Dynamics of a Simple Many-Body System of Hard Rods

Abstract

General formulas are given for the exact calculation of the nonequilibrium properties of the one-dimensional system of equal-mass hard rods both for a finite but large system and in the limit of infinite size. Only properties which depend upon labeling one or more of the particles are nontrivial in this system. Various results are obtained on Poincaré cycles, delocalization of a particle with time and electrical conductivity when one particle is charged.

Velocity correlations, diffusion, and stochasticity in a one-dimensional system

“…, N + ), with independent, identically distributed velocities V j drawn from normalized velocity distributions φ ± (V ) respectively. As the particles merely exchange velocities upon collision, at any instant of time the piston is on one of the "free" trajectories X j +V j t. This, together with the fact that the particles cannot move across each other, suffices to solve [2] the problem of determining, among other quantities, the phase space distribution function of the piston (or that of any of the other particles) at any time t > 0 by averaging over the initial positions and velocities of the gas particles on both sides of the piston [2], [3]. In the thermodynamic limit L → ∞ and N ± → ∞ such that one has finite densities lim N ± /L = n ± on the left and right of the piston, the conditional one-particle distribution function of the piston is found to be…”

“…One such model, consisting of identical hard-point particles moving on a line and interacting through elastic collisions, was introduced several decades ago [1]. Based on the observation that the particles merely exchange velocities in a collision, Jepsen [2] was able to calculate explicitly several properties of a gas of such particles. Subsequently, Lebowitz and co-workers [3]- [5] refined and extended these calculations to include, among other aspects, a comparison with the results of the Boltzmann approximation.…”

The motion of a large particle

“…Various authors (D. W. Jepsen [5], T. E. Harris [3], and others) have studied the above model and generalizations of it. F. Spitzer [8] has proved that if, instead of…”

Dynamics and Transport Near Quantum-Critical Points

“…The collisions of these thermally excited quasi-particles will lead to a broadening of the delta function pole in (51): the form of this broadening can be computed exactly in the limit T ≪ |∆| using a semiclassical approach similar to that employed for the ordered side [17]. The argument again employs a semiclassical path-integral approach to evaluating the correlator in (41). The key observation now is that we may consider the operator σ z to be given by…”

“…the trajectory of the −1 in Fig 11). Fortunately, precisely this correlator was considered three decades ago by Jepsen [41] and a little later by others [42]; they showed that, at sufficiently long times, this correlator has a Brownian motion form. Inserting their results into (78), we obtain the results presented below.…”