Merits and qualms of work fluctuations in classical fluctuation theorems

Work is one of the most basic notions in statistical mechanics, with work fluctuation theorems being one central topic in nanoscale thermodynamics. With Hamiltonian chaos commonly thought to provide a foundation for classical statistical mechanics, here we present general salient results regarding how (classical) Hamiltonian chaos generically impacts on nonequilibrium work fluctuations. For isolated chaotic systems prepared with a microcanonical distribution, work fluctuations are minimized and vanish altogether in adiabatic work protocols. For isolated chaotic systems prepared at an initial canonical distribution at inverse temperature $\beta$, work fluctuations depicted by the variance of $e^{-\beta W}$ are also minimized by adiabatic work protocols. This general result indicates that, if the variance of $e^{-\beta W}$ diverges for an adiabatic work protocol, it diverges for all nonadiabatic work protocols sharing the same initial and final Hamiltonians. Such divergence is hence not an isolated event and thus greatly impacts on the efficiency of using Jarzynski's equality to simulate free-energy differences. Theoretical results are illustrated in a Sinai model. Our general insights shall boost studies in nanoscale thermodynamics and are of fundamental importance in designing useful work protocols.

Figure 1

The chaotic Sinai billiard with a moving piston (via changing $L_v$). $\lambda$ in the main text refers to the free area enclosed by the billiard. For comparison, a rectangular billiard without the circular structure of radius $R$ is also considered as a nonchaotic test case. We set $L_h=40$ and $R=15$, with $L_v=40$ at the beginning of all work protocols.

Figure 2

Variance of work fluctuations, denoted $\mathrm{var}\left(W\right)$, vs. the duration $\tau$ of the work protocol ${\lambda }_0\to {\lambda }_{\tau }=1.2{\lambda }_0$, obtained from $2.4\times {10}^5$ individual work realizations. Regimes with very large $\tau$ refer to adiabatic situations. For the Sinai billiard case, $\mathrm{var}\left(W\right)$ approaches 0 as $\tau$ goes to infinity; for the nonchaotic case (square billiard) with the same ${\lambda }_{\tau }/{\lambda }_0, \mathrm{var}\left(W\right)$ stays well above zero even for large $\tau$. Here and elsewhere of the main text, all plotted quantities are scaled and hence dimensionless, with $E_0=50$ and $m=1$.

Figure 3

Work fluctuations depicted by the variance of $e^{-\beta W}$ vs. the duration $\tau$ of the work protocol ${\lambda }_0\to {\lambda }_{\tau }=1.2{\lambda }_0$, with $2.4\times {10}^5$ individual work realizations and $\beta =0.01$. In the chaotic case, the variance of $e^{-\beta W}$ reaches its minimal values as $\tau$ becomes very large (adiabatic regime), in agreement with theory. In the nonchaotic case, the variance of $e^{-\beta W}$ oscillates violently vs. $\tau$ [40]. Remaining parameters are as described in the caption of Fig. 2.

Figure 4

The 95% confidence interval width of the simulated average ${\sum }_{i=1}^n{e}^{-\beta W_i}/n$ over $2.4\times {10}^4$ statistical samples, as a function of the sample size $n$, with $\beta =0.01$. For the bottom curve, $s={\lambda }_{\tau }/{\lambda }_0=1/3$ and so $\mathrm{var}\left(e^{-\beta W}\right)$ being finite, the error scaling is in agreement with CLT. For the upper three curves, $s=3$ and hence $\mathrm{var}\left(e^{-\beta W}\right)$ diverges, with the error scaling given by $\sim n^{-\frac{1}{3}}$ in the adiabatic case ($\tau ={10}^4$) or even slower scalings in the other two cases.

Figure 5

Variance of exponential work, denoted $\mathrm{var}\left(e^{-\beta W}\right)$, vs. the duration $\tau$ of the work protocol mentioned under Eq. (D1), for a classical system of mixed phase space structure. Note the logarithmic scale used for both $\tau$ and $\mathrm{var}\left(e^{-\beta W}\right)$.