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 ΔF(λ_{A}→λ_{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 λ(t)] at the same temperature T_{0}. Using the generalized dimensionless work function ΔW_{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 ΔF(λ_{A}→λ_{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 ΔW_{G}. The results are demonstrated on two test cases: (i) a single particle quartic oscillator having a known closed form ΔF, and (ii) a one-dimensional ϕ^{4} chain. Each system is sampled from the canonical distribution at an arbitrary T^{'} with λ=λ_{A}, then subjected to a temperature gradient between its ends, and after steady state is reached, the protocol change λ_{A}→λ_{B} is effected in time τ, following which ΔW_{G} is computed. The reverse path likewise initiates in equilibrium at T^{'} with λ=λ_{B} and the protocol is time reversed leading to λ=λ_{A} and the reverse ΔW_{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.