The H-theorem, formulated by the physicist Ludwig Boltzmann in the context of statistical mechanics, provides a theoretical foundation for understanding the approach to thermodynamic equilibrium in a gas. The theorem states that, under certain conditions, the entropy of an isolated system will tend to increase over time, leading to a state of equilibrium.
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For the normalized isotropic Maxwell molecule collision operator, put and . The integral is over position, both velocities and collision direction. The symmetrized weak identity yields the displayed derivative of the Boltzmann H functional. It is nonpositive because for positive . Spatial transport contributes only a boundary flux, which must vanish for the total-space inequality.