Question

The Boltzmann equation for evolution of $f_1(\vec{p}_1, \vec{q}_1, t)$ is given by $\left[\frac{\partial}{\partial t} - \frac{\partial U}{\partial \vec{q}_1} \cdot \frac{\partial}{\partial \vec{p}_1} + \frac{\vec{p}_1}{m} \cdot \frac{\partial}{\partial \vec{q}_1}\right] f_1(\vec{p}_1) = - \int d^3 \vec{p}_2 d^2 \vec{b} |\vec{v}_1 - \vec{v}_2| [f_1(\vec{p}_1) f_1(\vec{p}_2) - f_1(\vec{p}_1') f_1(\vec{p}_2')]$ where it is implicitly understood that $f_1$ is also function of $\vec{q}_1$ and t. Show that for $H(t)$ defined as $H(t) = \int d^3 \vec{p}_1 d^3 \vec{q}_1 f_1(\vec{p}_1, \vec{q}_1, t) \ln f_1(\vec{p}_1, \vec{q}_1, t)$, $dH/dt \le 0$ for evolution of $f_1(\vec{p}_1, \vec{q}_1, t)$ given by the Boltzmann equation.

          The Boltzmann equation for evolution of $f_1(\vec{p}_1, \vec{q}_1, t)$ is given by
$\left[\frac{\partial}{\partial t} - \frac{\partial U}{\partial \vec{q}_1} \cdot \frac{\partial}{\partial \vec{p}_1} + \frac{\vec{p}_1}{m} \cdot \frac{\partial}{\partial \vec{q}_1}\right] f_1(\vec{p}_1) = - \int d^3 \vec{p}_2 d^2 \vec{b} |\vec{v}_1 - \vec{v}_2| [f_1(\vec{p}_1) f_1(\vec{p}_2) - f_1(\vec{p}_1') f_1(\vec{p}_2')]$
where it is implicitly understood that $f_1$ is also function of $\vec{q}_1$ and t. Show that for $H(t)$ defined as
$H(t) = \int d^3 \vec{p}_1 d^3 \vec{q}_1 f_1(\vec{p}_1, \vec{q}_1, t) \ln f_1(\vec{p}_1, \vec{q}_1, t)$, 
$dH/dt \le 0$ for evolution of $f_1(\vec{p}_1, \vec{q}_1, t)$ given by the Boltzmann equation.

The Boltzmann equation for evolution of f1(p⃗1, q⃗1, t) is given by
[(∂)/(∂ t) - (∂ U)/(∂q⃗1)·(∂)/(∂p⃗1) + (p⃗1)/(m)·(∂)/(∂q⃗1)] f1(p⃗1) = - ∫ d^3 p⃗2 d^2 b⃗ |v⃗1 - v⃗2| [f1(p⃗1) f1(p⃗2) - f1(p⃗1') f1(p⃗2')]
where it is implicitly understood that f1 is also function of q⃗1 and t. Show that for H(t) defined as
H(t) = ∫ d^3 p⃗1 d^3 q⃗1 f1(p⃗1, q⃗1, t) ln f1(p⃗1, q⃗1, t),
dH/dt ≤ 0 for evolution of f1(p⃗1, q⃗1, t) given by the Boltzmann equation.

Added by Dolores M.

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