Fractional Advection Dispersion Equation and Boltzmann equation: Difference between pages

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The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/>) to better account for super-diffusive spreading of tracer particles in aquifers.  It has quite a few variations at this point, but a couple of characteristic examples are
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


\[
\begin{equation*}
u_t = c\frac{\partial u}{\partial x} + p\frac{\partial^\alpha u}{\partial x^\alpha} + q\frac{\partial^\alpha u}{\partial (-x)^\alpha} ,
\int_A f(x,v,t)dxdy
\]
\end{equation*}
where the fractional derivatives are defined as
\begin{equation}
\frac{\partial^{\alpha}}{\partial e^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x-ye)}{y}y^{-\alpha}dy\ \ \text{and}\ \ \frac{\partial^{\alpha}}{\partial (-e)^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x+ye)}{y}y^{-\alpha}dy .
\end{equation}
It is important to point out that these fractional derivatives are examples of one-dimensional [[Linear integro-differential operator | linear integro-differential operators]] with non-symmetric kernels (in this case, e.g. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).


then $f(x,v,t)$ solves the non-local equation


== See also ==
\begin{equation*}
\partial_t f + v \cdot \nabla_x f = Q(f,f)
\end{equation*}


* [[Drift-diffusion equations]]
where $Q(f,f)$ is the Boltzmann collision operator, given by


== References ==
\begin{equation*}
{{reflist|refs=
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_*
<ref name="MBB1999">{{Citation | last1=Meerschaert | first1=M.M. | last2=Benson | first2=D. J. | last3=Bäumer | first3=B. | title=Multidimensional advection and fractional dispersion | publisher=APS | year=1999 | journal=[[Physical Review|Physical Review E]] | issn=1539-3755 | volume=59 | issue=5 | pages=5026}}</ref>
\end{equation*}
<ref name="BMW">{{Citation | last1=Benson | first1=D. J. | last2=Wheatcraft | first2=S.W. | last3=Meerschaert | first3=M.M. | title=Application of a fractional advection-dispersion equation | year=2000 | journal=Water Resources Research | volume=36 | issue=6 | pages=1403–1412}}</ref>
 
<ref name="MBB2001">{{Citation | last1=Meerschaert | first1=M.M. | last2=Benson | first2=D. J. | last3=Baeumer | first3=B. | title=Operator Lévy motion and multiscaling anomalous diffusion | publisher=APS | year=2001 | journal=[[Physical Review|Physical Review E]] | issn=1539-3755 | volume=63 | issue=2 | pages=021112}}</ref>
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
<ref name="BMW2001">{{Citation | last1=Benson | first1=D. J. | last2=Schumer | first2=R. | last3=Meerschaert | first3=M.M. | last4=Wheatcraft | first4=S.W. | title=Fractional dispersion, Lévy motion, and the MADE tracer tests | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2001 | journal=Transport in Porous Media | volume=42 | issue=1 | pages=211–240}}</ref>
 
}}
\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.

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