Flow of Energy in Thermal Transpiration for a Bose-Einstein and a Fermi-Dirac Gas

In the case of two chambers maintained at different temperatures containing an ideal gas (classical $\beta =0\mathrm{}$, Fermi-Dirac $\beta =-1\mathrm{}$, or Bose-Einstein $\beta =+1\mathrm{}$) and intercommunicating through an effusion-orifice, there is, in the steady state brought about by thermal transpiration, no net flow of matter but a continuous flow of energy (due to effusion of particles) from the chamber at the higher temperature to the other one. This energy flow is determined for a completely relativistic and a completely non-relativistic gas. Three possible cases are discussed for each of them: (a) gas non-degenerate in both chambers, (b) gas degenerate in both chambers, and (c) gas non-degenerate in one chamber and degenerate in the other. Case (b) is possible for a F.-D. gas only. In case (c) the rate of flow of energy to a first order is the same as in case (a) but with the temperature of the chamber at the lower temperature (i.e., the chamber containing the degenerate gas) put equal to zero, which is in accord with the properties of a degenerate gas. In case (a) for a relativistic classical gas ($\beta =0\mathrm{}$) the energy flow is proportional to the concentration $n$ which is the same in both the chambers and to the difference of temperature between them. For a non-relativistic classical gas the energy flow varies as $n_{1}T_1^{}{}_{}{}^{\frac{1}{2}}(T_1-T_2)$, $n_1$ being the concentration in the chamber at temperature $T_1$. For a F.-D. gas the flow is slightly less and for a B.-E. gas slightly greater than that for the classical gas under the same conditions. In case (b) when there is F.-D. degenerate gas in the two chambers, the energy flow to a first order both for the relativistic and non-relativistic cases, varies as $n_1^{}{}_{}{}^{\frac{2}{3}}(T_1^{}{}_{}{}^2-T_2^{}{}_{}{}^2)$; in the relativistic case, of course, $n_1=n_2$. For fixed temperatures, the energy flow is always greater when the concentrations are such that the gas in the two chambers is degenerate than when the gas in both of them is non-degenerate.