Jarzynski's [Phys.Rev.Lett.78,2690 (1997)] identity for the free-energy difference between two equilibrium states can be viewed as a special case of a more general procedure based on phase-space mappings.Solving a system's equation of motion by approximate means generates a mapping that is perfectly valid for this purpose,regardless of how closely the solution mimics true time evolution.We exploit this fact,using crudely dynamical trajectories to compute free-energy differences that are in principle exact.Numerical simulations show that Newton's equation can be discretized to low order over very large time steps (limited only by the computer's ability to represent resulting values of dynamical variables) without sacrificing thermodynamic accuracy.For computing the reversible work required to move a particle through a dense liquid,these calculations are more efficient than conventional fast-switching simulations by more than an order of magnitude.We also explore consequences of the phase-space mapping perspective for systems at equilibrium,deriving an exact expression for the statistics of energy fluctuations in simulated conservative systems.
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