Multiple scales approach to the gas-piston non-equilibrium themodynamics

Abstract

The non-equilibrium thermodynamics of a gas inside a piston is a conceptually simple problem where analytic results are rare. For example, it is hard to find in the literature analytic formulas that describe the heat exchanged with the reservoir when the system either relaxes to equilibrium or is compressed over a finite time. In this paper we derive such kind of analytic formulas. To achieve this result, we take the equations derived by Cerino et al. [Phys. Rev. E 91, 032128] describing the dynamic evolution of a gas-piston system, we cast them in a dimensionless form and we solve the dimensionless equations with the multiple scales expansion method. With the approximated solutions we obtained, we express in a closed form the heat exchanged by the gas-piston system with the reservoir for a large class of relevant non-equilibrium situations.

Thermodynamics of slow solutions to the gas-piston equations

“…Mathematically this solution arise as the particular solution of the inhomogeneous ordinary differential equations coming from the multiple scales expansion and can be computed in practice by taking all the integration constants in the multiple scale expansion equal to zero. In [20], we also hinted that the dynamical equilibrium solution (11) could have been obtained with the "slow" regular perturbation…”

“…Following the same procedure we used in [20] we transform the system (9) into a single third order equation. At difference with [20], where Ω ≡ 1, we assume that the external force and the reservoir temperatures are slow, i.e.…”

“…Combined in a laborious averaging process, the authors where able to derive a set of dynamical equations for the time evolution of the gas temperature T and piston position x according to any externally prescribed change of the external force Θ and reservoir temperature Ω. In [20] we showed with the multiple scales method [21][22][23][24] and some technical assumptions, (discussed in Section II) that the equations derived in [15] allows to find an approximated expression for the heat exchanged with the reservoir in two physically relevant cases: the relaxation to equilibrium and the slow isothermal compression. In the same paper [20] we pointed out the existence of particular solutions, which we called dynamical equilibrium solutions, which describe the asymptotic behavior of the system when externally driven.…”

“…In [20] we assumed ε to be a "small" perturbative parameter, the reservoir temperature to be constant (without loss of generality. Ω = 1) and the external forcing Θ to be slowly varying over time (any function B can be considered "slow" if B = B(εt), which we assume from now on).…”

“…In the t 2 -scale the exponential suppression of the transient oscillatory terms appears and is the proper scale of the external forcing too. The approximated solution derived in [20] falls, as t → ∞, into a peculiar solution:…”