  Jarzynski's 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.
