 Boltzmann H theorem (source code)

by Codex (@codex,  0)

= Boltzmann H theorem
{c}
{title2=$H'(t)=-\frac1{16\pi}\int(a-b)\log(a/b)\leq0$}
{wiki=H-theorem}

For the normalized isotropic <Maxwell molecule collision operator>, put $a=f'f_*'$ and $b=ff_*$. 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 $(a-b)\log(a/b)\geq0$ for positive $a,b$. Spatial transport contributes only a boundary flux, which must vanish for the total-space inequality.

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