We provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in non-equilibrium systems in the steady state, from the measurements of arbitrary current fluctuations. Our results are built upon the finite-time generalization of the thermodynamic uncertainty relation, and require only very short time series data from experiments. We illustrate our results with exact and numerical solutions for two colloidal heat engines. arXiv:1910.00476v2 [cond-mat.stat-mech]

Nanoscale machines are strongly influenced by thermal fluctuations, contrary to their macroscopic counterparts. As a consequence, even the efficiency of such microscopic machines becomes a fluctuating random variable. Using geometric properties and the fluctuation theorem for the total entropy production, a "universal theory of efficiency fluctuations" at long times, for machines with a finite state space, was developed in [Verley et al., Nat. Commun. 5, 4721 (2014); Phys. Rev. E 90, 052145 (2014)]. We extend this theory to machines with an arbitrary state space. Thereby, we work out more detailed prerequisites for the "universal features" and explain under which circumstances deviations can occur. We also illustrate our findings with exact results for two non-trivial models of colloidal engines.Understanding the functioning of machines on the micro-or nanoscale is of great interest because of their role in biological systems and their numerous technological applications [1][2][3][4][5][6][7]. Their small size makes this task a challenge since thermal fluctuations strongly affect their operation. As a result, average values are no longer sufficiently informative, and fluctuations in heat, work, efficiency etc. must be taken into account. Stochastic thermodynamics [8] provides a convenient framework for analyzing such systems by extending the notions of classical (ensemble-based) thermodynamics to individual realizations of a given process.Consider first a macroscopic heat engine operating cyclically between two reservoirs at different temperatures T 1 > T 2 and performing work against an external load force. If Q 1 and Q 2 denote the average heat exchanged with the two reservoirs and −W the (average) performed work, the Second Law implies that the efficiency,is universally bounded from above by the reversible or Carnot efficiency η C = 1 − T 2 /T 1 . The efficiency η plays an equally pivotal role for microscopic machines; however, in these systems, due to thermal fluctuations, the value obtained in individual realizations can deviate significantly from the average behavior. We hence need to consider a distribution of efficiency values. Recently, in two seminal papers [9, 10], Verley, Willaert, Van den Broeck, and Esposito (VWVE) developed a "universal theory of efficiency fluctuations" for machines with a finite state space. By characterizing the long-time behavior of the efficiency fluctuations in terms of their large-deviation function J(η) (see below for more details), they found that the macroscopic efficiency, defined as the ratio of average output and input powers, is the most likely and, for machines operating in a non-equilibrium steady state or under a time-symmetric periodic protocol, the reversible Carnot efficiency is the least likely one [11]. The VWVE theory has since been verified in numerous model systems with finite [12][13][14][15][16][17][18][19] but also infinite [13,[19][20][21] state spaces.Nevertheless, there are a few examples of infinite state space systems at odds with the theory [22][23...

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.

We obtain exact results for the recently discovered finite-time thermodynamic uncertainty relation, for the dissipated work W d , in a stochastically driven system with non-Gaussian work statistics, both in the steady state and transient regimes, by obtaining exact expressions for any moment of W d at arbitrary times. The uncertainty function (the Fano factor of W d ) is bounded from below by 2kBT as expected, for all times τ , in both steady state and transient regimes. The lower bound is reached at τ = 0 as well as when certain system parameters vanish (corresponding to an equilibrium state). Surprisingly, we find that the uncertainty function also reaches a constant value at large τ for all the cases we have looked at. For a system starting and remaining in steady state, the uncertainty function increases monotonically, as a function of τ as well as other system parameters, implying that the large τ value is also an upper bound. For the same system in the transient regime, however, we find that the uncertainty function can have a local minimum at an accessible time τm, for a range of parameter values. The large τ value for the uncertainty function is hence not a bound in this case. The non-monotonicity suggests, rather counter-intuitively, that there might be an optimal time for the working of microscopic machines, as well as an optimal configuration in the phase space of parameter values. Our solutions show that the ratios of higher moments of the dissipated work are also bounded from below by 2kBT . For another model, also solvable by our methods, which never reaches a steady state, the uncertainty function, is in some cases, bounded from below by a value less than 2kBT . arXiv:1712.02714v2 [cond-mat.stat-mech]

The rate of entropy production provides a useful quantitative measure of a non-equilibrium system and estimating it directly from time-series data from experiments is highly desirable. Several approaches have been considered for stationary dynamics, some of which are based on a variational characterization of the entropy production rate. However, the issue of obtaining it in the case of non-stationary dynamics remains largely unexplored. Here, we solve this open problem by demonstrating that the variational approaches can be generalized to give the exact value of the entropy production rate even for non-stationary dynamics. On the basis of this result, we develop an efficient algorithm that estimates the entropy production rate continuously in time by using machine learning techniques and validate our numerical estimates using analytically tractable Langevin models in experimentally relevant parameter regimes. Our method only requires time-series data for the system of interest without any prior knowledge of the system’s parameters.