Abstract
When do non-equilibrium forms of disordered energy qualify as heat? \textcolor{blue}{We address this question in the context of cyclically operating heat engines in contact with a non-equilibrium energy reservoir that defies the zeroth law of thermodynamics. To consistently address the latter as a heat bath requires the existence of a precise mapping to an equivalent cycle with an equilibrium bath at a time-dependent effective temperature. We identify the most general setup for which this can generically be ascertained and thoroughly discuss an analytically tractable, experimentally relevant scenario}: a Brownian particle confined in a \textcolor{blue}{periodically} modulated harmonic potential and coupled to some non-equilibrium bath of variable activity. We deduce formal limitations for its thermodynamic performance, including maximum efficiency, efficiency at maximum power, and maximum efficiency at fixed power. They can guide the design of new micro-machines and clarify how much these can outperform passive-bath designs, which has been a debated issue for recent experimental realizations. To illustrate the general principles for practical quasi-static and finite-rate protocols, we further analyze a specific realization of such an active heat engine based on the paradigmatic Active Brownian Particle (ABP) model. This reveals some non-intuitive features of the explicitly computed dynamical effective temperature, illustrates various conceptual and practical limitations of the effective-equilibrium mapping, and clarifies the operational relevance of various coarse-grained measures of dissipation.
Highlights
The study of heat engines is as old as the industrialization of the world
These parameters are used to extract work (“ordered” energy in the sense of the external handling) from the engine or to feed it from an external work source. Examples from this class of cyclic engines are various colloidal engines immersed in active fluids such as bacteria suspensions. For these machines, there is a well-defined regime, where energy extracted from the nonequilibrium bath and transformed to work can unambiguously and quantitatively be interpreted as heat—namely, if there exists a precise mapping to an equivalent setup with an equilibrium bath at a suitable time-dependent effective temperature Teff(t )
Active heat engines coupled to such nonequilibrium baths and Hamiltonians proportional to a single control parameter can always be reinterpreted in terms of equivalent engines in contact with equilibrium baths, at some dynamic effective temperature
Summary
Its practical importance has prompted physicists and engineers to persistently improve their experiments and theories to eventually establish the consistent theoretical framework of classical thermodynamics It allows to quantify very generally, on a phenomenological level, how work is transformed to heat, and to what extent this process can be reversed. The corresponding “active heat engines” utilizing such baths can outperform classical designs by evading the zeroth law of thermodynamics, which would require interacting degrees of freedom to mutually thermalize. Engines that exploit this unconventional property can operate between hugely different (effective) temperatures and thereby at unconventionally high efficiencies, without risking the evaporation or freezing of the laboratory. Various technical details have been deferred to an Appendix
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