Reexamination of the Statistical Derivations of Fourier's Law and Cattaneo's Equation

Abstract

The derivation of Fourier's law from the Boltzmann transport equation (BTE) under the assumptions: (1) the relaxation time approximation with a velocity-independent relaxation time, (2) local equilibrium, and (3) steady state has been accepted and well understood for many years. Alternatively, Cattaneo's equation can be derived from the BTE by removing the third—that is, steady-state—assumption. Because fewer approximations are necessary, the latter would be more accurate for transient problems, and the reduction of Cattaneo's equation to Fourier's law is only applicable under certain conditions. In a number of investigations, the local equilibrium assumption is formulated in a relatively ill-defined manner. Though this avoids complicated formulation, the simple expression can be somewhat misleading and subsequently yields erroneous conclusions if not interpreted properly. The objective of this investigation is to reexamine the statistical derivations, with an emphasis on the local equilibrium assumption, and to then generalize the results to applications involving transient analysis.