box (Fig.\[fig:piston\_Ftyp\_Rdom\]b). The probability of generating such an initial condition is $(1/2)^{n_p}$, hence the number of realizations needed to observe a single such event is $$N_c^R \sim 2^{n_p},$$ which is equal to $\exp(\beta \overline W_d^F)$, as predicted by Eq.\[eq:NcR\].

For the convergence of $\langle e^{-\beta W}\rangle_F$, the dominant realizations are represented by Fig.\[fig:piston\_Rtyp\_Fdom\]b: initially, half the particles are characterized by a thermal velocity profile, the other half with $v_x \approx 2u$. The probability of generating such a microstate is roughly $$\label{eq:roughEstimate}
\frac{n_p!}{(n_p/2)!(n_p/2)!} \cdot a^{n_p/2} .$$ The first factor counts the number of ways of choosing which $n_p/2$ particles start with $v_x \approx 2u$, and $a \equiv \exp[-\beta m (2u)^2/2]$ is an estimate of the probability for a single particle to have a such a large initial $v_x$. Using Stirling’s approximation for the factorials, Eq.\[eq:roughEstimate\] reduces to $2^{n_p} \cdot \exp(-n_p\beta mu^2)$. Taking the reciprocal, we get $$N_c^F \sim \exp[n_p(\beta mu^2 - \ln 2)],$$ which is equal to $\exp(\beta \overline W_d^R)$, as predicted by Eq.\[eq:NcF\]. Since the fast piston limit (Eq. \[eq:fastPiston\]) implies $\beta mu^2 \gg 1$, it is legitimate to drop the $\ln 2$ term in this result, obtaining $N_c^F \sim \exp(n_p\beta mu^2)$. This is consistent with the calculations of Lua 