Generalization of the Jarzynski and Crooks nonequilibrium work theorems in molecular dynamics simulations

The Jarzynski identity [C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)] and the Crooks equation [G. E. Crooks, J. Stat. Phys. 90, 1481 (1998)] relate thermodynamic free energy differences to the work done on a system during a collection of nonequilibrium transformations. In the present Rapid Communication we provide generalized versions of these nonequilibrium work theorems, which hold for dissipative transformations where the system may undergo simultaneously mechanical work and pressure-temperature or volume-temperature changes. The proof is valid in the context of dynamic systems that evolve with $NPT$-based equations of motion according to the Martyna-Tobias-Klein algorithm [Martyna et al. J. Chem. Phys. 101, 4177 (1994)]. An extension of the proof to dynamic systems that evolve through $NVT$-based equations of motion is also provided. The theorems may be effectively used in non-Hamiltonian molecular dynamics simulations for evaluating Helmholtz or Gibbs free energy differences, or the ratio of partition functions at different temperatures to be eventually used in thermodynamic cycles.