Nonequilibrium Statistical Mechanics for Adiabatic Piston Problem

Abstract

We consider the dynamics of a freely movable wall of mass $M$ with one degree of freedom that separates a long tube into two regions, each of which is filled with rarefied gas particles of mass $m$. The gases are initially prepared at equal pressure but different temperatures, and we assume that the pressure and temperature of gas particles before colliding with the wall are kept constant over time in each region. We elucidate the energetics of the setup on the basis of the local detailed balance condition, and then derive the expression for the heat transferred from each gas to the wall. Furthermore, by using the condition, we obtain the linear response formula for the steady velocity of the wall and steady energy flux through the wall. Using perturbation expansion in a small parameter $\epsilon\equiv\sqrt{m/M}$, we calculate the steady velocity up to order $\epsilon$.Comment: 17 pages, 2 figure

Exact Equalities and Thermodynamic Relations for Nonequilibrium Steady States

“…Here the nonequilibrium entropy S(α) is defined by (4.4) with the newly defined F (α). 22 It may be also reasonable to consider a model in which λ [45]). Then all the results in the present section remain valid.…”

Linear and nonlinear thermodynamics of a kinetic heat engine with fast transformations

“…When force and temperature are constant, the stochastic dynamics generated by this rule satisfies the detailed balance condition with respect to the canonical probability distribution ρ(Γ) ∝ exp(−βH). Even if the particles do not directly interact, there is an indirect but relevant interaction through the piston, making N and N m/M important parameters for the dynamics [23][24][25][26]. We study the system in a range of N m/M close to ∼ 1, meaning that there is a nontrivial interplay between the gas and the piston.…”

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

“…An interesting model, that keeps the system always in this last stage, considers two semiinfinite ideal gases in each sides of the piston having the same pressure and temperature difference. Even the fact that the net force to the piston is zero, the system reaches a steady state where the mean velocity of the piston is constant, different from zero, and towards the hotter gas [19,20,21,22,23,24]. When the mass of the piston goes to infinity, its mean velocity vanishes [25,26,27].…”