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

Abstract

We consider the motion of a test particle in a one-dimensional system of equal-mass point particles. The test particle plays the role of a microscopic "piston" that separates two hard-point gases with different concentrations and arbitrary initial velocity distributions. In the homogeneous case when the gases on either side of the piston are in the same macroscopic state, we compute and analyze the stationary velocity autocorrelation function C(t).Explicit expressions are obtained for certain typical velocity distributions, serving to elucidate in particular the asymptotic behavior of C(t). It is shown that the occurrence of a non-vanishing probability mass at zero velocity is necessary for the occurrence of a long-time tail in C(t). The conditions under which this is a t −3 tail are determined. Turning to the inhomogeneous system with different macroscopic states on either side of the piston, we determine * Permanent address: Department of Physics, Indian Institute of Technology-Madras, Chennai 600 036, India 1 its effective diffusion coefficient from the asymptotic behavior of the variance of its position, as well as the leading behavior of the other moments about the mean. Finally, we present an interpretation of the effective noise arising from the dynamics of the two gases, and thence that of the stochastic process to which the position of any particle in the system reduces in the thermodynamic limit.51.10+y, 02.50-r, 64.60.cn

Universal extremal statistics in a freely expanding Jepsen gas

“…The fact that n 2 c (t) − n c (t) 2 = n c (t) indicates, contrary to previous claims [33], that the collision process in the thermodynamic limit of the Jepsen gas is not Poissonian.…”

“…[24,25,26,27,28,29,30]. It has also proved very useful in the study of a class of outof-equilibrium problems like the evolution of the "adiabatic piston" [31,32,33,34], the Jarzynksi theorem [35], and a quasispecies biological evolution model [19,20,21]. Moreover, the Jepsen gas also turns out to have important applications in spin transport processes in the onedimensional nonlinear σ-model [36,37].…”

Analytic calculation of energy transfer and heat flux in a one-dimensional system

“…It is helpful to recapitulate in brief the relevant features of the model, in the notation used in earlier work [6,7]. The tagged particle, located at X = 0 at t = 0 with an initial velocity Among other quantities, the exact one-particle distribution function of the tagged particle, P (X, V, t|0, V 0 ), can be found.…”

“…The tagged particle, located at X = 0 at t = 0 with an initial velocity Among other quantities, the exact one-particle distribution function of the tagged particle, P (X, V, t|0, V 0 ), can be found. The complicated stochastic motion of the tagged particle (henceforth termed the "piston") can be interpreted [7] as being driven by two independent…”

“…In particular, it can be shown that the motion of this tagged particle becomes diffusive asymptotically, i.e., converges to Brownian motion. Recently, the model has been revisited and studied in greater detail [6,7], one of the motivating factors being its relationship to the adiabatic piston problem. The non-equilibrium situation implied by different velocity distributions for the gases to the left and right of the tagged particle has been analyzed.…”