of the expansion process is characterized by no piston-particle collisions, and a typical realization of the compression process is characterized by $\sim n_p/2$ collisions (Figs. \[fig:piston\_Ftyp\_Rdom\]a, \[fig:piston\_Rtyp\_Fdom\]a), for [*dominant*]{} realizations it is the other way around (Figs. \[fig:piston\_Rtyp\_Fdom\]b, \[fig:piston\_Ftyp\_Rdom\]b).

Derivation of central result {#sec:derivation}
============================

Consider a classical system described by a Hamiltonian $H(\Gamma;\lambda)$, where $\Gamma =({\bf q},{\bf p})$ denotes a point in the system’s phase space (a [*microstate*]{}), and $\lambda$ is an externally controlled parameter, such as the piston location in the above example. For every value of $\lambda$, assume $H$ is time-reversal invariant: $$\label{eq:tri}
H(\Gamma^*;\lambda) = H(\Gamma;\lambda)
\quad,\quad
\Gamma^*\equiv({\bf q},-{\bf p}).$$ This system can be prepared in an equilibrium state, by placing it in weak thermal contact with a sufficiently large heat reservoir at temperature $T$, holding the parameter fixed at a value $\lambda$, and then removing the reservoir after a sufficiently long relaxation time. This generates a microstate $\Gamma_0$ that is a random sample from the Boltzmann-Gibbs distribution, $$\label{eq:pBG}
p_\lambda(\Gamma_0) = \frac{1}{Z_\lambda} \exp[-\beta H(\Gamma_0;\lambda)],$$ where $Z_\lambda = \int d\Gamma\,\exp[-\beta H(\Gamma;\lambda)]$ is the partition function. The free energy associated with this equilibrium state is $$\label{eq:fdef}
F_\lambda = -\beta^{-1}\ln Z_\lambda.$$ In the analysis and discussions below, the dependence of 