The Fluctuation Theorem

Denis J. Evans,
Australian National University

The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behavior has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in thermostatted dissipative nonequilibrium systems. The relation was derived heuristically and applied to the special case of dissipative nonequilibrium systems subject to constant energy "thermostatting". These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been in the last couple of years that the generality of the Theorem has been appreciated.

The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quantifies the probability of observing Second Law violations in small systems observed for a short time. In other words, the Fluctuation Theorem generalizes the Second Law to finite systems observed for finite times. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observation that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversed 'anti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been confirmed experimentally, both using molecular dynamics computer simulation and very recently in laboratory experiments involving optical tweezers.