Equilibrium free-energy differences at different temperatures from a single set of nonequilibrium transitions

Crook's fluctuation theorem (CFT) and Jarzynski equality (JE) are effective tools for obtaining free-energy difference $\mathrm{\Delta }F({\lambda }_A\to {\lambda }_B,T_0)$ through a set of finite-time protocol driven nonequilibrium transitions between two equilibrium states $A$ and $B$ [parametrized by the time-varying protocol $\lambda \left(t\right)$] at the same temperature $T_0$. Using the generalized dimensionless work function $\mathrm{\Delta }{W}_G$, we extend CFT to transitions between two nonequilibrium steady states (NESSs) created by a thermal gradient. We show that it is possible, provided the period over which the transitions occur is sufficiently long, to obtain $\mathrm{\Delta }F({\lambda }_A\to {\lambda }_B,T_0)$ for different values of $T_0$, using the same set of finite-time transitions between these two NESSs. Our approach thus completely eliminates the need to make new samples for each new $T_0$. The generalized form of JE arises naturally as the average of the exponentiated $\mathrm{\Delta }{W}_G$. The results are demonstrated on two test cases: (i) a single particle quartic oscillator having a known closed form $\mathrm{\Delta }F$, and (ii) a one-dimensional ${\varphi }^4$ chain. Each system is sampled from the canonical distribution at an arbitrary $T^{\prime }$ with $\lambda ={\lambda }_A$, then subjected to a temperature gradient between its ends, and after steady state is reached, the protocol change ${\lambda }_A\to {\lambda }_B$ is effected in time $\tau$, following which $\mathrm{\Delta }{W}_G$ is computed. The reverse path likewise initiates in equilibrium at $T^{\prime }$ with $\lambda ={\lambda }_B$ and the protocol is time reversed leading to $\lambda ={\lambda }_A$ and the reverse $\mathrm{\Delta }{W}_G$. Our method is found to be more efficient than either JE or CFT when free-energy differences at multiple $T_0$'s are required for the same system.

Figure 1

Comparison of $\mathrm{\Delta }F$ obtained using theoretical and proposed approaches in test case 1. Notice, that the proposed approach provides a good approximation to the theoretical results.

Figure 2

Forward and reverse probability densities of generalized work function at $T_0=0.29$ calculated using 5000 forward and reverse trajectories using two sets of ($T_H,T_C$): (red) = (0.27,0.23) and (green) = (0.30,0.20). The intersection of forward and reverse $\mathrm{\Delta }{W}_G$ densities gives $\mathrm{\Delta }F$, and is not affected by the choice of $T_H,T_C$. $\mathrm{\Delta }F$ calculated using proposed fluctuation relation compares well with that of JE and CFT. Results obtained using the same data set for other $T_0$ values are similar, and agree well with CFT.