  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.
