Talk:To Do List and Boltzmann equation: Difference between pages
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The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by | |||
\begin{equation*} | |||
\int_A f(x,v,t)dxdy | |||
\end{equation*} | |||
then $f(x,v,t)$ solves the non-local equation | |||
\begin{equation*} | |||
\partial_t f + v \cdot \nabla_x f = Q(f,f) | |||
\end{equation*} | |||
where $Q(f,f)$ is the Boltzmann collision operator, given by | |||
= | \begin{equation*} | ||
Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_* | |||
\end{equation*} | |||
where $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write | |||
\begin{align*} | |||
v' & = v-(v-v_*,e)e\\ | |||
v'_* & = v_*+(v-v_*,e)e | |||
\end{align*} | |||
where $B$, known as the Boltzmann collision kernel, measures the strength of collisions in different directions. | |||
Revision as of 10:42, 20 November 2012
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The Boltzmann equation is an evolution equation used to describe the configuration of particles in a gas, but only statistically. Specifically, if the probability that a particle in the gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by
\begin{equation*} \int_A f(x,v,t)dxdy \end{equation*}
then $f(x,v,t)$ solves the non-local equation
\begin{equation*} \partial_t f + v \cdot \nabla_x f = Q(f,f) \end{equation*}
where $Q(f,f)$ is the Boltzmann collision operator, given by
\begin{equation*} Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_* \end{equation*}
where $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write
\begin{align*} v' & = v-(v-v_*,e)e\\ v'_* & = v_*+(v-v_*,e)e \end{align*}
where $B$, known as the Boltzmann collision kernel, measures the strength of collisions in different directions.