Numerical study of the steady state fluctuation relations far from equilibrium

A thermostatted dynamical model with five degrees of freedom is used to test the fluctuation relation of Evans and Searles (Ω-FR) and that of Gallavotti and Cohen (Λ-FR). 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 Ω-FR is verified by the simulation data while the Λ-FR 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 Λ-FR [Bonetto, Physica D 105, 226 (1997); Giuliani, J. Stat. Phys. 119, 909 (2005)]. It was conjectured that when the number of nontrivial Lyapunov exponents that are positive becomes less than the number of such negative exponents, then the form of the Λ-FR needs to be corrected. We show that there is no evidence for this conjecture in the empirical data. In fact, when the correction factor differs from unity, the corrected form of Λ-FR is less accurate than the uncorrected Λ-FR. Also as the field increases the uncorrected Λ-FR appears to be satisfied with increasing accuracy. The reason for this observation is likely to be that as the field increases, the argument of the Λ-FR more and more accurately approximates the argument of the Ω-FR. Since the Ω-FR works for arbitrary field strengths, the uncorrected Λ-FR appears to become ever more accurate as the field increases. The final piece 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 Λ-FR. We see no evidence for a discontinuity at this field strength.