Large deviations of the current for driven periodic diffusions

Abstract

We study the large deviations of the time-integrated current for a driven diffusion on the circle, often used as a model of nonequilibrium systems. We obtain the large deviation functions describing the current fluctuations using a Fourier-Bloch decomposition of the so-called tilted generator and also construct from this decomposition the effective (biased, auxiliary or driven) Markov process describing the diffusion as current fluctuations are observed in time. This effective process provides a clear physical explanation of the various fluctuation regimes observed. It is used here to obtain an upper bound on the current large deviation function, which we compare to a recently-derived entropic bound, and to study the low-noise limit of large deviations.

Exact Fluctuations of Nonequilibrium Steady States from Approximate Auxiliary Dynamics

“…This is illustrated by the discovery of the fluctuation theorems [1][2][3][4], thermodynamic uncertainty relations [5,6], and extensions of the fluctuationdissipation theorem to systems far from equilibrium [7][8][9][10]. Large deviation functions provide a general mathematical framework within which to characterize and understand nonequilibrium fluctuations [11], and their evaluation has underpinned much recent progress in understanding driven systems [12][13][14][15]. However, the current Monte Carlo methods, such as the cloning algorithm [16][17][18][19][20][21] or transition path sampling [22], exhibit low statistical efficiency when accessing rare fluctuations that are needed to compute them [21,23].…”

“…For this simple one particle system we can determine the optimal GDF by diagonalizing W λ in a plane wave basis [29,34]. To illustrate the behavior when an approximate GDF is used, we also consider a GDF obtained from an instantonic solution to the eigenvalue equation, which captures the correct limiting behavior of ΞðθÞ at large λ, where ΞðθÞ is just a constant [34].…”

“…To illustrate the behavior when an approximate GDF is used, we also consider a GDF obtained from an instantonic solution to the eigenvalue equation, which captures the correct limiting behavior of ΞðθÞ at large λ, where ΞðθÞ is just a constant [34].…”

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

“…Then, trajectories PX and TX both have J τ = −J, so dP J (PX) = 0 = dP J (TX). On the other hand, (23) means that trajectory PTX has J τ (PTX) = J and (18) implies that dP (PTX) = dP (X). Hence, using (26), one has…”

“…On the other hand, the physical interpretation of the conditioned process is that the particle borrows energy from the heat bath in order to overcome the barrier, before returning that energy to the heat bath as it falls back down again. For explicit computations on a similar system in the overdamped limit, see [23].…”