Fluctuation theorems for non-linear generalized Langevin systems

Abstract

The fluctuation theorems obtained in a stochastic Markovian process are generalized to a non-Markovian system governed by the non-linear generalized Langevin equation with a Gaussian colored noise. We derive the non-Markovian version of the Crooks fluctuation theorem that relates the statistical averages of the two different dynamics characterized by the forward process and the reverse process. In contrast to the similar study by Zamponi et al, ours does not assume a stationary state asymptotically in time so that the present fluctuation theorem can deal explicitly with the dependence of the initial condition and the transient behavior. The Jarzynski equality for the non-equilibrium work relation and the representation of the linear response in the non-equilibrium steady state are also discussed. The conditions for the memory kernel that the fluctuation theorems hold are examined by analyzing a solvable model and are confirmed by a direct derivation of the fluctuation theorems for the cases of an exponential decay and a power law decay of the memory kernel.

Anomalous Fluctuation Relations

“…This raises the question to which extent conventional FRs are valid for anomalous dynamics. Theoretical results for generalized Langevin equations [47,48,49,50], Lévy flights [51,52] and Continuous-Time Random Walk models [53] as well as computer simulations for glassy dynamics [54] showed both validity and violations of the various types of conventional FRs referred to above, depending on the specific type of anomalous dynamics considered and the nonequilibrium conditions that have been applied [55].…”

“…Probably inspired by the experiment of Ref. [27], typically Langevin dynamics in a harmonic potential moving with a constant velocity has been studied in the literature [48,49,50,61], cf. Fig.…”

“…FRs for this type of dynamics have more recently been analyzed in Refs. [47,48,49,50]. Examples of applications for this type of stochastic modeling are given by generalized elastic models [66], polymer dynamics [67] and biological cell migration [43].…”

Generalized fluctuation theorems for classical systems

“…Recent years have seen an upsurge of interest in generalized fluctuation dissipation like theorems for driven, nonequilibrium, dissipative systems [2][3][4][5][6][7][8]. Much of this renewed interest has concentrated on a new time dependent dynamical variable W (t) which characterizes the work done on the system by an external, time dependent force, The natural, conjugate (to the applied force) variable is the velocity or the current.…”

Symmetry for the duration of entropy-consuming intervals

“…In the last few years, the applicability of these relations have also been extended to systems exhibiting stochastic non-Markovian dynamics [12][13][14][15][16][17][18]. On the other hand, they have been widely tested in experiments [19][20][21][22][23][24].…”