Operator monotone function and Boltzmann equation: Difference between pages

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A function $f : [0, \infty) \to [0, \infty)$ is said to be an operator monotone function (complete Bernstein function, Nevanlinna-Pick function for the half-line) if $A \ge B \ge 0$ implies $f(A) \ge f(B) \ge 0$ for any self-adjoint matrices $A$, $B$. Many equivalent definitions can be given.<ref name="SSV"/>
{{stub}}
 
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"/>
 
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"/>.


==Representation==
== The classical Boltzmann equation ==  
A function $f$ is operator monotone if and only if
\[ f(z) = a z + b + \int_{(0, \infty)} \frac{z}{z + r} \, \frac{\rho(\mathrm d r)}{r} \]
for some $a, b \ge 0$ and a Radon measure $\rho$ such that $\int_{(0, \infty)} \min(r^{-1}, r^{-2}) \rho(\mathrm d r) < \infty$.


==Examples==
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
The following functions are operator monotone:
* $z^s$ for $s \in [0, 1]$,
* $\log(1 + z)$,
* $\frac{z}{r + z}$ for $r > 0$.


==Properties==
\begin{equation*}
If $f, f_1, f_2$ are operator monotone, then the following functions are also operator monotone:
\int_A f(x,v,t)dxdy.
* $c_1 f_1(z) + c_2 f_2(z)$ for $c_1, c_2 > 0$,
\end{equation*}
* $((f_1(z))^s + (f_2(z))^s)^{1/s}$ and $(f_1(z^s) + f_2(z^s))^{1/s}$ for $s \in (0, 1]$,
* $(f_1(z))^s (f_2(z))^{1-s}$ and $f_1(z^s) f_2(z^{1-s})$ for $s \in [0, 1]$,
* $f_1(f_2(z))$,
* $z^{1-s} f(z^s)$ and $(f(z^s))^{1/s}$ for $s \in (0, 1]$,
* $z / f(z)$, $z f(1/z)$ and $1 / f(1/z)$.


==Relation to Bernstein functions==
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
Operator monotone functions form a subclass of [[Bernstein function]]s. Namely, a Bernstein function $f$ is an operator monotone function if and only if the measure $\mu$ in the Bernstein representation of $f$:
 
\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
\begin{equation}\label{eqn: Cauchy problem}\tag{1}
has a [[completely monotone function|completely monotone]] density function. In this case
\left \{ \begin{array}{rll}
\[ \mu(\mathrm d t) = \left( \int_{(0, \infty)} e^{-t r} \rho(\mathrm d r) \right) \mathrm d t \]
\partial_t f + v \cdot \nabla_x f & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+,\\
This explains the name complete Bernstein functions.
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


==Holomorphic extension==
Every operator monotone function $f$ extends to a holomorphic function on $\C \setminus (-\infty, 0]$ such that
\begin{align*}
\begin{align*}
\Im f(z) & \ge 0 \qquad && \text{if } \Im z \ge 0 , \\
v'  & = v-(v-v_*,e)e\\
f(z) & \ge 0 \qquad && \text{if } \Im z = 0 , \\
v'_* & = v_*+(v-v_*,e)e
\Im f(z) & \le 0 \qquad && \text{if } \Im z \le 0 .
\end{align*}
\end{align*}
Conversely, any function $f$ with above properties is an operator monotone function.


Functions with nonnegative imaginary part in the upper half-plane are often called Nevanlinna-Pick functions, or Pick functions.
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 $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.  


==Operator monotone functions of the Laplacian==
Operator monotone functions of the Laplacian are particularly regular examples of translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for an operator monotone $f$ if and only if
\[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + z) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \]
for some $a, b \ge 0$ and $k(z)$ of the form
\begin{align*}
k(z) &= \int_0^\infty \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} e^{-t r} \mathrm d t \rho(\mathrm d r) \\
&= \frac{1}{(2 \pi)^{n/2}} \int_0^\infty \left(\frac{\sqrt{r}}{|z|}\right)^{n/2 - 1} K_{n/2 - 1}(\sqrt{r} |z|) \rho(\mathrm d r) .
\end{align*}
Here $K_\nu$ is the modified Bessel function of the second kind.


For $n = 1$, the above expression simplifies to
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).
\[ k(z) = \int_0^\infty \frac{e^{-\sqrt{r} |z|}}{2 \sqrt{r}} \, \rho(\mathrm d r) ; \]
that is, $k(z)$ can be an arbitrary [[completely monotone function]] of $|z|$, which satisfies the ususal integrability condition $\int_{-\infty}^\infty \min(1, z^2) k(z) dz < \infty$. In a similar way, for $n = 3$,
\[ k(z) = \frac{1}{4 \pi |z|} \int_0^\infty e^{-\sqrt{r} |z|} \rho(\mathrm d r) ; \]
hence, $|z| k(z)$ can be an arbitrary [[completely monotone function]] of $|z|$, provided that $\int_{\R^3} \min(1, |z|^2) k(z) dz < \infty$.


==References==
== References ==
{{reflist|refs=
{{reflist|refs=
<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
 
<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>
}}
}}
{{stub}}

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