Application of the Gallavotti-Cohen fluctuation relation to thermostated steady states near equilibrium

The fluctuation relation of the Gallavotti-Cohen fluctuation theorem (GCFT) concerns fluctuations in the phase-space compression rate of dissipative, reversible dynamical systems. It has been proven for Anosov systems, but it is expected to apply more generally. This raises the question of which non-Anosov systems satisfy the fluctuation relation. We analyze time-dependent fluctuations in the phase space compression rate of a class of N-particle systems that are at equilibrium or in near equilibrium steady states. This class does not include Anosov systems or isoenergetic systems; however, it includes most steady-state systems considered in molecular-dynamics simulations of realistic systems. We argue that the fluctuations of the phase-space compression rate of these systems at or near equilibrium do not satisfy the fluctuation relation of the GCFT, although the discrepancies become somewhat smaller as the systems move further from equilibrium. In contrast, similar fluctuation relations for an appropriately defined dissipation function appear to hold both near and far from equilibrium.