Numerical study of the Steady State Fluctuation Relations Far from   Equilibrium
  A thermostatted dynamical model with five degrees of freedom is used to test both the Evans-Searles and the Gallavotti-Cohen fluctuation relations (ESFR and GCFR respectively). In the absence of an external driving field, the model generates a time independent ergodic equilibrium state with two conjugate pairs of Lyapunov exponents. Each conjugate pair sums to zero. The fluctuation relations are tested numerically both near and far from equilibrium. As expected from previous work, near equilibrium the ESFR is verified by the simulation data while the GCFR is not confirmed by the data. Far from equilibrium where a positive exponent in one of these conjugate pairs becomes negative, we test a conjecture regarding the GCFR made by Gallavotti and co-workers. They conjectured that where the number of nontrivial Lyapunov exponents that are positive becomes less than the number of such negative exponents, then the form of the GCFR needs to be corrected. We show that there is no evidence for this conjecture in the empirical data. In fact as the field increases, the uncorrected form of the GCFR appears to become more accurate. The real reason for this observation is likely to be that as the field increases, the argument of the GCFR more and more accurately approximates the argument of the ESFR. Since the ESFR works for arbitrary field strengths, the uncorrected GCFR appears to become ever more accurate as the field increases. The final point of evidence against the conjecture is that when the smallest positive exponent changes sign, the conjecture predicts a discontinuous change in the "correction factor" for GCFR. We see no evidence for a discontinuity at this field strength; only a gradual improvement of degree of agreement as the field increases. 
Stephen R. Williams, Debra J. Searles and Denis J. Evans