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Chemical Dynamics in Condensed PhasesRelaxation, Transfer and Reactions in Condensed Molecular Systems$
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Abraham Nitzan

Print publication date: 2006

Print ISBN-13: 9780198529798

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780198529798.001.0001

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Stochastic Equations Of Motion

Stochastic Equations Of Motion

Chapter:
(p.255) 8 Stochastic Equations Of Motion
Source:
Chemical Dynamics in Condensed Phases
Author(s):

Abraham Nitzan

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198529798.003.0014

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Schrödinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underlying dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. The time evolution of stochastic processes can be described in two ways: 1. Time evolution in probability space. In this approach we seek an equation (or equations) for the time evolution of relevant probability distributions. In the most general case we deal with an infinite hierarchy of functions, P(zntn; zn−1tn−1; . . . ; z1t1) as discussed in Section 7.4.1, but simpler cases exist, for example, for Markov processes the evolution of a single function, P(z, t; z0t0), fully characterizes the stochastic dynamics. Note that the stochastic variable z stands in general for all the variables that determine the state of our system. 2. Time evolution in variable space. In this approach we seek an equation of motion that describes the evolution of the stochastic variable z(t) itself (or equations of motion for several such variables). Such equations of motions will yield stochastic trajectories z(t) that are realizations of the stochastic process under study. The stochastic nature of these equations is expressed by the fact that for any initial condition z0 at t = t0 they yield infinitely many such realizations in the same way that measurements of z(t) in the laboratory will yield different such realizations. Two routes can be taken to obtain such stochastic equations of motions, of either kind: 1. Derive such equations from first principles. In this approach, we start with the deterministic equations of motion for the entire system, and derive equations of motion for the subsystem of interest. The stochastic nature of the latter stems from the fact that the state of the complementary system, “the rest of the world,” is not known precisely, and is given only in probabilistic terms.

Keywords:   Langevin equation, Markovian limit, Ohmic bath, Wiener process, birth-and-death processes, fluctuation–dissipation theorem, generating functions, memory kernel, overdamped limit

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