Thermodynamics of slow solutions to the gas-piston equations

Despite its historical importance, a perfect gas enclosed by a pistons and in contact with a thermal reservoirs is a system still largely under study. Its thermodynamic properties are not yet well understood when driven under nonequilibrium conditions, and analytic formulas that describe the heat exchanged with the reservoir are rare. In this paper we prove a power series expansions for the heat when both the external force and the reservoir temperature are slowly varying over time but the overall process is not quasistatic. To do so, we use the dynamical equations from [Cerino et al., Phys. Rev. E 91, 032128 (2015)] and an uncommon application of the regular perturbation technique.

Figure 1

A schematic representation of the gas closed by a piston and in contact with a thermal reservoir. The quantities shown are the piston position $\overline{x}$, the gas temperature $\overline{T}$, the external force $\overline{\mathrm{\Omega }}$, and the reservoir temperature $\overline{\mathrm{\Theta }}$.

Figure 2

Sample trajectories for $\varepsilon =1/16$ with the first three nonzero terms of $x$ and $T$ expansions for $\mathrm{\Omega }\left(s\right)=1+0.1sin\left(s\right)$ and $\mathrm{\Theta }\left(s\right)=1+0.1cos\left(s\right)$.

Figure 3

Plot of $R$ as a function of $K$ for different values of $\varepsilon$.