Extended heat-fluctuation theorems for a system with deterministic and stochastic forces

Abstract

Heat fluctuations over a time tau in a nonequilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic potential which is moved with constant velocity. Using a Langevin equation, we find the exact Fourier transform of the distribution of these fluctuations for all tau. By a saddle-point method we obtain analytical results for the inverse Fourier transform, which, for not too small tau, agree very well with numerical results from a sampling method as well as from the fast Fourier transform algorithm. Due to the interaction of the deterministic part of the motion of the particle in the mechanical potential with the stochastic part of the motion caused by the fluid, the conventional heat fluctuation theorem is, for infinite and for finite tau, replaced by an extended fluctuation theorem that differs noticeably and measurably from it. In particular, for large fluctuations, the ratio of the probability for absorption of heat (by the particle from the fluid) to the probability to supply heat (by the particle to the fluid) is much larger here than in the conventional fluctuation theorem.

Chaotic Hypothesis, Fluctuation Theorem and Singularities

“…it is of the form ∼ e −κF , for some constant κ > 0. This is a valuable and interesting remark brought up for the first time, and correctly interpreted, already in [4] and in the following papers [24,25]. The analysis of [4,24,25] applies to cases where the unbounded fluctuations are driven by an external white noise.…”

“…Outside this interval ζ(a) does not satisfy the FR and in particular for a ≥ a + it is ζ(a) − ζ(−a) = const., as already described in [24]. Eq.…”

“…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.…”

“…The analysis of [4,24,25] applies to cases where the unbounded fluctuations are driven by an external white noise. In the following we extend the theoretical analysis in [24,25] to cases in which the unbounded fluctuations do not arise from a Gaussian noise but from a deterministic evolution like the ones in [3,15]: this is a simple extension of the main idea and method of [24] and provides an alternative interpretation to the analysis in [3,15].…”

Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

“…Interestingly, it has been shown that unlike their entropy production, the heat dissipation of Langevin processes does not satisfy an AFT, at least in the conventional sense [39,40], displaying importance of the boundary terms distinguishing these quantities. Along similar lines, Rákos and Harris have recently shown [30] that infinite state spaces can result in the divergence of the boundary terms, causing a breakdown of the Gallavotti-Cohen symmetry.…”