The paradox of fourier heat equation: A theoretical refutation

The Fourier equation shows infinitesimal heat disturbances that propagate at an infinite speed. To suppress this paradox, a great number of non-Fourier heat conduction models were introduced. The Fourier equation is a macroscopic description at some macroscopic length scale L large with respect to the microscopic scale l, here the atomic scale. In the paper, we investigate the Fourier equation in the light of the underlying homogenization process between these two scales. The Fourier equation then appears to be valid within an approximation O(ϵ),ϵ=l/L≪1. That enables us to point out the inconsistency of the paradox. Finally, the above investigations suggest to pay attention to the eventual consequences of the approximative character of the Fourier equation and different other macroscopic models on their further upscaling to a larger scale.