Non-Markoffian Theory of Activated Rate Processes

Abstract

The Brownian motion of a general classical anharmonic oscillator is studied in the lowviscosity limit for a general non-Markoffian interaction with a heat bath. Memory effects are shown to have a profound influence on the rate of energy accumulation and relaxation.PACS numbers: 05.40. + J, 82.20.FdThe dynamics of activated rate processes plays a central role in many areas of physics and chemistry. Following Kramers, 1 most studies use as a model a particle moving in a potential well under the influence of a thermal bath and distinguish between three cases: The high-viscosity limit corresponds to a diffusive motion of the overdamped oscillator described by the Smoluchowski equation. The intermediate-viscosity case focuses on the diffusive motion near the potential barrier and yields transition-state theory as the low-viscosity limit. Finally, for very low viscosities, the dynamics of the energy accumulation by the particle becomes important and the rate approaches zero as the viscosity decreases. Obviously, the dynamics of energy accumulation is always important for nonsteady-state process-.es, e.g., when the kinetics is monitored following a temperature jump.The starting point in the Kramers model is the Langevin equation* = ~(l/M)dV(x)/Qx-yx +(1/ M)R(t) 9 where x is the coordinate of the particle of mass M moving in the potential V{x) and where y and R are the damping rate and the (assumed Gaussian) fluctuating force associated with the coupling to the thermal bath [y and R are related by the fluctuation-dissipation theorem (R(t 1 )R(t 2 j) = 2yMkT6{t 1 -t 2 ), k being the Boltzmann constant and T the temperature]. In many cases the correlation time associated with the bath is longer than the characteristic period of the particle (though still much shorter than the rate of energy exchange).In this case we use the model described by the generalized Langevin equation (1) For molecular dynamics problems we usually have (oo is the oscillator frequency) 2 y«l/r c «co.The existence of these vastly different time scales makes the problem very difficult to solve numerically though such solutions have been ob-where (R) = 0 and r <&(t 1 )R(t 2 ))=Z(t 1~t2 )MkT; £dtZ(t) = y.(2)The correlation function Z(t) is characterized by the correlation time r c . For specificity we shall refer to the simple model Z(t) = (y/T c )exp(-t/r c ).(3)

Temperature-driven irreversible generalized Langevin equation can capture the nonequilibrium dynamics of two dissipated coupled oscillators

“…It forms the basis for the description of a large number of chemical reactions using a chosen reaction coordinate coupled to a statistical bath [29][30][31][32][33][34][35][36][37]. Moreover, the bath can be colored [38][39][40][41], spacedependent [42], or even nonstationary [19,43,44]. This begs the question as to whether this framework could also be of use in describing the dynamics of chemical reactions (and more general systems) when the environments are far from equilibrium.…”

Stochastic resonance in underdamped, bistable systems

“…Appropriate modification to the rate formula in the very weak dissipation regime was obtained by Kramers [16,17], Carmeli et.al. [18] and by Buttiker et.al. [19], in which the transition rate was found to be proportional to η…”

Dynamical simulation of dipolar Janus colloids: Dynamical properties

“…This technique has been suggested as a general solution to the accelerated dynamics of CG models 14 and has been successfully applied to several CG systems. [12][13][14] In general, η can be colored noise, [15][16][17] and even include nonstationarity depending on space, [18][19][20][21][22] time, 23 or both. 24 Presently, we only consider uniform (in both space and time) Markovian memory friction kernels.…”