Entropy production of a Rayleigh piston separating gases of different temperature

For thermodiffusion of polymers, it was shown that the transport coefficient has two contributions of different origins: first, dispersion forces between solvent and solute that grow it to cold areas, and secondly, a gradient of chemical potential of the solvent that acts as generalized force on the solute and promotes to hot areas. The competition of these opposite effects explains the variation with the molecular weight of polymer diffusion coefficient, which can change sign. As a toy model for this problem, we looked at the example of the Rayleigh piston: in a cylinder, a piston, assumed adiabatic (representing the solute as a macro particle without internal structure) fluctuates because of collisions with the two gases which it separates. Even if the pressures in the two semiinfinite reservoirs are equal, i.e., even if there is macroscopic equilibrium, when the temperatures are different, the system is out of equilibrium and acquires a nonzero average velocity, proportional to temperature gradient. The piston thus acts as a rectifier of fluctuations of Brownian motion. Therefore, there is a heat transfer between the two gases, causing a flow of entropy that we have calculated, together with the work of the ”generalized force”.