Fundamental aspects of steady-state conversion of heat to work at the nanoscale

The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors and some artificial micro-engines operate. As described by stochastic thermodynamics, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit. Here we report an experimental realization of a Carnot engine with a single optically trapped Brownian particle as the working substance. We present an exhaustive study of the energetics of the engine and analyse the fluctuations of the finite-time efficiency, showing that the Carnot bound can be surpassed for a small number of non-equilibrium cycles. As its macroscopic counterpart, the energetics of our Carnot device exhibits basic properties that one would expect to observe in any microscopic energy transducer operating with baths at different temperatures. Our results characterize the sources of irreversibility in the engine and the statistical properties of the efficiency-an insight that could inspire new strategies in the design of efficient nano-motors.

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Brownian Carnot engine

Heat engines should ideally have large power output, operate close to Carnot efficiency and show constancy, ie, exhibit only small fluctuations in this output For steady-state heat engines, driven by a constant temperature difference between the two heat baths, we prove that out of these three requirements only two are compatible Constancy enters quantitatively the conventional trade-off between power and efficiency Thus, we rationalize and unify recent suggestions for overcoming this simple trade-off Our universal bound is illustrated for a paradigmatic model of a quantum dot solar cell and for a Brownian gyrator delivering mechanical work against an external force

Universal Trade-Off between Power, Efficiency, and Constancy in Steady-State Heat Engines.

The power of an engine scales with the number of particles that make up its working fluid, a generalization that has proven true down to a single atom

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A single atom heat engine

Stochastic thermodynamics extends the traditional laws of thermodynamics to microscopic systems where thermal and quantum fluctuations cannot be ignored. This review summarizes progress in this field with a look at several experimental and theoretical results and a look toward potential applications in biology and nanotechnology.

https://link.aps.org/pdf/10.1103/PhysRevX.7.021051

Experiments in Stochastic Thermodynamics: Short History and Perspectives

Carnot efficiency is the highest theoretically possible efficiency that a heat engine can have. Verley et al. use the fluctuation theorem to show that the Carnot value is the least likely efficiency in the long time limit.

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The unlikely Carnot efficiency

Using the fluctuation theorem supplemented with geometric arguments, we derive universal features of the (long-time) efficiency fluctuations for thermal and isothermal machines operating under steady or periodic driving, close or far from equilibrium. In particular, the probabilities for observing the reversible efficiency and the least likely efficiency are identical to those of the same machine working under the time-reversed driving. For time-symmetric drivings, this reversible and the least probable efficiency coincide.

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Universal theory of efficiency fluctuations.

We show that the scaled cumulant generating and large deviation function, associated to a two-state Markov process involving two processes, obey a symmetry relation reminiscent of the fluctuation theorem, independent from any conditions on the transition rates. The Legendre transform leading from the scaled cumulant generating function to the large deviation function is performed in an ingenious way, avoiding the sign problem associated to taking a square root. Applications to the theory of random walks and to the stochastic thermodynamics for a quantum dot are presented.

Fluctuation symmetry in a two-state Markov model

Detailed fluctuation theorems are statements about the probability distribution for the stochastic entropy production along a trajectory. It involves the consideration of a suitably transformed dynamics, such as the time reversed, the adjoint, or a combination of these. We identify specific, typically unique, initial conditions, called echo states, for which the final probability distribution of the transformed dynamics reproduces the initial distribution. In this case the detailed fluctuation theorems relate the stochastic entropy production of the direct process to that of the transformed one. We illustrate our results by an explicit analytical calculation and numerical simulations for a modulated two-state quantum dot.

Tim Willaert

Papers

The unlikely Carnot efficiency

Universal theory of efficiency fluctuations.

Fluctuation symmetry in a two-state Markov model

Echo states for detailed fluctuation theorems.

Fluctuation symmetry in a two-state Markov model