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The Journal of Chemical Physics

On the Decay of Initial Correlations in Stochastic Processes

Irwin Oppenheim, Kurt E. Shuler, George H. Weiss

Published May 15, 1967• 16 citations
Publisher

Abstract

We study the relaxation of a noninteracting, initially correlated many-particle system in contact with an infinite reservoir. We use the master equation to study the time development of the r-particle distribution function Pr(n;t) and assume that the relaxation process is Markovian. We study the decay of the correlations by investigating the time development of the r-particle Ursell functions, Ur(n;t). We show that the correlation function Ur(n;t) goes to zero much more rapidly with time than the r-particle distribution function approaches its equilibrium value Pr(n; ∞)=Π lim i=1rP(ni; ∞).The exact forms of the relaxation of Pr(n;t) and Ur(n;t) depend upon the eigenvalue spectrum of the transition rate matrix of the master equation. The general theory is developed and then applied to a number of examples.

DOI
10.1063/1.1840492

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We study the relaxation of a noninteracting, initially correlated many-particle system in contact with an infinite reservoir. We use the master equation to study the time development of the r-particle distribution function Pr(n;t) and assume that the relaxation process is Markovian. We study the decay of the correlations by investigating the time development of the r-particle Ursell functions, Ur(n;t). We show that the correlation function Ur(n;t) goes to zero much more rapidly with time than the r-particle distribution function approaches its equilibrium value Pr(n; ∞)=Π lim i=1rP(ni; ∞).The exact forms of the relaxation of Pr(n;t) and Ur(n;t) depend upon the eigenvalue spectrum of the transition rate matrix of the master equation. The general theory is developed and then applied to a number of examples.
We have studied the relaxation of the n-particle momentum distribution function fn(pn; t) and the corresponding n-particle Ursell function Un(pn; t) for both an isolated infinite chain of coupled harmonic oscillators and an infinite chain of coupled harmonic oscillators in contact with a heat bath. The time development of the Ursell function gives us the desired information on the decay of initial momentum correlations. For both of these systems we have found the asymptotic relaxation laws fn(pn; t) − fn(0)(pn) ∼ f1(p; t) − f1(0)(p) and Un(pn; t)→0 as [f1(p; t) − f1(0)(p)]n, where the superscript zero denotes the equilibrium distribution. The momentum correlation function Un(pn; t) thus goes to zero (no correlations) much more rapidly than the n-particle distribution function approaches its equilibrium value. These results are identical with those obtained by us previously for initially correlated, noninteracting many-particle systems in contact with a heat bath and by Glauber for the relaxation of spins in an Ising lattice. It is interesting to speculate on the reason for this identity of results in such apparently different relaxation models, but no answer can be provided as yet.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. Weapproach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non- Markovian characteristics are demonstrated.
Interaction of a small system S with a large reservoir R amounts to thermal relaxation of the reduced system density operator ρS(t). The presence of the reservoir enters the equation of motion for ρS(t) through the reservoir correlation functions fkl(τ) (defined in the text), which decay to zero for τ→∞ on a time scale τc. Commonly, this τc is much smaller than the inverse relaxation constants for the time evolution of ρS(t). Then a series of approximations can be made, which lead to a Markovian equation of motion for ρS(t). In this paper the assumption of a small reservoir correlation time is removed. The equation of motion for ρS(t) is solved, and it appears that the memory effect, due to τc≂/0, can be incorporated in a frequency dependence of the relaxation operator Γ̃(ω). Subsequently, (unequal-time) quantum correlation functions of two system operators are considered, where explicit expressions for (the Laplace transform of) the correlation functions are obtained. They involve again the relaxation operator Γ̃(ω), which accounts for the time regression. Additionally it is found that an initial-correlation operator Υ̃(ω) arises, as a consequence of the fact that the equal-time correlation functions do not factorize as ρS(t) times the reservoir density operator. It is pointed out that the frequency dependence of Γ̃(ω) and the occurrence of a nonzero T̃(ω) both arise as a result of τc≂/0, and should therefore be treated on an equal footing. Explicit evaluation of Γ̃(ω) and Υ̃(ω) shows that their matrix elements can be expressed entirely in f̃kl(ω), just as in the Markov approximation. Hence no essential complications appear if one should go beyond the limits of a small reservoir correlation time τc.
We consider a classical point particle bilinearly coupled to a harmonic bath. Assuming that the evolution of the particle is monitored on a timescale which is longer than the characteristic bath correlation time, we derive the Markovian master equation for the probability density of the particle. The relaxation operator of this master equation is evaluated analytically, without invoking the perturbation theory and the approximation of weak system-bath coupling. When the bath correlation time tends to zero, the Fokker-Planck equation is recovered. For a finite bath correlation time, the relaxation operator contains contributions of all orders in the system-bath coupling.
We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn–Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and the jump operators. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.