Asymmetry relations and effective temperatures for biased Brownian gyrators

Abstract

We focus on a paradigmatic two-dimensional model of a nanoscale heat engine, -the so-called Brownian gyrator -whose stochastic dynamics is described by a pair of coupled Langevin equations with different temperature noise terms. This model is known to produce a curl-carrying non-equilibrium steady-state with persistent angular rotations. We generalize the original model introducing constant forces doing work on the gyrator, for which we derive exact asymmetry relations, that are reminiscent of the standard fluctuation relations. Unlike the latter, our relations concern instantaneous and not time averaged values of the observables of interest. We investigate the full two-dimensional dynamics as well as the dynamics projected on the x-and y-axes, so that information about the state of the system can be obtained from just a part of its degrees of freedom. Such a state is characterized by effective "temperatures" that can be measured in nanoscale devices, but do not have a thermodynamic nature. Remarkably, the effective temperatures appearing in full dynamics are distinctly different from the ones emerging in its projections, confirming that they are not thermodynamic quantities, although they precisely characterize the state of the system. While in the past statistical physics has been mainly devoted to the microscopic basis of the macroscopic behaviour, present day research is largely addressing "small" or non-thermodynamic systems, which either evolve spontaneously or are subjected to external drivings and constraints. Unlike macroscopic systems, that are by and large described by thermodynamics and linear response theory, these systems are still hard to be framed within a comprehensive theory.In fact, macroscopic observations amount to drastic projections from highly dimensional spaces to spaces of observables that consist of just a few dimensions. This is the reason why thermodynamics is so universal; those projections loose an enormous amount of information about the microscopic dynamics, hence the properties of observables only minimally depend on such microscopic details, provided a few conditions are met. Basically, it suffices that atomic forces are short ranged and repulsive. Then, universality as well as the equivalence of ensembles are established, and the resulting theory of macroscopic objects very generally holds. On the contrary, the behavior of systems made of a non-thermodynamic number of elementary constituents strongly depends on all defining parameters, and a theory as widely applicable as thermodynamics can hardly be envisaged.Nevertheless, one common facet of such systems is that their observables undergo non-negligible fluctuations. Therefore, the seminal paper [1] represents a pioneering attempt towards a unified theory of fluctuating phenomena [2,3]. Its chief result is called Fluctuation Relation (FR), and it constitutes one of the first exact results obtained for systems which are almost arbitrarily far from equilibrium. Close to equilibrium the FR reproduces the Green-Kubo and Onsager re...

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