Boltzmann Grad limit - davidar/scholarpedia GitHub Wiki
The Boltzmann-Grad limit is the limiting procedure by which the Boltzmann equation of the kinetic theory of gases is deduced from the $N$-body Hamiltonian dynamics, i.e. from Newton's equations governing the dynamics of gas molecules. It is named after L. Boltzmann and H. Grad, who identified the scaling regime in which this derivation can be rigorously justified (Grad, 1949). While Newton's equations are a deterministic description for arbitrary mechanical systems, kinetic theory is a statistical model for the evolution of very large systems of identical microscopic constituents (such as neutral or ionized gases, plasmas...) Because of the very different nature of both theories, the limiting process relating them involves serious conceptual difficulties.
Consider a system of $N$ identical spherical particles with radius $r$. In the absence of external forces, each particle moves freely in the Euclidean space
$\mathbf{R}^3$, until it collides with another particle. Collisions between particles are assumed to be perfectly elastic, and the torque of each particle will
be considered negligible. Moreover, collisions involving three or more particles will also be neglected in the Boltzmann-Grad limit. Denoting by $x_i(t)$ and
$v_i(t)$ the position and velocity at time $t$ of the center of the $i$-th particle, the motion equations for this $N$ particle system are
\begin{equation}\label{NBodyEqNoColl}
\dot{x}_i(t)=v_i(t)\,,\qquad\dot{v}_i(t)=0\,,
\end{equation}
whenever $|x_i(t)-x_j(t)|>2r$ for all $j\not=i$, while
\begin{equation}\label{HardSphColl}
\begin{array}{ll}
&x_i(t+0)=x_i(t-0)\,,\qquad &v_i(t+0)=v_i(t-0)-(v_i(t-0)-v_j(t-0))\cdot n_{ji}(t)n_{ji}(t)\,,
\\
&x_j(t+0)=x_j(t-0)\,,\qquad &v_j(t+0)=v_j(t-0)-(v_j(t-0)-v_i(t-0))\cdot n_{ji}(t)n_{ji}(t)\,,
\end{array}
\end{equation}
whenever $|x_i(t)-x_j(t)|=2r$, with the notation
\begin{equation}
n_{ji}:=\frac{x_i-x_j}
\,.
\end{equation}
(See chapter 2 and chapter 4, section 2 in (Cercignani and al., 1994), and section 4.1 in (Gallagher and al. 2012).)
The set of all admissible positions for such an $N$-particle system is
\begin{equation}
\Omega^r_N:=\{(x_1,\ldots,x_N)\in(\mathbf{R^3})^N\hbox{ s.t. }|x_i-x_j|>2r\hbox{ for all }i\not=j\}\,.
\end{equation}
Denote by $\Gamma^r_N:=\Omega^r_N\times(\mathbf{R}^3)^N$ the $N$-particle phase-space, i.e. the set of all $N$-tuples of admissible positions and velocities of a system of $N$ spherical particles of radius $r$, and by $m_N$ the Lebesgue measure on $(\mathbf{R^3})^N\times(\mathbf{R^3})^N$, i.e. the volume element $dm_N(x_1,\ldots,x_N,v_1,\ldots,v_N):=dx_1\ldots dx_Ndv_1\ldots dv_N$.
Given $(x^{in}_1,\ldots,x^{in}_N,v^{in}_1,\ldots,v^{in}_N)\in\Gamma^r_N$, the solution of the system \ref{NBodyEqNoColl}-\ref{HardSphColl} with initial data $x_i(0)=x^{in}_i$ and $v_i(0)=v^{in}_i$ for $i=1,\ldots,N$ is henceforth denoted
\begin{equation}\label{DefSNrt}
(x_1(t),\ldots,x_N(t),v_1(t),\ldots,v_N(t))=:S_t^{N,r}(x^{in}_1,\ldots,x^{in}_N,v^{in}_1,\ldots,v^{in}_N)\,.
\end{equation}
In general, the transformation $S_t^{N,r}$ is not defined for all $t\in\mathbf{R}$ on the whole phase space $\Gamma^r_N$; however there exists a subset $E\subset\Gamma^r_N$ such that $m_N(E)=0$ and $S^{N,r}_t$ is a one-parameter group of transformations defined on $\Gamma^r_N\setminus E$ for all $t\in\mathbf{R}$: see Theorem 4.2.1 and Appendix 4.A in (Cercignani and al., 1994), Proposition 4.1.1 in (Gallagher and al., 2012), and Proposition 4.3 in (Alexander, 1976). Thus, for all $t\in\mathbf{R}$, the transformation $S^{N,r}_t$ maps $\Gamma^r_N\setminus E$ into itself and
\begin{equation}
S^{N,r}_{t+s}(x^{in}_1,\ldots,x^{in}_N,v^{in}_1,\ldots,v^{in}_N)=S^{N,r}_{t}(S^{N,r}_{s}(x^{in}_1,\ldots,x^{in}_N,v^{in}_1,\ldots,v^{in}_N))
\end{equation}
for all $t,s\in\mathbf{R}$ and $(x^{in}_1,\ldots,x^{in}_N,v^{in}_1,\ldots,v^{in}_N)\in\Gamma^r_N\setminus E$. (The $m_N$-negligible set $E$ includes in particular all initial data leading to triple collisions in finite time.)
The Boltzmann equation
The Boltzmann equation governs the evolution of ideal gases in kinetic theory. In kinetic theory, the state of a monatomic gas at time $t$ is described by its distribution function $f(t,x,v)\ge 0$, a measurable function that is the density of molecules located at the position $x$ with velocity $v$. Assuming that molecular interactions can be viewed as hard sphere collisions, the Boltzmann equation takes the form (see (Cercignani and al., 1994) on p. 31, or (Sone, 2007) on p. 3)
\begin{equation}\label{BoltzEq}
\frac{\partial f}{\partial t}+v\cdot\nabla_xf=\mathcal{C}(f)\,.
\end{equation}
The right-hand side in the Boltzmann equation is the Boltzmann collision integral, defined by the formula
$$
\mathcal{C}(f)(t,x,v)\!=\!\lambda\!\iint_{\mathbf{S}^2\times\mathbf{R}^3}\!(f(t,x,v')f(t,x,v_*')-f(t,x,v)f(t,x,v_*))((v-v_*)\cdot n)_+dv_*dn
$$
where $\lambda>0$ is some constant involving the molecular radius, and where
$$
v'\equiv v'(v,v_*,n):=v-(v-v_*)\cdot nn\,,\qquad v'_*\equiv v'(v,v_*,n):=v_*-(v_*-v)\cdot nn\,.
$$
Assuming that $f$ decays rapidly enough as $|x|,|v|\to\infty$, the collision integral satisfies (see chapter 3, sections 1 and 3 in (Cercignani and al., 1994), or (Sone, 2007) on p. 6)
$$
\int_{\mathbf{R}^3}\mathcal{C}(f)(t,x,v)dv=\int_{\mathbf{R}^3}\mathcal{C}(f)(t,x,v)v_jdv=\int_{\mathbf{R}^3}\mathcal{C}(f)(t,x,v)\tfrac12|v|^2dv=0
$$
for $j=1,2,3$, so that
$$
\frac{d}{dt}\iint_{\mathbf{R}^3\times\mathbf{R}^3}f(t,x,v)dxdv=\frac{d}{dt}\iint_{\mathbf{R}^3\times\mathbf{R}^3}v_j\,f(t,x,v)dxdv
=\frac{d}{dt}\iint_{\mathbf{R}^3\times\mathbf{R}^3}\tfrac12|v|^2f(t,x,v)dxdv=0\,.
$$
The first relation is equivalent to the conservation of the total number of gas molecules, while the second and third relations are equivalent to the conservation of the total momentum and kinetic energy of the gas molecules respectively. Without loss of generality, one can assume that $f$ is a probability density, i.e. that
\begin{equation}\label{ProbaNorma}
\iint_{\mathbf{R}^3\times\mathbf{R}^3}f(t,x,v)dxdv=1\,.
\end{equation}
Assuming that $f$ decays rapidly enough while $\ln f$ has at most polynomial growth as $|x|,|v|\to\infty$ (an example of such a function $f$ being $f(t,x,v)=e^{-|x|^2-|v|^2}$), the collision integral satisfies
$$
\int_{\mathbf{R}^3}\mathcal{C}(f)(t,x,v)\ln f(t,x,v)dv\le 0\,,
$$
with equality if and only if $f$ is a Maxwellian, i.e. is of the form ((Sone, 2009), pp. 6-7)
\begin{equation}\label{Maxwell}
x_i-x_j>2r\hbox{ unless }\{i,j\}=\{k,l\}\hbox{ and }x_k-x_l=2r\,,
\end{equation}
where $(v'_k,v'_l)$ are defined in terms of $(v_k,v_l)$ by the relations \ref{HardSphColl}. The partial differential equation \ref{Liouville} and the boundary condition \ref{BCLiouville} are completed with the initial condition
\begin{equation}\label{CILiouville}
F(0,x_1,\ldots,x_N,v_1,\ldots,v_N)=F^{in}(x_1,\ldots,x_N,v_1,\ldots,v_N)\,.
\end{equation}
It is assumed that the boundary condition and the initial condition are compatible. In other words, the initial data $F^{in}$ should verify the boundary condition \ref{BCLiouville}, i.e.
$$
F^{in}(x_1,\ldots,x_N,v_1,\ldots,v_N)=F^{in}(x_1,\ldots,x_N,v_1,\ldots,v_{k-1},v'_k,v_{k+1},\ldots,v_{l-1},v'_l,v_{l+1},\ldots,v_N)\,,
\quad x_i-x_j>2r\hbox{ unless }\{i,j\}=\{k,l\}\hbox{ and }x_k-x_l=2r\,,
$$
with the same notation as above.
The boundary condition \ref{BCLiouville} corresponds to the relations \ref{HardSphColl} describing hard sphere elastic collisions. In the Boltzmann-Grad limit, the interplay between the Liouville equation \ref{Liouville} and the boundary condition \ref{BCLiouville} leads to the Boltzmann collision integral $\mathcal{C}(f)$ on the right hand side of the Boltzmann equation. Thus, the boundary condition \ref{BCLiouville}, although localized on a Lebesgue-negligible set, is an essential feature of the derivation of the Boltzmann equation from the system of Newton's equations for the $N$-particle system.
It is therefore crucial that the $N$-particle distribution function $F$ satisfies \ref{BCLiouville} for all times. This can be proved by the following argument. Let $\mathfrak{H}_N=L^2(\Gamma^r_N;dm_N)$ be the Hilbert space of square-integrable functions defined a.e. on $\Gamma^r_N$, and denote by $A$ the advection operator
$$
AF= \sum_{k=1}^Nv_k\cdot\nabla_{x_k}F\,.
$$
Assume that $F^{in}\in\mathfrak{H}_N$. Specializing the equality \ref{LiouvThm} to the case where $\Phi(z)=z^2$ shows that $\tilde S^{N,r}_tF^{in}\in\mathfrak{H}_N$ for all $t\in\mathbf{R}$. Then $A\tilde S^{N,r}_tF^{in}\in\mathfrak{H}_N$ and $\tilde S^{N,r}_tF^{in}$ satisfies \ref{BCLiouville} for all $t\in\mathbf{R}$ if and only if $AF^{in}\in\mathfrak{H}_N$ and $F^{in}$ satisfies \ref{BCLiouville}. (This equivalence comes from the fact that $\tilde S^{N,r}_t$ is a $1$-parameter group of operators on $\mathfrak{H}_N$, whose generator is the advection operator $A$ with domain the set of $f\in\mathfrak{H}_N$ such that $Af\in\mathfrak{H}_N$ and $f$ satisfies \ref{BCLiouville}: see (Bardos, 1970) and Theorem 4 (i) on p. 421 in (Lax, 2002).)
The BBGKY hierarchy
The description of a gas by the Liouville equation \ref{Liouville} with its boundary condition \ref{BCLiouville} - equivalent to the transformation of
pre- to post-collision velocity pairs \ref{HardSphColl} - involves a number $N$ of gas molecules too large to be of any practical interest. For instance,
$N\simeq 6.02\cdot 10^{23}$ is needed in order to describe the state of $1$ gram of hydrogen. Therefore, one should look for some asymptotic limit of
\ref{Liouville}-\ref{BCLiouville}-\ref{CILiouville} as $N\to\infty$. Since the distribution function $F$ is a function of $6N+1$ variables, one should
also find a way to reduces the complexity of this description by retaining only finitely many variables in the large $N$ limit.
As explained above, in kinetic theory, the state of a gas at time $t$ is described by its $1$-particle distribution function $f(t,x,v)$, in other words, the
probability of finding, at time $t$, one gas molecule in a small volume $dx$ centered at the position $x$, with velocity in a small volume $dv$ around $v$,
is $f(t,x,v)dxdv$. All the information in the $N$-particle distribution function $F$ satisfying the Liouville equation is thus contained in the $1$-particle
distribution function $f$ after taking the Boltzmann-Grad limit.
A first ingredient in this reduction of the number of variables from $6N+1$ to $7$ (the number of variables in the $1$-particle distribution function $f$)
is the fact that the gas molecules are undistinguishable. Thus the $N$-particle distribution function $F$ satisfies the following symmetry relation
\begin{equation}\label{UndistF}
F(t,x_1,\ldots,x_N,v_1,\ldots,v_N)=F(t,x_{\sigma(1)},\ldots,x_{\sigma(N)},v_{\sigma(1)},\ldots,v_{\sigma(N)})
\end{equation}
for each $t\in\mathbf{R}$, each $(x_1,\ldots,x_N,v_1,\ldots,v_N)\in\Gamma^r_N$ and each permutation $\sigma$ of $\{1,\ldots,N\}$. If $F$ satisfies
the symmetry relation \ref{UndistF} at some instant of time $t_0$ - for instance at $t_0=0$ - and is a solution of \ref{Liouville}-\ref{BCLiouville},
then it satisfies the symmetry relation \ref{UndistF} for all times $t\in\mathbf{R}$. (See (Cercignani and al., 1994) on pp. 18-19.)
Henceforth, we shall use the following elements of notation: $X_N:=(x_1,\ldots,x_N)$ and $V_N:=(v_1,\ldots,v_N)$, while $X_{k,N}:=(x_k,\ldots,x_N)$
and $V_{k,N}:=(v_k,\ldots,v_N)$. If $F_N:\equiv F_N(t,X_N,V_N)$ is a nonnegative measurable function defined a.e. on $\mathbf{R}^{6N+1}$, one
defines
\begin{equation}\label{Marginal}
F_{N:k}(t,X_k,V_k):=\int F_N(t,X_N,V_N)dX_{k+1,N}dV_{k+1,N}
\end{equation}
for all $k$ such that $1\le k\le N$. If $k>N$, one sets $F_{N:k}\equiv 0$. Whenever $F_N$ is a nonnegative measurable function satisfying the symmetry \ref{UndistF}, $F_{N:k}$ is a nonnegative measurable function that also satisfies the symmetry relation \ref{UndistF} - where the number of particles is $k$ instead of $N$. When $F_N$ is a probability distribution, $F_{N:k}$ is referred to as the $k$-th marginal distribution of $F_N$ (see (Cercignani and al., 1994) on p. 26, (Bouchut and al., 2000) on p. 131, and section 4.2 in (Gallagher and al., 2012)).
Let $F$ be the solution of \ref{Liouville}-\ref{BCLiouville}-\ref{CILiouville}; denote by $F_N$ be the extension of $F$ by $0$ in the complement of
$\Gamma^r_N$:
\begin{equation}\label{DefFN}
F_N(t,X_N,V_N)=\left\{\begin{array}{ll}F(t,X_N,V_N)&\hbox{ if }X_N\in\Gamma^r_N\\ 0&\hbox{ if }X_N\notin\Gamma^r_N\end{array}\right.
\end{equation}
The function $F_N$ satisfies
\begin{equation}\label{LiouvilleDist}
\frac{\partial F_N}{\partial t}+\sum_{k=1}^Nv_k\cdot\nabla_{x_k}F_N
=\sum_{1\le i<j\le t}+v_1\cdot\nabla_{x_1}F_{N:1}\right)(t,x_1,v_1)
="(N-1)(2r)^2\int_{\mathbf{S}^2\times\mathbf{R}^3}F_{N:2}(t,x_1,x_1+2rn,v_1,v_2)(v_2-v_1)\cdot" $i="&amp;quot;1$&amp;quot;" $j="&amp;quot;3$:&quot;" \delta_<table></table>&="&quot;&amp;#10;\int&quot;" \delta_<table></table>\\
\displaystyle&="&quot;&amp;#10;&#45;\int&quot;" \delta_<table></table>\\
&="&quot;&amp;#10;\displaystyle\int&quot;" $x_2="&amp;quot;x_1+2rn$.&amp;#10;&amp;#10;Finally,&quot;" ndv_2dn
="&quot;&amp;#10;\int_&#123;(v_2&#45;v_1)\cdot&quot;&gt;&lt;/j\le&gt;0}F_{N:2}(t,x_1,x_1+2rn,v_1,v_2)(v_2-v_1)\cdot" t}+v_1\cdot\nabla_{x_1}F_{N:1}="(N-1)(2r)^2\mathcal{C}_N^{12}(F_{N:2})\,, \end{equation} where"
$$
\mathcal{C}_N^{12}(F_{N:2})(t,x_1,v_1):="\int_{\mathbf{S}^2\times\mathbf{R}^3}(F_{N:2}(t,x_1,x_1+2rn,T_{12}[n](v_1,v_2)) -F_{N:2}(t,x_1,x_1-2rn,v_1,v_2))((v_2-v_1)\cdot" n)_+dv_2dn
$$
and
$$
T_{12}[n](v_1,v_2):="(v_1-(v_1-v_2)\cdot" $k="&amp;quot;2,\ldots,N$.&quot;" t}+\sum_{j="1}^k\mathcal{C}_N^{j,k+1}(F_{N:k+1}) +\sum_{1\le" x_i-x_j="2r&#125;(v_j&#45;v_i)\cdot" n_{ij}F_{N:k}\Big_{\partial\Gamma^r_k}></table>\end{equation}
where
$$
\mathcal{C}_N^{j,k+1}(F_{N:k+1})(t,X_k,V_k):="&quot;\int_&#123;\mathbf&#123;S&#125;^2\times\mathbf&#123;R&#125;^3&#125;(F_&#123;N:k+1&#125;(t,X_k,x_k+2rn,T_&#123;j,k+1&#125;&#91;n&#93;V_&#123;k+1&#125;)&amp;#10;&#45;F_&#123;N:k+1&#125;(t,X_k,x_k&#45;2rn,V_&#123;k+1&#125;))((v_&#123;k+1&#125;&#45;v_j)\cdot&quot;" n)_+dv_{k+1}dn
$$
and
$$
T_{j,k+1}[n](V_{k+1}):="&quot;v_&#123;k+1&#125;&#45;(v_&#123;k+1&#125;&#45;v_j)\cdot&quot;" $F_&#123;N:N&#125;="&amp;quot;F_N$&amp;quot;" data
\begin{equation}\label{CGIn}
F(0,x,\omega,s,h)="&quot;f^&#123;in&#125;(x,\omega)\int_s^\infty\int_&#123;&#45;1&#125;^1P(\tau,hh)dhd\tau\,,&amp;#10;\end&#123;equation&#125;&amp;#10;where&quot;&gt;&lt;/j\le&gt;léquation" 383–392.
