Numerical study of the steady state fluctuation relations far from equilibrium

Abstract

Nonchaotic attractor with a highly fluctuating finite-time Lyapunov exponent in a hybrid optical system AIP Conf. Proc. 501, 342 (2000); 10.1063/1.59945Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. A thermostatted dynamical model with five degrees of freedom is used to test the fluctuation relation of Evans and Searles ͑⍀-FR͒ and that of Gallavotti and Cohen ͑⌳-FR͒. 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 ⍀-FR is verified by the simulation data while the ⌳-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 ⌳-FR ͓Bonetto et al., Physica D 105, 226 ͑1997͒; Giuliani et al., J. Stat. Phys. 119, 909 ͑2005͔͒. 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 ⌳-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 ⌳-FR is less accurate than the uncorrected ⌳-FR. Also as the field increases the uncorrected ⌳-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 ⌳-FR more and more accurately approximates the argument of the ⍀-FR. Since the ⍀-FR works for arbitrary field strengths, the uncorrected ⌳-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 ⌳-FR. We see no evidence for a discontinuity at this field strength.

Fluctuation Relations for Diffusion Processes

“…Numerical (see e.g. [7,45,82,38,85]) and experimental testing of the fluctuation relations (see e.g. [15,37,17,2,54,47]) has attracted over years a lot of attention, inspiring further developments.…”

Time Reversibility, Correlation Decay and the Steady State Fluctuation Relation for Dissipation

“…This supports the use of thermostatted dynamics in this context. However, the understanding of the physical mechanisms underlying the validity of the steady state FRs is less complete [6,[14][15][16]. The present paper clarifies some of these outstanding issues.…”

“…Following the approach developed by Evans and Searles, we demonstrate how time reversibility, ergodic consistency and a form of the decay of correlations, known as T-mixing [14,15,17], lead to the steady state Ω-FR for systems of arbitrary size, near or far from equilibrium.…”

On the probability of violations of Fourier’s law for heat flow in small systems observed for short times

“…All expressions will satisfy asymptotic steady state fluctuation relations, but ⍀ and ⌺ therm will converge more rapidly than the other two functions. 13,20,21 The TFR obtained using ⌺ therm can be used in real laboratory experiments. One simply needs some way to measure the instantaneous heat fluxes across planes that have sufficient area that they can be regarded as being in local thermodynamic equilibrium ͑as the areas increase, the fluxes become smaller, and the local equilibrium approximation becomes more accurate͒.…”

“…͓Note: we refer to a fluctuation relation when the mathematical form of a FT is proposed in conjunction with the substitution of a variable ͑usually with the same average value͒ for which the corresponding theorem has not been proved.͔ This is a standard problem with fluctuating boundary terms and is related to the well known convergence problems for the Gallavotti-Cohen FT for thermostatted steady states as equilibrium is approached. 13,15,20,21 The ultimate fluxes into and out of our system are given by the energy gain or loss by the thermostats themselves. These are the only nonconservative elements of our system.…”