Multiple scales approach to the Gas-Piston non-equilibrium themodynamics

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 \textit{et al.} [Phys. Rev. E \textbf{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.