Efficiency Fluctuations in Microscopic Machines

Abstract

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...

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