Levy processes and Boltzmann equation: Difference between pages

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A Lévy process is an important type of [[stochastic process]] (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$).  In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations.
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Informally speaking, a Lévy process is a random trajectory, generalizing the concept of Brownian motion, which may contain jump discontinuities. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard [[Brownian motion]] and $N(t)$ is a [[Compound Poisson process]], the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths.
The Boltzmann equation is a nonlinear evolution equation first put forward by Ludwig Boltzmann to describe the configuration of particles in a gas, but only statistically. However, this equation and related equations are used in other physical situations, such as in optics. The corresponding linear inverse problem is also used in tomography <ref name="Bal2009"/>


== Definition ==
In reality, the Boltzmann equation is not a single equation but a family of equations, where one obtains different equations depending on the nature of the interaction between particles (see below). Although there has been a lot of progress in the analysis of the Cauchy problem under many circumstances, the broad understanding of the equation and the dynamics of its solutions remains largely incomplete. For an overview of the mathematical issues revolving around this equation see for instance <ref name="Vil2002"/>. A basic reference is also <ref name="CerIllPul1994"/>.


A stochastic process $X=\{X(t)\}_{t \geq 0}$ with values in $\mathbb{R}^d$ is said to be a Lévy process if
== The classical Boltzmann equation ==


1.For any sequence $0 \leq t_1 < t_2 <...<t_n$ the random variables $X(t_0),X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})$ are independent.
As explained originally by Boltzmann in the probabilistic description of a gas, we assume that the probability that a particle in a gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by some function


2.For any positive times $s\leq t$ the random variables $X(t-s)$ and $X(t)-X(s)$ have the same probability law.
\begin{equation*}
\int_A f(x,v,t)dxdy.
\end{equation*}


3.Almost surely, the trajectory of $X(t)$ is continuous from the right, with limit from the left also known as "càdlàg" for its acronym in french.
Then, under certain (natural) physical assumptions, Boltzmann derived an evolution equation for $f(x,v,t)$. In particular, if one imposes $f$ at time $t=0$ then $f$ should  solve the Cauchy problem


== Lévy-Khintchine Formula ==
\begin{equation}\label{eqn: Cauchy problem}\tag{1}
\left \{ \begin{array}{rll}
\partial_t f + v \cdot \nabla_x f  & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+,\\
f  & = f_0 & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \{ 0 \}.
\end{array}\right.
\end{equation}


It follows from the first two properties above that if $X$ is a Lévy process and we further assume $X(0)=0$ a.s. then for each fixed positive $t$ the random variable $X(t)$ is infinitely divisible, that is, it can be written as the sum of $n$ independent and identically distribued random variables, for all $n\in\mathbb{N}$. Indeed, let $h=\tfrac{t}{n}$, then
where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by


\[X(t) = \left( X(h)-X(0)\right)+\left( X(2h)-X(h)\right)+...+\left( X(t)-X((n-1)h)\right)\]
\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*}


and by the above definition the differences $X(kh)-X((k-1)h)$ are independent and distributed the same as $X(h)$. From the infinite divisibility of $X(t)$ it follows by a theorem of Lévy and Khintchine that  for any $\xi \in \mathbb{R}^d$ we have
here $d\sigma$ 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


\[ \mathbb{E} \left [ e^{i\xi\cdot X_t}\right ] = e^{t\eta(\xi)}\]
\begin{align*}
v'  & = v-(v-v_*,e)e\\
v'_* & = v_*+(v-v_*,e)e
\end{align*}


the function $\eta(\xi)$ given by
and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.


\[\eta(\xi)=i y\cdot b  -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{i \xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y)  \]
== Collision Invariants ==


where $b$ is a vector, $A$ is a positive matrix, $B_1$ is the unit ball and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that
The Cauchy problem \ref{eqn: Cauchy problem} enjoys several conservation laws, which in the Boltzmann literature are known as collision invariants. Take $\phi(v)$ to be any of the following functions


\[ \int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty. \]
\begin{equation*}
\phi(v) = 1, \;\;v,\;\; \tfrac{|v|^2}{2}
\end{equation*}
\begin{equation*}
\text{(the first and  third ones are real valued functions, the second one is vector valued)}
\end{equation*}


The interpretation of this measure $\mu$ is that ''jumps'' from some point $x$ to $x+y$ with $y$ in some set $A$ occur as a Poisson process with intensity $\mu(A)$.
and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have


== Connection with linear integro-differential operators ==
\begin{equation*}
\frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0
\end{equation*}


Any Lévy process $X(t)$ such that $X(0)=0$ defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows
according to what $\phi$ we pick this equation corresponds to conservation of mass, conservation momentum or conservation of energy.


\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]\]
== The Landau Equation ==


Given the initial assumption on $X(0)$ it is clear that $U_0$ is the identity, and given that $X(t)-X(s)$ is distributed as $X(t-s)$ it follows that $U_t \circ U_s = U_{t+s}$.
A closely related evolution equation is the [[Landau equation]]. For Coulumb interactions, the corresponding collision kernel $B$ always diverges, instead in this case, one uses an equation (which is an asymptotic limit of Boltzmann equation) first derived by Landau,


As a semigroup, $U_t$ has an infinitesimal generator which turns out to be a [[Linear integro-differential operator]]. More precisely, if we let $f(x,t):=(U_tf)(x)$, then, assuming that $f(x,t)$ has enough regularity it can be checked that
\begin{equation*}
f_t + x\cdot \nabla_y f = Q_{L}(f,f)
\end{equation*}


\[\partial_t f = Lf \;\;\;\mbox{ for all } (x,t)\in\mathbb{R}^d\times \mathbb{R}_+\]
where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as


where for any smooth function $\phi$, we have
\begin{equation*}
Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2
\end{equation*}


\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]
where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.


== [[Stochastic control]] and fully non-linear integro-differential operators ==


A similar connection holds via the [[Isaacs equation|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of [[viscosity solutions]] to build it.
Note that when $f$ is independent of $x$ the above equation becomes second-order parabolic equation where the coefficients depend non-locally on $f$, in particular, one has an apriori estimate for all higher derivatives of $f$ in terms of its $L^\infty$ and $L^1$ norms (via a bootstrapping argument).
 
== References ==
{{reflist|refs=
 
<ref name="Bal2009">{{Citation | last1=Bal | first1=G. | title=Inverse transport theory and applications | publisher=IOP Publishing | year=2009 | journal=Inverse Problems | volume=25 | issue=5 | pages=053001}}</ref>
 
<ref name="Vil2002">{{Citation | last1=Villani | first1=C. | title=A review of mathematical topics in collisional kinetic theory | publisher=[[Elsevier]] | year=2002 | journal=Handbook of mathematical fluid dynamics | volume=1 | pages=71–74}}</ref>
 
<ref name="CerIllPul1994">{{Citation | last1=Cercignani | first1=Carlo | last2=Illner | first2=R. | last3=Pulvirenti | first3=M. | title=The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106) | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1994}}</ref>
}}

Revision as of 20:11, 18 October 2013

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The Boltzmann equation is a nonlinear evolution equation first put forward by Ludwig Boltzmann to describe the configuration of particles in a gas, but only statistically. However, this equation and related equations are used in other physical situations, such as in optics. The corresponding linear inverse problem is also used in tomography [1]

In reality, the Boltzmann equation is not a single equation but a family of equations, where one obtains different equations depending on the nature of the interaction between particles (see below). Although there has been a lot of progress in the analysis of the Cauchy problem under many circumstances, the broad understanding of the equation and the dynamics of its solutions remains largely incomplete. For an overview of the mathematical issues revolving around this equation see for instance [2]. A basic reference is also [3].

The classical Boltzmann equation

As explained originally by Boltzmann in the probabilistic description of a gas, we assume that the probability that a particle in a gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by some function

\begin{equation*} \int_A f(x,v,t)dxdy. \end{equation*}

Then, under certain (natural) physical assumptions, Boltzmann derived an evolution equation for $f(x,v,t)$. In particular, if one imposes $f$ at time $t=0$ then $f$ should solve the Cauchy problem

\begin{equation}\label{eqn: Cauchy problem}\tag{1} \left \{ \begin{array}{rll} \partial_t f + v \cdot \nabla_x f & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+,\\ f & = f_0 & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \{ 0 \}. \end{array}\right. \end{equation}

where $Q(f,f)$ is the Boltzmann collision operator, a non-local 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*}

here $d\sigma$ 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*}

and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.

Collision Invariants

The Cauchy problem \ref{eqn: Cauchy problem} enjoys several conservation laws, which in the Boltzmann literature are known as collision invariants. Take $\phi(v)$ to be any of the following functions

\begin{equation*} \phi(v) = 1, \;\;v,\;\; \tfrac{|v|^2}{2} \end{equation*} \begin{equation*} \text{(the first and third ones are real valued functions, the second one is vector valued)} \end{equation*}

and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have

\begin{equation*} \frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0 \end{equation*}

according to what $\phi$ we pick this equation corresponds to conservation of mass, conservation momentum or conservation of energy.

The Landau Equation

A closely related evolution equation is the Landau equation. For Coulumb interactions, the corresponding collision kernel $B$ always diverges, instead in this case, one uses an equation (which is an asymptotic limit of Boltzmann equation) first derived by Landau,

\begin{equation*} f_t + x\cdot \nabla_y f = Q_{L}(f,f) \end{equation*}

where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as

\begin{equation*} Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2 \end{equation*}

where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.


Note that when $f$ is independent of $x$ the above equation becomes second-order parabolic equation where the coefficients depend non-locally on $f$, in particular, one has an apriori estimate for all higher derivatives of $f$ in terms of its $L^\infty$ and $L^1$ norms (via a bootstrapping argument).

References

  1. Bal, G. (2009), "Inverse transport theory and applications", Inverse Problems (IOP Publishing) 25 (5): 053001 
  2. Villani, C. (2002), "A review of mathematical topics in collisional kinetic theory", Handbook of mathematical fluid dynamics (Elsevier) 1: 71–74 
  3. Cercignani, Carlo; Illner, R.; Pulvirenti, M. (1994), The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106), Berlin, New York: Springer-Verlag