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.

Spectral fingerprints of non-equilibrium dynamics: The case of a Brownian gyrator

“…These expressions were also recently re-examined from a different perspective in Ref. [65]. We also remark that this latter setting, with constant forces applied on the BG, is mathematically equivalent to the one-dimensional beadspring model studied via Brownian-dynamics simulations in Ref.…”

“…This observation motivated extensive investigations of the BG model, see, e.g., Refs. [37][38][39][56][57][58][59][60][61][62][63][64][65][66]. In particular, by considering the response of this BG to external nonrandom forces it was possible to establish a non-trivial fluctuation theorem and to provide explicit expressions for the effective temperatures [63,64].…”

Frequency–frequency correlations of single-trajectory spectral densities of Gaussian processes

“…Various aspects of the BG dynamical behaviour and of its steady-state properties have been studied, see, e.g., [24,25] and [40][41][42][43][44][45][46][47][48][49][50][51]. In particular, a non-trivial fluctuation theorem was established by considering the response of the BG to external regular forces, what allowed to define explicitly an effective temperature [46][47][48]. We also remark that the setting with constant forces exerted on the BG is mathematically identical to the one-dimensional bead-spring model studied via Brownian-dynamics simulations in [52] and analytically in [53,54].…”

Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes

“…This explains the name of the model. Various aspects of the BG dynamical behaviour and of its steady-state properties have been studied, see, e.g., [24,25] and [39][40][41][42][43][44][45][46][47][48][49]. In particular, a non-trivial fluctuation theorem was established by considering the response of the BG to external regular forces, what allowed to define explicitly an effective temperatures [45][46][47].…”

“…Various aspects of the BG dynamical behaviour and of its steady-state properties have been studied, see, e.g., [24,25] and [39][40][41][42][43][44][45][46][47][48][49]. In particular, a non-trivial fluctuation theorem was established by considering the response of the BG to external regular forces, what allowed to define explicitly an effective temperatures [45][46][47]. We also remark that the setting with constant forces exerted on the BG is mathematically identical to the one-dimensional bead-spring model studied via Brownian-dynamics simulations in [50] and analytically in [51] and [52].…”