Heat and work distributions for mixed Gauss–Cauchy process

Abstract

Abstract. We analyze energetics of a non-Gaussian process described by a stochastic differential equation of the Langevin type. The process represents a paradigmatic model of a nonequilibrium system subject to thermal fluctuations and additional external noise, with both sources of perturbations considered as additive and statistically independent forcings. We define thermodynamic quantities for trajectories of the process and analyze contributions to mechanical work and heat. As a working example we consider a particle subjected to a drag force and two statistically independent Lévy white noises with stability indices α = 2 and α = 1. The fluctuations of dissipated energy (heat) and distribution of work performed by the force acting on the system are addressed by examining contributions of Cauchy fluctuations (α = 1) to either bath or external force acting on the system.

Fluctuation relations for anomalous dynamics generated by time-fractional Fokker–Planck equations

“…Continuous lines represent the result of equation(38) and agree for larger W values with the exact results denoted by circles. Equation(38) additionally contains a timedependent scaling factor that is proportional t (2 2) ( 2) α α − − . This factor is positive for the subdiffusive type B FFPE and negative for type A FFPE.…”

“…These results have been reproduced and generalized by a stochastic thermodynamics approach [32]. For non-Gaussian PDFs generated by Langevin equations with non-Gaussian noise, such as Lévy noise or Poissonian shot noise, violations of conventional steady state and transient fluctuation relations (TFRs) have been reported [33][34][35][36][37][38]. For a CTRW model with a power law waiting time distribution it was found that the steady state FRs may or may not hold depending on the exponent of the waiting time distribution [39].…”

Peculiarities of escape kinetics in the presence of athermal noises

“…Divergent moments of Lévy statistics and Lévy motion seem to stay in conflict with energetic and thermodynamics of the stochastic differential equation of the Langevin type 23,44,57,58 . Yet, accumulating evidence shows that Markovian Lévy flights (LFs) with distribution of jumps emerging from the generalized version of the central limit theorem are well suited representations of complex phenomena, to name just a few recent applications of LFs in description of mental searches 61 , analysis of free neutron output in a fusion experiment with a deuteron plasma 56 , investigations of generegulatory networks 62 or examination of self-regulatory motion of insects 63 .…”

“…The central part of the jump length distribution control short jumps which are responsible for penetration of the potential barrier 4 . Therefore, in the current subsection, we assume that the particle is driven by two stochastic forces [55][56][57][58] , so that the Langevin equation assumes the form…”

Phase transitions and entropies for synchronizing oscillators

“…which leads toṠ(t) = 1/(2t) [27]. That the diffusion entropy follows S ∝ 1 2 ln t can also be understood from more intuitive arguments.…”