (s−iL)−^1 Pi ...

602 Langevin and generalized Langevin equations =F(t) + ∫t 0 dτ〈iLF(τ)A†〉〈AA†〉−^1 eiL(t−τ)A(0) =F(t) + ∫t 0 dτ〈iLF(τ)A†〉〈AA†〉−^1 ...

Mori–Zwanzig theory 603 to generate the orthogonal dynamics of exp(QiLt) (Darveet al., 2009). Taken as a phenomenological theory ...