of its kinetic energy. At the moment the piston reaches $B$, the container is uniformly filled with a gas characterized by a Maxwellian velocity distribution at temperature $T$. Needless to say, this “anti-shock” wave represents an exotic sequence of events! When $n_p\gg 1$ and $u\gg v_{\rm th}$, the probability of sampling an initial microstate for which half the particles have $v_x \approx +2u$ is fantastically small. However, according to the central result of this paper, rare realizations of this sort are precisely the dominant ones that contribute most to $\langle e^{-\beta W}\rangle_F$.

Similarly, the dominant realizations of the compression process are the conjugate twins of typical realizations of the expansion process. Thus to achieve convergence of $\langle e^{-\beta W}\rangle_R$, we must observe realizations during which the bulk of the gas happens to be localized in the left half of the box at $t=0$. Then, as the piston moves rapidly from $B$ to $A$, it sweeps through a largely empty region, as in Fig.\[fig:piston\_Ftyp\_Rdom\]b. Again, this represents an unusual scenario: it is unlikely that the randomly sampled initial conditions are such that virtually all the particles are found in the left half of the box.

To summarize, while a typical realization 