Steady State Solution for a Rayleigh's Piston in a Temperature Gradient

International audienceAs a toy model for the thermodiffusion of polymers, we look at the Rayleigh piston problem in which a solute macroparticle is represented by a piston wall in a 1D canal. This piston wall is assumed to be adiabatic (without internal degrees of freedom) and fluctuates owing to collisions with the two gases or solvents that it separates.If the pressures in the two semi-infinite reservoirs are equal, i.e., even if there is macroscopic equilibrium, the system is out of equilibrium when the temperatures of the two semi-infinite reservoirs are different: the piston acquires a nonzero average velocity. This is due to the gradient of the chemical potential of the solvent, which acts as a generalized force on the solute (piston) and directs it to hot areas. This generalized force acts as a rectifier of the fluctuations of Brownian motion of the piston of finite mass.A pressure difference between the two semi-infinite reservoirs can be imposed to compensate this generalized force. The piston is then again at mechanical equilibrium. Even if the piston is adiabatic, because of its fluctuation in the temperature and pressure gradients, there is still heat transfer between the two gases, causing a flow of entropy that can be calculated using the fact that both the gases and the piston are at rest