Abstract

This article introduces an extension of the H theorem to an arbitrary order d ≥1 . The extended H theorem defines the extended entropy Hd  for a given distribution and identifies the distribution function f_d(v) that maximizes entropy H_d . When d=1,  H_1  corresponds to the original entropy defined by Boltzmann, and the distribution that maximizes it is the Boltzmann distribution. For d=2 , the maximizing distribution is the Rayleigh distribution. For d=3 , the maximizing distribution is the Maxwell distribution. Therefore, in the three-dimensional physical world, the correct speed distribution of gas particles is the Maxwell distribution, which maximizes the extended entropy H_3  rather than the original entropy H_1 . The result distribution f_d(v)  that maximizes the extended entropy H_d  for any d≥1  is derived and applied to gases that include rotational and strain energy in addition to translational energy. Numerical validation supporting these findings is also presented.

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