Determination of boundary conditions in extended thermodynamics via fluctuation theory

A system of equations from extended thermodynamics is proposed for the calculation of the temperature field in a rarefied gas at rest. Numerical simulations suggest that the temperature field in this case should exhibit boundary layers superposed on the classical Fourier solution. In order to derive such a field from the extended thermodynamics of moments, one needs boundary values for higher moments, which in practice cannot be assigned and controlled. We suggest that such boundary data emerge as mean values of thermal fluctuations and thus calculate them. The result agrees qualitatively with numerical simulations and it is quantitatively of the same order of magnitude.