M. Polettini, G. Verley and M. Esposito,

*Efficiency Statistics at All Times: Carnot Limit at Finite Power,*

Phys. Rev. Lett. **114**, 050601 (2015), arXiv:1409.4716

Enhancing the efficiency of energy conversion is the ultimate mission of thermodynamics. According to the paradigms of the phenomenological theory, it has long been believed that high efficiency can only be reached in processes that are infinitely slow and therefore useless for all practical purposes. However, there is no fundamental reason for this, and today we can investigate this question with the tools of a modern and powerful theory called Stochastic Thermodynamics. Stochastic Thermodynamics is capable of treating small systems that fluctuate; while fluctuations usually worsen the accuracy of processes, there exist critical conditions (e.g. near phase transitions) whereby small fluctuations are greatly enhanced, so to become structural properties. Following the proposal of Verley et al. [Verley 2014] to investigate efficiency in a probabilistic way, in this paper we investigate an analytically treatable model, that might be considered as the second-order approximation of any more complex model, where the thermodynamic currents are Gaussian and the efficiency statistics can be calculated exactly. The theory gives remarkable results, in particular there exists a nonvanishing probability that the system operates in reverse mode, i.e. that the power source is actually fed by the device it is supposed to power. This results in a bimodal probability distribution, a prediction of the theory that has already found partial experimental confirmation in an experiment on a small-scale Carnot engine [Martinez 2016]. Furthermore, at critical conditions our analysis shows that the Carnot efficiency can be maintained for long times at finite power output; hence we produced the first theoretical analysis of a system that has high efficiency but that needs not be infinitely slow. This observation has found further substantiation in a recent work relating critical scaling and efficiency enhancement [Campisi 2015]. Finally, other properties of the probability distribution of the efficiency can be shown to be universal, such as its power-law tails which imply that there is no average efficiency but only more probable efficiency. This explains the effect of the divergent average efficiency observed by certain authors in numerical simulations.

[Verley 2014] Verley, G., et al. “The unlikely Carnot efficiency. Nature communications, 5 (2014).

[Martinez 2016] Martínez, I. A., et al. “Brownian carnot engine.” Nature Physics 12.1 (2016): 67-70.

[Campisi 2015] Campisi, M. and Fazio, R. “Universality and scaling of optimal heat engines”, arXiv:1510.06183.

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