Linear integro-differential operator and Boltzmann equation: Difference between pages

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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

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

[Bal2009-1] Bal, G. (2009), "Inverse transport theory and applications", Inverse Problems (IOP Publishing) 25 (5): 053001

[Vil2002-2] Villani, C. (2002), "A review of mathematical topics in collisional kinetic theory", Handbook of mathematical fluid dynamics (Elsevier) 1: 71–74

[CerIllPul1994-3] Cercignani, Carlo; Illner, R.; Pulvirenti, M. (1994), The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106), Berlin, New York: Springer-Verlag

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-The linear integro-differential operators that we consider ''in this wiki'' are the generators of [[Levy processes]]. According to the Levy-Kintchine formula, they have the general form
+{{stub}}
-\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
+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"/>
-where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
-\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
-The above definition is very general. In most cases we are interested in some subclass of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made.
+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"/>.
-== Absolutely continuous measure ==
+== The classical Boltzmann equation ==
-In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
+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
-We keep this assumption in all the examples below.
+\begin{equation*}
+\int_A f(x,v,t)dxdy.
+\end{equation*}
-== Purely integro-differential operator ==
+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
-In this case we neglect the local part of the operator
+\begin{equation}\label{eqn: Cauchy problem}\tag{1}
-\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]
+\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}
-== Symmetric kernels ==
+where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by
-If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.
-In the purely integro-differentiable case, it reads as
+\begin{equation*}
-\[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]
+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*}
-The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
+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
-== Translation invariant operators ==
+\begin{align*}
-In this case, all coefficients are independent of $x$.
+v'  & = v-(v-v_*,e)e\\
-\[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]
+v'_* & = v_*+(v-v_*,e)e
+\end{align*}
-== The fractional Laplacian ==
+and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.
-The [[fractional laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.
+== Collision Invariants ==
-\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
+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
-== Uniformly elliptic of order $s$ ==
+\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*}
-This corresponds to the assumption
+and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have
-\[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
-The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
+\begin{equation*}
+\frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0
+\end{equation*}
-An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.
+according to what $\phi$ we pick this equation corresponds to  conservation of mass, conservation momentum or conservation of energy.
-== Smoothness class $k$ of order $s$ ==
+== The Landau Equation ==
-This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded
-\[ D^r K(x,y) \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]
-== Order strictly below one ==
+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,
-If a non symmetric kernel $K$ satisfies the extra local integrability assumption
+\begin{equation*}
-\[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
+f_t + x\cdot \nabla_y f = Q_{L}(f,f)
-then the extra gradient term is not necessary in order to define the operator.
+\end{equation*}
-\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
+where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as
-The modification in the integro-differential part of the operator becomes an extra drift term.
+\begin{equation*}
+Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2
+\end{equation*}
-A uniformly elliptic operator of order $s<1$ satisfies this condition.
+where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.
-== Order strictly above one ==
-If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
+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).
-\[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
-then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
-\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
+== References ==
+{{reflist|refs=
-The modification in the integro-differential part of the operator becomes an extra drift term.
+<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>
-A uniformly elliptic operator of order $s>1$ satisfies this condition.
+<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>
-== Indexed by a matrix ==
+<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>
-In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
+}}
-\[ K_A(y) =  \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]