Asymptotics of work distributions in a stochastically driven system

Abstract

Abstract. We determine the asymptotic forms of work distributions at arbitrary times T , in a class of driven stochastic systems using a theory developed by Nickelsen and Engel (EN theory) [D. Nickelsen and A. Engel, Eur. Phys. J. B 82, 207 (2011)], which is based on the contraction principle of large deviation theory. In this paper, we extend the theory, previously applied in the context of deterministically driven systems, to a model in which the driving is stochastic. The models we study are described by overdamped Langevin equations and the work distributions in path integral form, are characterised by having quadratic augmented actions. We first illustrate EN theory, for a deterministically driven system -the breathing parabola model, and show that within its framework, the Crooks fluctuation theorem manifests itself as a reflection symmetry property of a certain characteristic polynomial, which also determines the exact moment-generating-function at arbitrary times. We then extend our analysis to a stochastically driven system, studied in references [S. Sabhapandit, EPL 89, 60003 (2010) (2014)], for both equilibrium and non-equilibrium steady state initial distributions. In both cases we obtain new analytic solutions for the asymptotic forms of (dissipated) work distributions at arbitrary T . For dissipated work in the steady state, we compare the large T asymptotic behaviour of our solution to the functional form obtained in reference [New J. Phys. 16, 095001 (2014)]. In all cases, special emphasis is placed on the computation of the pre-exponential factor and the results show excellent agreement with numerical simulations. Our solutions are exact in the low noise (β → ∞) limit.

Counting statistics of energy transport across squeezed thermal reservoirs

“…A brief description of two methods that can be used to solve this class of equations is given in the appendix. It is to be noted that similar type of partial differential equations also appeared in the studies of heat current fluctuations through classical harmonic chains 39,40 and work statistics of driven classical harmonic oscillators subjected to thermal noise 41,42 . Also, a related partial differential equation is encountered in the study of work statistics of degenerate parametric amplification process 43 .…”

Fluctuations in heat engines

“…Namely, for a piece-wise constant protocol, where k(t) and T (t) involve a single jump [136] or two jumps [7,137]; in the slow-driving limit ( k(t) small compared to the relaxation time of the system) where the work PDF is Gaussian for any k(t) [37,138]; and if k(t) is a rational function of time [40,41]. In addition to exact approaches, theories were developed to predict asymptotics of work PDFs for small and large values of work [40,133,[139][140][141][142]. Also an Onsager-Machlup-type theory has been applied to obtain approximate solutions of the problem in Ref.…”

Statistics of work performed by optical tweezers with general time-variation of their stiffness

“…Analytical results for the work distribution could be derived for specific protocols in both modes of operation. Moreover, theories were developed to predict the exact asymptotic behaviour of work distributions [29,[36][37][38][39][40]. For the moving parabola with a potential minimum changing with time, the work distribution was calculated in Refs.…”