Approach to thermal equilibrium in a system with adiabatic constraints

Abstract

The problem of prediction of the equilibrium state in an isolated composite system with an adiabatic internal wall is a delicate problem, whose solution is easily seen to be not entirely determined in the frame of elementary thermodynamics. We show how this indeterminacy can be removed by introducing a suitable kinetic model, in which the influence of the finite velocity of the wall on the change of momentum of the gas molecules impinging on it plays a relevant role. An interesting feature of the entropy behavior of the system is discussed.

Built-in reduction of statistical fluctuations of partitioning objects

“…(1)], consider the situation in which all the less probable asymmetric collisional events are neglected. This means neglecting large pressure fluctuations and corresponds to the quasiequilibrium regime conjectured in adiabatic piston models, for which the pressure in the two sections is assumed equal [8][9][10][11][12][13]. In turn, equal pressures imply that the maximum energy of the single particles in A and B is correlated to the position x of the object, because its value is now proportional to (L/2 + x) in A and to (L/2 − x) in B.…”

A simple piston problem in one dimension

“…In the thermodynamic limit M, n 1 , n 2 → ∞ while n 1 /M and n 2 /M are held fixed, Gruber et al [9] concluded that in the first stage above, the piston performs damped oscillatory motion, where the damping is strong if n 1 /M, n 2 /M > 1, and weak if n 1 /M, n 2 /M < 1. Using kinetic theory, Crosignani et al [5] had already derived similar equations describing this damped oscillatory motion for the adiabatic piston.…”

The Adiabatic Piston and the Second Law of Thermodynamics

“…The adiabatic-piston problem is a peculiar example in thermodynamics [1], which has recently been the object of renewed interest [2], [3]. We refer to the model case of an adiabatic cylinder divided into two regions A and B by a movable, frictionless, perfectly insulating piston.…”

“…We assume the piston to be held up to time t=0 by latches, so that the gases in A and B are initially characterized by well-defined equilibrium states, respectively corresponding to temperatures and volumes TA=Ti(0) Once the latches are released, the piston starts moving and its dynamical evolution is described by means of a suitable kinetic model (see Eqs. (6), (7), (8) of [2]). In particular, the solution of this set of coupled equations allows one to predict the final position reached by the piston for large values of time, as well as the final temperatures and the common value of the final pressure, quantities which are not entirely determinate in the frame of elementary thermodynamics [1].…”

“…In particular, the solution of this set of coupled equations allows one to predict the final position reached by the piston for large values of time, as well as the final temperatures and the common value of the final pressure, quantities which are not entirely determinate in the frame of elementary thermodynamics [1]. The model considered in [2] describes situations in which the macroscopic pressure difference in A and B constitutes the main driving force on the piston, so that microscopic pressure fluctuations are consistently neglected.…”