Simplest piston problem. I. Elastic collisions

Abstract

We study the dynamics of three elastic particles in a finite interval where two light particles are separated by a heavy "piston". The piston undergoes surprisingly complex motion that is oscillatory at short time scales but seemingly chaotic at longer scales. The piston also makes long-duration excursions close to the ends of the interval that stem from the breakdown of energy equipartition. Many of these dynamical features can be understood by mapping the motion of three particles on the line onto the trajectory of an elastic billiard in a highly skewed tetrahedral region. We exploit this picture to construct a qualitative random walk argument that predicts a power-law tail, with exponent −3/2, for the distribution of time intervals between successive piston crossings of the interval midpoint. These predictions are verified by numerical simulations.

Simulations of Transport in Hard Particle Systems

“…The scale of these simulations is so large that we had to restrict the region of parameter space explored. In particular, we perform simulations for these large-N values, global densities η = 0.5, 1, 2, 3, a strong temperature gradient defined by T 0 = 20 and T L = 1, and two intermediate mass ratios μ = 3 and μ = 2.2 for which relaxation (and correlation) time scales are slightly shorter (note that for both small and large μ the fluid's relaxation and correlation times increase drastically [158,159]). Figure 29b shows the collapse of density profiles for μ = 2.2 and μ = 3 obtained by using the measured anomaly exponent α(μ) in each case, namely α(μ = 2.2) = 0.308 and α(μ = 3) = 0.297, see Table 1, once the new data for N = 31623 and N = 10 5 +1 have been added.…”

Mesoscopic description of the adiabatic piston: kinetic equations and $\mathcal H$-theorem

“…Specifically, when there is no energy source supplying the dissipation of the inelastic collisions, each of the gases reaches a state close to the homogeneous cooling state [44], and the "thermal" equilibration occurs when the cooling rates (or rates of energy loss) of both gases and the piston are the same. The new "equilibrium" criterion, as opposite of the temperatures being the same for the elastic case, explains the emergence of non-equilibrium phase transitions [45,46,47]. See also [43,48,49,50,50,51] for theoretical and numerical studies relevant for experimental situations.…”

“…This has been done rigorously [29,30] and approximately [31,32,33,34,35,36] for the case of non-interacting and similar models. For more realistic ones, such as models with hard spheres or disks, most of the existing results are numerical [37,38], or use rude approximations [39]. As in the case of non-interacting particles, the numerical simulations show an exponential relaxation of the system towards thermal equilibrium [40,41].…”

Critical behavior of two freely evolving granular gases separated by an adiabatic piston

“…Nevertheless, when fluctuations are taken into account, it follows that the piston moves until the system relaxes to mechanical and thermal equilibrium with equal pressures and temperatures in both compartments. In the last years, the problem has attracted a lot of attention [2][3][4], mainly stimulated by the seminal paper by Lieb [5] and the suitability of the model to investigate fundamental issues in mesoscopic systems. An illuminating review of the adiabatic piston is given in ref.…”