Heat, temperature and Clausius inequality in a model for active Brownian particles

Abstract

Methods of stochastic thermodynamics and hydrodynamics are applied to a recently introduced model of active particles. The model consists of an overdamped particle subject to Gaussian coloured noise. Inspired by stochastic thermodynamics, we derive from the system’s Fokker-Planck equation the average exchanges of heat and work with the active bath and the associated entropy production. We show that a Clausius inequality holds, with the local (non-uniform) temperature of the active bath replacing the uniform temperature usually encountered in equilibrium systems. Furthermore, by restricting the dynamical space to the first velocity moments of the local distribution function we derive a hydrodynamic description where local pressure, kinetic temperature and internal heat fluxes appear and are consistent with the previous thermodynamic analysis. The procedure also shows under which conditions one obtains the unified coloured noise approximation (UCNA): such an approximation neglects the fast relaxation to the active bath and therefore yields detailed balance and zero entropy production. In the last part, by using multiple time-scale analysis, we provide a constructive method (alternative to UCNA) to determine the solution of the Kramers equation and go beyond the detailed balance condition determining negative entropy production.

Clausius Relation for Active Particles: What Can We Learn from Fluctuations

“…Recently it has been shown that the above fallacy is bypassed in a model of active particles where Gaussian colored noise plays the role of self-propulsion [19][20][21]. Even if thermal fluctuations are neglected, a notion of "coarse-grained heat" can be introduced, together with a spatial-dependent effective temperature, such that the original Clausius relation is fully recovered.…”

“…Based upon such a bare diffusivity, the "active bath temperature" T a = γD a is usually defined. In [21] a "mass-less" active temperature T τ = D a /τ was defined. In this paper we show that it is not necessary, if an "effective mass" µ = γτ is used, as in [23].…”

“…In order to simplify the discussion, the main discussion is focused on the 1d case with a time-independent potential, which is sufficient to show the main difference between the definitions of heat and entropy production in the three works considered [21][22][23]. The generalisation of the proposal in [21] to multi-dimensional cases with a time-dependent potential is also discussed at the end.…”

“…Apart from this comparison, two novelties are present here with respect to [21]: (1) a single trajectory level of description is adopted, while in [21] an ensemble average had been considered; (2) a generalization of [21] to more than one dimension and interacting particles is presented. In order to simplify the discussion, the main discussion is focused on the 1d case with a time-independent potential, which is sufficient to show the main difference between the definitions of heat and entropy production in the three works considered [21][22][23]. The generalisation of the proposal in [21] to multi-dimensional cases with a time-dependent potential is also discussed at the end.…”

“…In Section 3, we introduce the active model with Gaussian colored noise with thermal fluctuations, where a "microscopic Clausius relation" is trivially satisfied, and then show what happens at the coarse-grained level, when inertia and thermal fluctuations are neglected. The three recipes appeared in [21][22][23] to connect entropy production, dissipated heat and temperature are reviewed and compared.…”