Fluctuation Relation beyond Linear Response Theory

Abstract

The Fluctuation Relation (FR) is an asymptotic result on the distribution of certain observables averaged over time intervals τ as τ → ∞ and it is a generalization of the fluctuation-dissipation theorem to far from equilibrium systems in a steady state which reduces to the usual Green-Kubo (GK) relation in the limit of small external non conservative forces. FR is a theorem for smooth uniformly hyperbolic systems, and it is assumed to be true in all dissipative "chaotic enough" systems in a steady state. In this paper we develop a theory of finite time corrections to FR, needed to compare the asymptotic prediction of FR with numerical observations, which necessarily involve fluctuations of observables averaged over finite time intervals τ . We perform a numerical test of FR in two cases in which non Gaussian fluctuations are observable while GK does not apply and we get a non trivial verification of FR that is independent of and different from linear response theory. Our results are compatible with the theory of finite time corrections to FR, while FR would be observably violated, well within the precision of our experiments, if such corrections were neglected.

Chaotic Hypothesis, Fluctuation Theorem and Singularities

“…The numerical results of [34][35][36] agree with the prediction that FR for the rate function of a 0 is valid even beyond a 0 = σ + . The prediction that (at least near equilibrium) the rate function of a should satisfy FR only up to a = σ + and that should become linear for a ≥ a + at the moment has been experimentally confirmed only in Gaussian cases [4,24,25].…”

“…It would be very interesting to investigate in detail the structure of ζ(a) even in non Gaussian cases. Note that this is far from being an easy task (in particular the analysis in [36] was not sophisticated enough to study this problem). In fact, as discussed in detail in [35], the presence in the definition of σ of a total derivative of an unbounded function may enlarge of 2 orders of magnitudes the times needed for the probability distribution of a to reach its asymptotic shape: even in the Gaussian region (small fluctuations of a around σ + ) the convergence times for ζ(a) are found to be of order 1000 decorrelation times, versus a time of order 10 decorrelation times needed for ζ 0 (a 0 ) to converge to its asymptotic shape [35].…”

“…Clearly, for times of order 1000 decorrelation times, it is very hard to observe fluctuations of a larger than a + − σ + . In order to verify the prediction for the shape of ζ(a) beyond a = a + an experiment specifically designed for this purpose would be needed, together with a detailed investigation of the finite time corrections to ζ(a), along the lines in [36].…”

Is it possible to experimentally verify the fluctuation relation? A review of theoretical motivations and numerical evidence

“…In [49] it was noted that, using Eq.s (19) and (20), it turns out that the non Gaussian term in (19) is proportional to J (3) ∞ E 3 (p − 1) 3 ; this suggested that, keeping fixed E to have a small σ + , one could increase the non Gaussian tails of ζ ∞ (p) by increasing J (3) ∞ , which is related to the nonlinear part of the transport coefficient. In a fluid of Lennard-Jones like particles, the nonlinear response is observed to increase on lowering the temperature: this fact was exploited in [49] where it was possible to verify the fluctuation relation in a numerical simulation on a non Gaussian ζ ∞ (p), see figure 2.…”

“…Unfortunately, 1) is limited by the fact that the fluctuation relation holds only for large τ , i.e. τ ≫ τ 0 ; so we can reduce τ , but at best we can use τ ∼ 100τ 0 if we do not want to observe finite τ effects [48,49]. Note also that in experiments τ is strongly constrained by the acquisition bandwidth.…”