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The Journal of Chemical Physics

Decay of Correlations. II. Relaxation of Momentum Correlations and Momentum Distributions in Harmonic Oscillator Chains

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

Published May 1, 1969• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.1671610

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arXiv:1201.0578•Jan 3, 2012•1 citation
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.
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.
This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T1 coupled to an infinite copy of itself at a temperature T2 and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter γ, takes the value γ = 1. The general case for arbitrary γ values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J. Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.
A system of noninteracting classical spins coupled to the modes of a chain of one-dimensional classical harmonic oscillators via −S z Σc k x k was investigated to see whether the spin-bath coupling could relax the spins toward equilibrium. We considered two different cases for the initial conditions on the classical harmonic oscillator variables when we took the harmonic oscillator bath variable averages for the total spin components. In the first case, the harmonic oscillators were initially in equilibrium while they did not recognize the presence of the spin, whereas in the second case, the spins were in equilibrium and did recognize the presence of the spin. For the first case, the bath variable averages of the x- and the y-components of the total spin showed that the effective angular velocity transiently slowed down after an initial increase; then, it recovered its initial angular velocity continually. In the second case, the effective angular velocity was fixed. For both cases, the z-component of the total spin vector remained constant. If the x- and the y-components of the single spins were randomly distributed, we would get equilibrium values. For the z-components of single spins, unless they are at equilibrium from the beginning, they do not attain equilibrium.
Abstract We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t−13t^{-\frac{1}{3}}t−31​ t - 1 3 for position and momentum correlations and as t−23t^{-\frac{2}{3}}t−32​ t - 2 3 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t−14t^{-\frac{1}{4}}t−41​ t - 1 4 for position and momentum correlators and with rate t−12t^{-\frac{1}{2}}t−21​ t - 1 2 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly.
The approach to thermal equilibrium of a free particle and a harmonic oscillator interacting with a heat bath can be understood from a semiclassical point of view in terms of the time evolution of a quantum mechanical state function appropriate to each system. The state functions are parameterized such that expectation values comply with the predictions of the classical theory of Brownian motion, with the consequence that the Fokker-Planck equation for a free particle in velocity space, the Montroll-Shuler equation for a quantum oscillator in discrete energy space, and their solutions are arrived at by a common procedure. These relaxation processes become visualizable as the progressive development from a coherent, relatively ordered state to that of equilibrium, so that the loss of information as equilibrium is reached and the consequences of random phasing can be traced dynamically in some detail. The unity of approach is reinforced by the demonstration that since both equations represent Markoffian processes, the Montroll-Shuler equation predicts distributions in coordinate space and momentum space which satisfy a Fokker-Planck equation.
This preprint contains the English translation of the paper "Relaxation dynamics of a quantum chain of harmonic oscillators", which was published by the author in 1980 in the Ukrainian Physical Journal. Comments describing its motivations, background ideas and results are presented as well. This paper addressed to the problem of approach to the thermal equilibrium in an isolated macroscopic quantum system, which was studied on the example of the quantum chain of weakly interacting harmonic oscillators. In the initial state, macroscopic energy was supplied to one oscillator (atom) in the chain. Subsequent evolution of the quantum state of the whole chain was determined due to the model integrability. The main subject of interest was the time evolution of the reduced density operators characterizing the quantum state of a particular atom. On the short time-scale, the energy perturbation expands along the chain with the velocity of the fastest phonon mode. On the long-time scale, the single-atom density operators display strong fluctuations around the canonical Gibbs distribution. These fluctuations are caused by the degeneracy of the energy level differences (presence of resonances) in the model of coupled harmonic oscillators. After lifting this degeneracy, fluctuations are suppressed providing, that the reduced density operator of each atom in the chain becomes very close to the Gibbs distribution at almost any time moment.