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The Journal of Chemical Physics

Non-Markovian theory of open systems in classical limit

ANATOLE A. NEUFELD

Published Jul 23, 2004• 4 citations
Publisher

Abstract

A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.

DOI
10.1063/1.1769353

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A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.
A quantum-classical approximation, capable of describing the evolution of open quantum systems well beyond the applicability limits of Redfield theory is suggested. The theory is based on the short lifetime of the quantum correlations between the quantum and the classical subsystem, caused by energy dispersion (per degree of freedom) of the canonical bath. The resulting quantum-classical approximation has the form of two auxiliary differential equations and fully accounts both for the arbitrary long memory of the heat reservoir and detailed balance. These equations allow direct solution in the time domain without constructing/diagonalizing Liouville space operators, and, in combination with molecular dynamics techniques to simulate bath dynamics, may be applied to quantum subsystems with a fairly large number of levels. A simple example of a two-level system, coupled to a single correlation time canonical bath, was considered to demonstrate different regimes of approaching the canonical equilibrium state.
We generalize the classical theory of Brownian motion so as to reckon with non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a free particle and a harmonic oscillator, it is shown that such non-Markovian effects account for the differentiability of the Brownian trajectories as well as the breakdown of the energy equipartition of statistical mechanics at short times in some physical situations. This non-Markovian approach is also extended to look at anomalous diffusion. Next, we bring in the dynamical-quantization method for investigating open quantum systems, which does consist in quantizing the classical Brownian motion starting directly from our non-Markovian Klein-Kramers and Smoluchowski equations, without alluding to any model Hamiltonian. Accordingly, quantizing our non-Markovian Klein-Kramers in phase space gives rise to a non-Markovian quantum master equation in configuration space, whereas quantizing our non-Markovian Smoluchowski equation in configuration space leads to a non-Markovian quantum Smoluchowski equation in phase space. In addition, it is worth noticing that non-Markovian quantum Brownian motion takes place in presence of a generic environment (e.g. a non-thermal quantum fluid). As far as the special case of a heat bath comprising of quantum harmonic oscillators is concerned, a non-Markovian Caldeira-Leggett master equation and a thermal quantum Smoluchowski equation are derived and extended to bosonic and fermionic heat baths valid for all temperatures.
A close examination of the validity of the Markovian approximation in the context of relaxation theory is presented. In particular, we examine the question of positivity of various approximations to the reduced dynamics of an open system in interaction with a heat reservoir. It is shown that the Markovian equations of motion obtained in the weak coupling limit (Redfield equations) are a consistent approximation to the actual reduced dynamics only if supplemented by a slippage in the initial conditions. This slippage captures the effects of the non-Markovian evolution that takes place in a short transient time, of the order of the relaxation time of the isolated bath.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.
Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).
We develop a generalized quantum Fokker-Planck theory in a non-Gaussian-Markovian model bath. The semiclassical bath adopted in this work is charactered by three parameters. One denotes the strength of system-bath coupling and the other two are chosen to interpolate smoothly the solvation dynamics between the long- and short-time regimes. The fluctuation-dissipation relation in this model bath is analyzed in detail. Based on this model bath, we derive two sets of coupled Fokker-Planck equations. These two equation sets are equivalent in the second order of system-bath coupling but different in the higher orders. The corresponding reduced Liouville equation in one set of the Fokker-Planck formulation is characterized by a memory relaxation kernel, while that in the other is by a local-time relaxation tensor. Each resulting set of Fokker-Planck equations involves only the reduced density operator and a series of well-characterized Hilbert-space relaxation operators. The present theory is valid for arbitrary time-dependent Hamiltonians and is applicable to the study of quantum coherence and relaxation in various dynamic systems.