The Problem of Engines in Statistical Physics

Abstract

Engines are open systems that can generate work cyclically at the expense of an external disequilibrium. They are ubiquitous in nature and technology, but the course of mathematical physics over the last 300 years has tended to make their dynamics in time a theoretical blind spot. This has hampered the usefulness of statistical mechanics applied to active systems, including living matter. We argue that recent advances in the theory of open quantum systems, coupled with renewed interest in understanding how active forces result from positive feedback between different macroscopic degrees of freedom in the presence of dissipation, point to a more realistic description of autonomous engines. We propose a general conceptualization of an engine that helps clarify the distinction between its heat and work outputs. Based on this, we show how the external loading force and the thermal noise may be incorporated into the relevant equations of motion. This modifies the usual Fokker–Planck and Langevin equations, offering a thermodynamically complete formulation of the irreversible dynamics of simple oscillating and rotating engines.

The Josephson junction as a quantum engine

“…Conversely, the working substance must be at a lower temperature (T d < 0) when it is more coupled to the colder bath (into which it rejects heat δQ < 0). Thus, in any heat engine there must be positive feedback between the working substance's instantaneous T d and its coupling to the external baths (which gives the instantaneous δQ) [18]. In [18] this general result is called the "Rayleigh-Eddington criterion".…”

“…Thus, in any heat engine there must be positive feedback between the working substance's instantaneous T d and its coupling to the external baths (which gives the instantaneous δQ) [18]. In [18] this general result is called the "Rayleigh-Eddington criterion". 2 If T d = 0 (as in the case of the JJ, where the Cooper pairs are always at the ambient temperature), then Eq.…”

“…Thus, we see that in a chemical engine (i.e., in an engine which runs on a disequilibrium of the external chemical potentials, rather than on a thermal disequilibrium) there must be a positive feedback such that, during the cycle of operation, matter enters the engine's working substance at a higher chemical potential (µ d > 0 when dN > 0) and exits at a lower chemical potential (µ d < 0 when dN < 0). In the models for the "putt-putt" steam engine presented in [18,37], as well as in the "electron shuttle" and the "Quincke rotor" described in [18] the engine's dynamics is formulated in terms of two coupled ordinary differential equations: a mechanical equation of motion (second order in time) for a macroscopic degree of freedom (the "tool"), and a kinetic equation (first order in time) for the amount of matter in the working substance. The kinetic equation is explicitly irreversible (i.e., non-invariant under t → −t).…”

“…The results of this paper will not depend on the detailed form of the g kk form factors in Eq. (18).…”

“…Papers [15,16] were inspired by this remark. [17] The present work pursues, in a quantum context, the program suggested long ago by Fabry and Le Corbeiller -and advanced more recently by some of the present authors-of describing self-oscillators as engines that generate work cyclically and irreversibly, at the expense of an external thermodynamic disequilibrium [8,18].…”