Asymmetry relations and effective temperatures for biased Brownian gyrators

TL;DR

This paper analyzes a two-dimensional Brownian gyrator model with external forces, deriving exact asymmetry relations for non-equilibrium steady states and exploring how effective temperatures differ between full and projected dynamics.

Contribution

It introduces generalized asymmetry relations for biased Brownian gyrators and investigates the non-thermodynamic effective temperatures in different projections of the system.

Findings

Derived exact asymmetry relations for the model.

Found that effective temperatures differ between full and projected dynamics.

Confirmed that these temperatures are not thermodynamic quantities.

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.