Methods of stochastic thermodynamics and hydrodynamics are applied to the arecently introduced model of active particles. The model consists of anoverdamped particle subject to Gaussian coloured noise. Inspired by stochasticthermodynamics, we derive from the system's Fokker-Planck equation the averageexchanges of heat and work with the active bath and the associated entropyproduction. We show that a Clausius inequality holds, with the local(non-uniform) temperature of the active bath replacing the uniform temperatureusually encountered in equilibrium systems. Furthermore, by restricting thedynamical space to the first velocity moments of the local distributionfunction we derive a hydrodynamic description where local pressure, kinetictemperature and internal heat fluxes appear and are consistent with theprevious thermodynamic analysis. The procedure also shows under whichconditions one obtains the unified coloured noise approximation (UCNA): such anapproximation neglects the fast relaxation to the active bath and thereforeyields detailed balance and zero entropy production. In the last part, by usingmultiple time-scale analysis, we provide a constructive method (alternative toUCNA) to determine the solution of the Kramers equation and go beyond thedetailed balance condition determining negative entropy production.
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