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Journal of the Korean Physical Society

Noninteracting classical spins coupled to a heat bath of one-dimensional classical harmonic oscillators: Exact bath variable average

Suhk Kun Oh

Published Nov 1, 2013• 0 citations
Publisher

Abstract

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.

DOI
10.3938/jkps.63.1892

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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.
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.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.
We report extensive numerical studies of the dynamics of a classical particle in an anharmonic one-dimensional potential while it is harmonically coupled to three different many-particle systems. The studies address the comparatively simpler dynamical problem when the energy of the anharmonic oscillator is sufficiently low. The first model is one in which the many-particle system is a chain of harmonic oscillators that have nearest-neighbor harmonic interactions. The anharmonic oscillator is connected to every harmonic oscillator via harmonic springs. The second model is identical to the first except that each oscillator is subjected to an additional one-body harmonic potential. The third model is identical to the second except that the nearest-neighbor couplings between the individual harmonic oscillators are absent, i.e., the oscillators are decoupled with respect to one another. In the first model we find that the lowest frequency of the anharmonic oscillator, while possessing a lower bound, increases as $\ensuremath{\omega}\ensuremath{\propto}\sqrt{\mathrm{NK}},$ where NNN is the number of oscillators in the bath and KKK is the force constant describing the strength of interaction between the anharmonic oscillator and the interacting harmonic many-body system. In the second model we find aspects of dynamical correlations as seen in the canonical ensemble and a dominant frequency that increases as NK.\sqrt{\mathrm{NK}}.NK​. We recover the dynamical correlations of the anharmonic oscillator as obtained within the framework of a canonical ensemble [S. Sen, R. S. Sinkovits, and S. Chakravarti, Phys. Rev. Lett. 77, 4855 (1996)] in the third model for sufficiently many harmonic oscillators. We comment briefly on an alternate way to correctly model heat baths for numerical studies on the dynamics of physical systems using canonical ensembles.
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Starting from quantum Langevin equations for operators we study thermal properties of a one-dimensional harmonic chain to whose ends independent heat baths are attached. In this paper, we mainly discuss the thermal equilibrium state that the chain eventually approaches if the heat baths are at equal temperatures. In the classical limit, this state is determined by the Gibbs ensemble of the free chain, whereas in the quantal case, this is only true if the strength of coupling between chain and heat baths is made infinitely small. We find that corrections for finite coupling strength are appreciable only in boundary layers near both ends of the chain. The thickness of the boundary layers depends only on the temperature and not on the damping constant. Outside these boundary layers we find an analogy between thermal properties of the chain and a discrete random walk.
The time-dependent behavior of an isolated system consisting of many spins in a solid lattice is studied. An interaction of the form Bjkσjxσkx is assumed between any pair (j, k) of spins. The system is treated without making any approximations; it is one of the few many-body systems whose time-evolution can be treated exactly. The behavior is fully determined by the initial density matrix and the distribution of values Bjk. By considering a variety of distributions for the Bjk, and a variety of initial density matrices, one finds cases (initial random phasing and particular continuous spectra) where the system relaxes monotonically to a metastable equilibrium; in other cases (coherent initial state or nearest-neighbor coupling only) the system oscillates indefinitely without relaxation; in still other cases (initial random phasing and N → ∞) the system relaxes in an oscillatory way to the metastable equilibrium.