Effects of long range interactions in harmonically coupled systems. I. Equilibrium fluctuations and diffusion

Abstract

Systems of harmonically coupled identical particles at thermal equilibrium provide dynamical models for studies of diffusion due to equilibrium fluctuations. The velocity autocorrelation function and mean square displacement of a particle selected from a given system are investigated for various models which have the common feature that the particle is directly coupled to L > 1 neighbors, reflecting the influence of long range interactions. Theorems are developed which indicate how the time course of diffusion is dictated by analytic properties of the vibrational frequency distribution as well as by quantum fluctuations whose presence is betrayed by the increasingly important role at progressively lower temperatures of τq = ℏ/πk T, the quantum transient time. The formalism is first applied to a system for which the long range couplings are so parametrized by a range parameter z that when z=0 the frequency distribution is identical to that for nearest neighbor coupling only (L=1), while as z approaches unity (L→∞) the frequency distribution becomes identifiable with that of Ford, Kac, and Mazur which served as the starting point for their dynamical theory of Brownian motion. Consequences of this model are: (1) when z <0.5, the classical velocity autocorrelation functions exhibit similar qualitative features to those computed for molecular diffusion in simple liquids; (2) as z approaches unity, the classical velocity autocorrelation function approaches the e-λτ Gaussian Markoffian form, and the mean square displacement in the same limit is identical to that predicted by the Langevin equation; (3) at low temperatures such that λτq>1, quantum fluctuations tend to dominate thermal fluctuations, resulting in severe departures from Gaussian Markoffian behavior. The low temperature effects are analyzed in some detail, and it is suggested that the predicted departure of the mean square displacement from its classical behavior might be displayed by a particle of macroscopic size suspended in a superfluid. Other models are developed which yield mean square displacements which depart even at high temperature from the linear dependence upon time characteristic of classical diffusion. The reasons and possible physical implications of these behaviors are discussed, together with a brief consideration of Poincaré cycles, whose neglect is implicit in any dynamical theory of irreversible processes.