Thermodynamic asymmetries in dual-temperature Brownian dynamics

Abstract

Recent work by Cerasoli et al (2018 Phys. Rev. E 98 042149) on a two-dimensional model of biased Brownian gyrators driven in part by temperature differences along distinct Cartesian axes, x and y, has revealed interesting asymmetries in the steady-state distribution of particle positions. These asymmetries are said to be reminiscent of the more conventional asymmetries associated with the fluctuation theorems of far-from-equilibrium thermodynamics. In the present paper, working within a path integral formalism, we derive the exact time-dependent propagator of this same 2D dual-temperature system, and show that it does in fact also satisfy several conventional fluctuation theorems, including the Crooks relation, the Jarzynski equality, the detailed fluctuation theorem, and the integral fluctuation theorem. For these theorems to be satisfied, however, we find that a parameter that we identify as an ‘effective temperature’ must bear a definite relation to the two temperatures that control particle dynamics in the x and y directions. This effective temperature turns out to be the harmonic mean of two analogous temperatures introduced by Cerasoli et al.