Nonlocal minimal surfaces and Boltzmann equation: Difference between pages

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In broad and vague terms, these surfaces arise as the boundaries of domains $E \subset \mathbb{R}^n$ that are minimizers or critical points (within a class of given admissible configurations) of the energy functional:
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"/>


\[ J_s(E)= C_{n,s}\int_{E}\int_{E^c}\frac{1}{|x-y|^{n+s}}dxdy,\;\; s \in (0,1) \]
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 <ref name="Vil2002"/>. A basic reference is also <ref name="CerIllPul1994"/>.


It can be checked easily that this agrees (save for a factor of $2$) with  norm of the characteristic function $\chi_E$ in the homogenous Sobolev space  $\dot{H}^{\frac{s}{2}}$. The dimensional constant $C_{n,s}$ blows up as $s \to 1^-$, in which case (at least when the boundary of $E$ is smooth enough) one can check that $J_s(E)$ converges to the perimeter of $E$.
== The classical Boltzmann equation ==


Classically,  [[minimal surfaces]] (or generally [[surfaces of constant mean curvature]] ) arise in physical situations where one has two phases interacting (eg. water-air, water-ice ) and the energy of interaction is proportional to the area of the interface, which is due to the interaction between particles/agents in both phases being negligible when they are far apart.
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


Nonlocal minimal surfaces then describe physical phenomena where the interaction potential does not decay fast enough as particles get farther and farther apart, so that two particles on different phases contribute a non-trivial amount to the total interaction energy even if they are away from the interface. In particular, one may consider much more general energy functionals corresponding to different interaction potentials
\begin{equation*}
\int_A f(x,v,t) \, \mathrm d x \mathrm dv.
\end{equation*}


\[ J_K(E)= \int_{E}\int_{E^c}K(x,y) dxdy \]
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


== Definition ==
\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}


Following the most accepted convention for [[minimal surfaces]],  a (classical) nonlocal minimal surface is (given $s\in (0,1)$) the boundary $\Sigma$ of an open set $E \subset \mathbb{R}^n$ such that $\Sigma$ is at least $C^{1,s+\epsilon}$ and more importantly,  
where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by


\[ H_s(x): = -C_{n,s}\int_{\mathbb{R}^n} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+s}}dy=0 \;\;\forall\; x \in \Sigma\]
\begin{equation*}
Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) \bigg( f(v'_*)f(v') - f(v_*)f(v) \bigg) d\sigma(e) dv_*.
\end{equation*}


In this case we say that $\Sigma$ is a nonlocal minimal surface in $\Omega$. The quantity $H_s(x)$ is called the "Nonlocal mean curvature of order $s$ of $\Sigma$ at $x$", or briefly, "Nonlocal mean curvature".
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


Example: Suppose that $E$ and $\Omega$ are such that for any other set $F$ such that $F \Delta E \subset \subset \Omega$ (i.e. $F$ agrees with $E$ outside $\Omega$) we have
\begin{align*}
v'  & = v-(v-v_*,e)e\\
v'_* & = v_*+(v-v_*,e)e
\end{align*}


\[J_s(E) \leq J_s(F) \]
and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.


Then, if it is the case that $E$ has a smooth enough boundary, one can check that $E$ is a nonlocal minimal surface in $\Omega$.
The notation $f' = f(v')$, $f_\ast = f(v_\ast)$ and $f'_\ast = f(v'_\ast)$ is customary.


<div style="background:#DDEEFF;">
== Collision Invariants ==
<blockquote>
'''Note''' For this definition to make sense, $\Sigma$ must be the boundary of some open set $E$, in this article, we will often refer to the set $E$ itself as "the" minimal surface, and no confusion should arise from this.
</blockquote>
</div>


== Nonlocal mean curvature ==
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


The scalar quantity
\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*}


\[ H_s : \Sigma \to \mathbb{R} \]
and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have
\[ H_s (x) := -C_{n,s} \int_{\mathbb{R}^n}\frac{\chi_E(y)-\chi_{E^c(y)}}{|x-y|^{n+s}}dy \]


Is called the nonlocal mean curvature of $\Sigma$ (or $E$) at $x$, its a real valued function defined on $\Sigma$. Like the usual mean curvature, it measures in some average sense the deviation of $\Sigma$ from its tangent hyperplane at $x$ (note that if $\Sigma$ is a hyperplane, then trivially $H_s \equiv 0$).
\begin{equation*}
\frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0
\end{equation*}


Mean curvature vs Nonlocal mean curvature: Whereas the standard mean curvature measures ''mean deviation from flatness'' at the infinitesimal scale, the nonlocal mean curvature takes into account all scales (infinitesimal and positive scales).  Note further that for the mean curvature to be classically defined $\Sigma$ must be at least a $C^2$ surface, however, for $H_s$ one needs only a bit more than $C^{1,s}$ regularity.
according to what $\phi$ we pick this equation corresponds to  conservation of mass, conservation momentum or conservation of energy.


As the kernel is invariant under Euclidean symmetries, we conclude for instance that any sphere  $\partial B_r(x_0)$ has constant nonlocal mean curvature. Moreover, via a change of variables in the integral defining $H_s$  ($x \to x_0 \to rx$) one can see that a sphere of radius $r$ has mean curvature equal to $c_{n,s}r^{-s}$ (Note that $s=1$ gives the local mean curvature).
== Typical collision kernels ==


An alternative expression for the nonlocal mean curvature can be obtained via integration by parts (when $\Sigma$ is smooth enough). It is based on the identity
The collision kernel $B(v_*-v,e)$ should depend only on the distance $|v-v_*|$ and the deviation angle $\theta$ given by
\[ \cos \theta = \frac{(v - v_*) \cdot \sigma }{|v-v_*|}.\]
We abuse notation by writing $B(v_*-v,e) = B(|v_*-v|,\theta)$.


\[ \nabla_y \cdot \left ( \frac{y-x}{|y-x|^{n+s}} \right ) = \frac{n}{|x-y|^{n+s}}-(n+s) \frac{(y-x) \cdot (y-x) }{|x-y|^{n+s+2}} = -\frac{s}{|y-x|^{n+s}} \]
The case $B(r,\theta) = Kr$ is known as Maxwell molecules.


We use integration by parts (compute the integrals outside a ball $B_\epsilon(x)$ and then letting $\epsilon \to 0$). If $\nu$ denotes the ''outer normal'' to $\Sigma$, we get
From particle models interacting by inverse $s$-power force, the collision kernel has the form
\[ B(r,\theta) = r^\gamma \theta^{-d+1-\nu} b(\theta),\]
where
\begin{align*}
\gamma &= \frac{s - (2d - 1)}{s-1}, \\
\nu &= \frac 2 {s-1}, \\
\end{align*}
$b(\theta)$ is some positive bounded function which is not known explicitly.


\[ H_s(x) = -\frac{2C_{n,s}}{s}\int_{\Sigma} \frac{(x-y) \cdot \nu(y)}{|x-y|^{n+s}} d\sigma(y) \]
The operator $Q(f,f)$ may not make sense even for $f$ smooth if $\gamma$ is too negative or $\nu$ is too large. The operator does make sense if $\gamma \in (-d,0)$ and $\nu \in (0,2)$.


== Surfaces minimizing non-local energy functionals ==
In the case $s = 2d-1$, we obtain that the collision kernel $B(r,\theta)$ depends only on the angular variable $\theta$. This case is called ''Maxwellian molecules''.


A variational formulation of the Plateau problem for the nonlocal energy functional $J_s$ is as follows:
=== Grad's angular cutoff assumption ===


"Given a bounded Lipschitz domain $\Omega$ and a  (possibly unbounded) set $E_0$, minimize $J_s$ among all sets which agree with $E_0$ outside $\Omega$."
This assumption consists in using a collision kernel that is integrable in the angular variable $\theta$. That is
\[ \int_{S^{d-1}} B(r,e) \mathrm d \sigma(e) < +\infty \text{ for all values of } r. \]


That there exists a unique minimizer is always true by the Sobolev embedding and the lower-semicontinuity of $J_s$ with respect to $L^1$ convergence. Of course to carry the argument one needs to assume that there is at least on set $E$ such that $\chi_E \in H^s$ and $E$ agrees with $E_0$ outside $\Omega$ (namely $E \Delta E_0 \subset \Omega$).
The purpose of this assumption is to simplify the mathematical analysis of the equation. Note that for particles interaction by inverse power forces this assumption never holds.


In contrast to the perimeter functional, which is local, the functional Js(E) has the remarkable feature of  behaving like a quadratic form. In particular, one has the following identity
== Stationary solutions ==


\[ J_s(E) = L(E,E^c) \]
The Gaussian (or Maxweillian) distributions in terms of $v$, which are constant in $x$, are stationary solutions of the equation. That is, any function of the form
\[ f(t,x,v) = a e^{-b|v-v_0|^2}, \]
is a solution. In fact, for this function one can check that the integrand in the definition of $Q$ is identically zero since $f' \, f'_* = f \, f_*$.


where for any pair of measurable sets $A,B$ we define
== Entropy ==


\[ L(A,B) := C_{n,s}\int_A \int_B \frac{1}{|x-y|^{n+s}}dxdy = L(B,A) \]  
The following quantity is called the entropy and is monotone decreasing along the flow of the Boltzmann equation
\[ H(f) = \iint_{\R^d_x \times \R^d_v} f \ \log f \ \mathrm d v \, \mathrm d x. \]


Therefore, if we denote $A^- = E \setminus F, A^+= F\setminus E$ then
The derivative of the entropy is called ''entropy dissipation'' and is given by the expression
\[ D(f) = -\frac{\mathrm d H(f)}{\mathrm d t} = \frac 14 \iint_{\R^{2d}} \int_{S^{d-1}} B(v-v_\ast,\sigma) (f' f'_\ast - f f_\ast) \left( \log (f'f'_\ast) - \log(f f_\ast) \right) \, \mathrm d \sigma \, \mathrm d v \, \mathrm d v_\ast \geq 0. \]


\[ J_s(F) - J_s(E) = L(F,F^c) - L(E,E^c ) \]
Without Grad's angular cut-off assumption, and for a rather general family of cross-sections $B$, the entropy dissipation is bounded below by a fractional Sobolev norm minus a small correction <ref name="ADVW2000" />.
\[ D(f) \geq c_1 \|f\|_{H^{\nu/2}}^2 - c_2 \|f\|_{L^1_2}^2,\]
where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.


\[ = \left [ L(A^-,E \setminus A^-) -L(A^-,E^c) \right ]- \left [ L(A^+,E)-L(A^+,(E \cup A^+)^c) \right ] +2 L(A^-,A^+) \geq 0 \]
== The Landau Equation ==


== The Caffarelli-Roquejoffre-Savin Regularity Theorem==
A closely related evolution equation is the [[Landau equation]]. For Coulomb 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,


In (reference), a regularity theory for solutions of the above variational problem was developed. It draws inspiration from geometric measure theory, as developed by De Giorgi, Federer, Almgren and many others. In particular, the $C^{1,\alpha}$ regularity of minimal surfaces near flat points is proved ("improvement of flatness") as well as a first estimate on the dimensional set. The main result is as follows:
\begin{equation*}
f_t + x\cdot \nabla_y f = Q_{L}(f,f)
\end{equation*}


"Let $E$ be a minimal surface in $\Omega$, then there is a $C^{1,\alpha}$ submanifold $\partial^* E$ such that $\partial^*E \subset \partial E$ and $\partial E \setminus \partial^* E$ is a closed set of Hausdorff dimension at most $n-2$."
where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as


<div style="background:#DDEEFF;">
\begin{equation*}
<blockquote>
Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2
Open question: The dimensional estimate for the singular set $n-2$ is not expected to be sharp (at least when $s$ is not too small). It has been shown that if $s$ is close to $1$ then the estimate becomes $n-7$, which is the same (sharp) bound known for classical minimal surfaces. However, for nonlocal minimal surfaces it is known what the sharp bound is. Even more, there are no known explicit examples of singular cones which are nonlocal minimal surfaces.
\end{equation*}
</blockquote>
 
</div>
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).{{citation needed}}
 
== 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>
 
<ref name="ADVW2000">{{Citation | last1=Alexandre | first1= R. | last2=Desvillettes | first2= L. | last3=Villani | first3= C. | last4=Wennberg | first4= B. | title=Entropy dissipation and long-range interactions | url=http://dx.doi.org/10.1007/s002050000083 | journal=Arch. Ration. Mech. Anal. | issn=0003-9527 | year=2000 | volume=152 | pages=327--355 | doi=10.1007/s002050000083}}</ref>
}}

Revision as of 13:06, 26 December 2013

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) \, \mathrm d x \mathrm dv. \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) \bigg( f(v'_*)f(v') - f(v_*)f(v) \bigg) 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.

The notation $f' = f(v')$, $f_\ast = f(v_\ast)$ and $f'_\ast = f(v'_\ast)$ is customary.

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.

Typical collision kernels

The collision kernel $B(v_*-v,e)$ should depend only on the distance $|v-v_*|$ and the deviation angle $\theta$ given by \[ \cos \theta = \frac{(v - v_*) \cdot \sigma }{|v-v_*|}.\] We abuse notation by writing $B(v_*-v,e) = B(|v_*-v|,\theta)$.

The case $B(r,\theta) = Kr$ is known as Maxwell molecules.

From particle models interacting by inverse $s$-power force, the collision kernel has the form \[ B(r,\theta) = r^\gamma \theta^{-d+1-\nu} b(\theta),\] where \begin{align*} \gamma &= \frac{s - (2d - 1)}{s-1}, \\ \nu &= \frac 2 {s-1}, \\ \end{align*} $b(\theta)$ is some positive bounded function which is not known explicitly.

The operator $Q(f,f)$ may not make sense even for $f$ smooth if $\gamma$ is too negative or $\nu$ is too large. The operator does make sense if $\gamma \in (-d,0)$ and $\nu \in (0,2)$.

In the case $s = 2d-1$, we obtain that the collision kernel $B(r,\theta)$ depends only on the angular variable $\theta$. This case is called Maxwellian molecules.

Grad's angular cutoff assumption

This assumption consists in using a collision kernel that is integrable in the angular variable $\theta$. That is \[ \int_{S^{d-1}} B(r,e) \mathrm d \sigma(e) < +\infty \text{ for all values of } r. \]

The purpose of this assumption is to simplify the mathematical analysis of the equation. Note that for particles interaction by inverse power forces this assumption never holds.

Stationary solutions

The Gaussian (or Maxweillian) distributions in terms of $v$, which are constant in $x$, are stationary solutions of the equation. That is, any function of the form \[ f(t,x,v) = a e^{-b|v-v_0|^2}, \] is a solution. In fact, for this function one can check that the integrand in the definition of $Q$ is identically zero since $f' \, f'_* = f \, f_*$.

Entropy

The following quantity is called the entropy and is monotone decreasing along the flow of the Boltzmann equation \[ H(f) = \iint_{\R^d_x \times \R^d_v} f \ \log f \ \mathrm d v \, \mathrm d x. \]

The derivative of the entropy is called entropy dissipation and is given by the expression \[ D(f) = -\frac{\mathrm d H(f)}{\mathrm d t} = \frac 14 \iint_{\R^{2d}} \int_{S^{d-1}} B(v-v_\ast,\sigma) (f' f'_\ast - f f_\ast) \left( \log (f'f'_\ast) - \log(f f_\ast) \right) \, \mathrm d \sigma \, \mathrm d v \, \mathrm d v_\ast \geq 0. \]

Without Grad's angular cut-off assumption, and for a rather general family of cross-sections $B$, the entropy dissipation is bounded below by a fractional Sobolev norm minus a small correction [4]. \[ D(f) \geq c_1 \|f\|_{H^{\nu/2}}^2 - c_2 \|f\|_{L^1_2}^2,\] where $c_1$ and $c_2$ depend only on mass, entropy and energy of $f$.

The Landau Equation

A closely related evolution equation is the Landau equation. For Coulomb 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).[citation needed]

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 
  4. Alexandre, R.; Desvillettes, L.; Villani, C.; Wennberg, B. (2000), "Entropy dissipation and long-range interactions", Arch. Ration. Mech. Anal. 152: 327--355, doi:10.1007/s002050000083, ISSN 0003-9527, http://dx.doi.org/10.1007/s002050000083