Boltzmann-Grad limit
| Francois Golse (2013), Scholarpedia, 8(10):9141. | doi:10.4249/scholarpedia.9141 | revision #143362 [link to/cite this article] |
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.
The $N$-body problem of classical mechanics
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}\tag{1} \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}\tag{2} \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}{|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 (1)-(2) 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}\tag{3} (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}\tag{4} \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}\tag{5} \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}\tag{6} f(t,x,v)=\mathcal{M}_{(\rho(t,x),u(t,x),\theta(t,x))}(v):=\frac{\rho(t,x)}{(2\pi\theta(t,x))^{3/2}}e^{-\frac{|v-u(t,x)|^2}{2\theta(t,x)}} \end{equation} for some $\rho(t,x),\theta(t,x)>0$ and $u(t,x)\in\mathbf{R}^3$. (See chapter 3, sections 2 and 4 in (Cercignani and al., 1994), (Bouchut and al., 2000) on pp. 46-49, especially Theorem 2.1 and Lemma 2.2, and section 2.2 in (Gallagher and al., 2012) .)
In particular \begin{equation}\tag{7} H(f)(t):=\iint_{\mathbf{R}^3\times\mathbf{R}^3}f(t,x,v)\ln f(t,x,v)dxdv\hbox{ is a nonincreasing function,} \end{equation} a statement known as Boltzmann's H Theorem.
The Boltzmann-Grad limit is the mathematical procedure by which the Boltzmann equation (4) can be derived from the $N$-body problem of classical mechanics (1)-(2).
The Liouville equation
In order to predict the evolution of some amount of gas (such as $1$ gram of hydrogen), it is obviously irrelevant to know the exact positions and velocities of each gas molecule at any instant of time. Thus, the system of Newton's equations (1)-(2) is replaced with the following statistical description (see for instance (Cercignani and al., 1994) on pp. 13-18, or sections 1.1 and 4.2 in (Gallagher and all, 2012)).
First, the measure $m_N$ (the $N$-particle phase space volume element) is invariant under the $1$-parameter group $S^{N,r}_t$: $$ m_N(S^{N,r}_t(A))=m_N(A)\,,\quad\hbox{ for each measurable }A\subset\Gamma^r_N\hbox{ and }t\in\mathbf{R}\,. $$ (Indeed, on each interval of time without collisions, the velocity of each particle remains constant while its position is transformed by a translation; whenever there is a collision involving a particle pair, the position of each particle is the same before and after the collision, while (2) shows that the transformation from pre- to post-collision velocity pairs is a linear isometry in $\mathbf{R}^3\times\mathbf{R}^3$.)
In particular, $S^{N,r}_t$ maps $m_N$-negligible sets on $m_N$-negligible sets. Therefore, for each nonnegative measurable function $F^{in}$ defined $m_N$-a.e. on $\Gamma^r_N$, the formula \begin{equation}\tag{8} \tilde S^{N,r}_tF^{in}(x_1,\ldots,x_N,v_1,\ldots,v_N):=F^{in}(S^{N,r}_{-t}(x_1,\ldots,x_N,v_1,\ldots,v_N)) \end{equation} defines a nonnegative measurable function $\tilde S^{N,r}_tF^{in}$ a.e. on $\Gamma^r_N$ for all $t\in\mathbf{R}$, satisfying \begin{equation}\tag{9} \int_{\Gamma^r_N}\Phi(\tilde S^{N,r}_tF^{in})dm_N=\int_{\Gamma^r_N}\Phi(F^{in})dm_N \end{equation} for each continuous $\Phi:\,[0,\infty)\to[0,\infty)$.
The most important example is the case where $F^{in}$ is the $N$-particle distribution function of the gas at time $t=0$. This distribution function is defined as follows: the infinitesimal quantity $F^{in}(x_1,\ldots,x_N,v_1,\ldots,v_N)dx_1\ldots dx_Ndv_1\ldots dv_N$ is the joint probability that, at time $t=0$, the first particle is located in a volume $dx_1$ around $x_1$ with velocity in a volume $dv_1$ around $v_1$, the second particle is located in a volume $dx_2$ around $x_2$ with velocity in a volume $dv_2$ around $v_2$ and so on until the $N$-th particle.
In that case, the function $F$ defined by $F(t,\cdot)=\tilde S^{N,r}_tF^{in}$, i.e. $$ F(t,x_1,\ldots,x_N,v_1,\ldots,v_N):=F^{in}(S^{N,r}_{-t}(x_1,\ldots,x_N,v_1,\ldots,v_N)) $$ is the $N$-particle distribution function of the gas at time $t$. This distribution function satisfies the Liouville equation \begin{equation}\tag{10} \left(\frac{\partial F}{\partial t}+\sum_{k=1}^Nv_k\cdot\nabla_{x_k}F\right)(t,x_1,\ldots,x_N,v_1,\ldots,v_N)=0\,, \qquad(x_1,\ldots,x_N,v_1,\ldots,v_N)\in\Gamma^r_N\,,\quad t\in\mathbf{R} \end{equation} in the sense of distributions, with the boundary condition on $\partial\Gamma^r_N$ \begin{equation}\tag{11} F(t,x_1,\ldots,x_N,v_1,\ldots,v_N)=F(t,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\,, \end{equation} where $(v'_k,v'_l)$ are defined in terms of $(v_k,v_l)$ by the relations (2). The partial differential equation (10) and the boundary condition (11) are completed with the initial condition \begin{equation}\tag{12} 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 (11), 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 (11) corresponds to the relations (2) describing hard sphere elastic collisions. In the Boltzmann-Grad limit, the interplay between the Liouville equation (10) and the boundary condition (11) leads to the Boltzmann collision integral $\mathcal{C}(f)$ on the right hand side of the Boltzmann equation. Thus, the boundary condition (11), 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 (11) 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 (9) 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 (11) for all $t\in\mathbf{R}$ if and only if $AF^{in}\in\mathfrak{H}_N$ and $F^{in}$ satisfies (11). (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 (11): 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 (10) with its boundary condition (11) - equivalent to the transformation of pre- to post-collision velocity pairs (2) - 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 (10)-(11)-(12) 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}\tag{13} 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 (13) at some instant of time $t_0$ - for instance at $t_0=0$ - and is a solution of (10)-(11), then it satisfies the symmetry relation (13) 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}\tag{14} 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 (13), $F_{N:k}$ is a nonnegative measurable function that also satisfies the symmetry relation (13) - 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 (10)-(11)-(12); denote by $F_N$ be the extension of $F$ by $0$ in the complement of $\Gamma^r_N$: \begin{equation}\tag{15} 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}\tag{16} \frac{\partial F_N}{\partial t}+\sum_{k=1}^Nv_k\cdot\nabla_{x_k}F_N =\sum_{1\le i<j\le N}\delta_{|x_i-x_j|=2r}(v_j-v_i)\cdot n_{ij}F_N\Big|_{\partial\Gamma^r_N} \end{equation} in the sense of distributions on $\mathbf{R}^{6N+1}$. The Dirac notation $\delta_{|x-x_0|=2r}$ on the right hand side of (16) designates the surface measure concentrated on the sphere of radius $2r$ centered at $x_0$. Integrating both sides of this equality in $x_2,\ldots,x_N, v_2,\ldots,v_N$, one obtains a equation for $F_{N:1}$: \begin{equation}\tag{17} \left(\frac{\partial F_{N:1}}{\partial 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 ndv_2dn\,. \end{equation}
Indeed, whenever $2\le i<j\le N$ $$ \int \delta_{|x_i-x_j|=2r}(v_j-v_i)\cdot n_{ij}F_N(t,X_N,V_N)dX_{2,N}dV_{2,N}=0\,. $$ (Consider for instance the case where $i=2$ and $j=3$: using (11) and (2) shows that $$ \begin{array}{l} \displaystyle\int \delta_{|x_2-x_3|=2r}(v_3-v_2)\cdot n_{23}F_N(t,X_N,V_N)dX_{2,N}dV_{2,N} &= -\int \delta_{|x_2-x_3|=2r}(v'_3-v'_2)\cdot n_{23}F_N(t,X_N,v_1,v'_2,v'_3,V_{4,N})dX_{2,N}dv_2dv_3dV_{4,N} \\ \displaystyle&= -\int \delta_{|x_2-x_3|=2r}(v_3-v_2)\cdot n_{23}F_N(t,X_N,V_N)dX_{2,N}dV_{2,N} \end{array} $$ after the substitution $(v_2,v_3)\mapsto(v'_2,v'_3)$.) On the other hand, whenever $i=1$ and $3\le j\le N$ $$ \begin{array}{l} \displaystyle\int \delta_{|x_1-x_j|=2r}(v_j-v_1)\cdot n_{1j}F_N(t,X_N,V_N)dX_{2,N}dV_{2,N} &= \int \delta_{|x_1-x_2|=2r}(v_2-v_1)\cdot n_{12}F_N(t,X_N,V_N)dX_{2,N}dV_{2,N} \\ &= \displaystyle\int \delta_{|x_1-x_2|=2r}(v_2-v_1)\cdot n_{12}F_{N:2}(t,x_1,x_2,v_1,v_2)dx_2dv_2\,, \end{array} $$ where the first equality follows from the substitution $(x_2,v_2,x_j,v_j)\mapsto(x_j,v_j,x_2,v_2)$ and the symmetry (13), while the second follows from integrating out the variables $x_3,\ldots,x_N,v_3,\ldots,v_N$. This last integral is transformed into the one on the right hand side of (17) by using spherical coordinates $(r,n)$ centered at $x_1$, so that $x_2=x_1+2rn$.
Finally, the integral on the right hand side of (17) is split as $$ \int_{\mathbf{S}^2\times\mathbf{R}^3}F_{N:2}(t,x_1,x_1+2rn,v_1,v_2)(v_2-v_1)\cdot ndv_2dn = \int_{(v_2-v_1)\cdot n>0}F_{N:2}(t,x_1,x_1+2rn,v_1,v_2)(v_2-v_1)\cdot ndv_2dn + \int_{(v_2-v_1)\cdot n<0}F_{N:2}(t,x_1,x_1+2rn,v_1,v_2)(v_2-v_1)\cdot ndv_2dn $$ One uses (11) to replace $F_{N:2}(t,x_1,x_1+2rn,v_1,v_2)$ with $F_{N:2}(t,x_1,x_1+2rn,v'_1,v'_2)$ in the first integral on the right hand side of the equality above, where $v'_1,v'_2$ are expressed in terms of $v_1,v_2$ by (2). Likewise, one changes $n$ into $-n$ in the second integral. After these manipulations, the equation for $F_{N:1}$ reads \begin{equation}\tag{18} \frac{\partial F_{N:1}}{\partial 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 $$ T_{12}[n](v_1,v_2):=(v_1-(v_1-v_2)\cdot nn,v_2-(v_2-v_1)\cdot nn)\,. $$ This is not an equation in closed form for $F_{N:1}$, as its right hand side involves $F_{N:2}$ which is not known in terms of $F_{N:1}$.
Therefore, one seeks an equation for $F_{N:2}$ by integrating out $x_3,\ldots,x_N,v_3,\ldots,v_N$ in (10). Proceeding as for $F_{N:1}$, one obtains an equation for $F_{N:2}$ in terms of $F_{N:3}$. More generally, for each $k=2,\ldots,N-1$, one obtains by the same procedure an equation for $F_{N:k}$ in terms of $F_{N:k+1}$. This equation takes the form \begin{equation}\tag{19} \frac{\partial F_{N:k}}{\partial t}+\sum_{j=1}^kv_j\cdot\nabla_{x_j}F_{N:k}=(N-k)(2r)^2\sum_{j=1}^k\mathcal{C}_N^{j,k+1}(F_{N:k+1}) +\sum_{1\le i<j\le k}\delta_{|x_i-x_j|=2r}(v_j-v_i)\cdot n_{ij}F_{N:k}\Big|_{\partial\Gamma^r_k} \end{equation} where $$ \mathcal{C}_N^{j,k+1}(F_{N:k+1})(t,X_k,V_k):=\int_{\mathbf{S}^2\times\mathbf{R}^3}(F_{N:k+1}(t,X_k,x_k+2rn,T_{j,k+1}[n]V_{k+1}) -F_{N:k+1}(t,X_k,x_k-2rn,V_{k+1}))((v_{k+1}-v_j)\cdot n)_+dv_{k+1}dn $$ and $$ T_{j,k+1}[n](V_{k+1}):=(v_1,\ldots,v_{j-1},v'_j,v_{j+1},\ldots,v_k,v'_{k+1}) $$ with $$ v'_j=v_j-(v_{k+1}-v_j)\cdot nn\,,\qquad v'_{k+1}=v_{k+1}-(v_{k+1}-v_j)\cdot nn\,. $$ Finally, the equation for $F_{N:N}=F_N$ coincides with the original Liouville equation in the sense of distributions (16).
Therefore, the sequence of marginals $F_{N:k}$ of the distribution function $F_N$ is governed by the hierarchy of equations (18)-(19) for $k=2,\ldots,N$. This hierarchy of equations is referred to as the BBGKY hierarchy, named after N.N. Bogoliubov, M. Born, H.S. Greene, J.G. Kirkwood and J. Yvon. Since the BBGKY hierarchy is deduced from the Liouville equation satisfied by $F_N$, and since the last equation of the BBGKY hierarchy coincides precisely with the Liouville equation for $F_N$, the BBGKY hierarchy is equivalent to the system (1)-(2) of Newton's equations for the $N$ identical gas molecules.
The presentation above slightly differs from chapter 2, section 4 in (Cercignani and al., 1994), sections 4.2 and 4.3 in (Gallagher and al., 2012) and (Bouchut and al., 2000) on pp. 131-133. The Dirac terms on the right hand side of (19), together with the condition $F_{N:k}(t,X_k,V_k)=0$ if $X_k\in\mathbf{R}^{3k}\setminus\overline{\Omega^r_k}$ inherited from (15) corresponds to the boundary conditions satisfied by $F_{N:k}$ on $\partial\Gamma^r_k$ following equation (4.3.3) in section 4.3 of (Gallagher and al., 2012).
The Boltzmann hierarchy
To derive the Boltzmann equation of the kinetic theory of gases from the $N$-body problem of classical mechanics, one seeks to pass to the limit for $F_{N:k}$ in each equation of the BBGKY hierarchy (18)-(19), letting $N\to\infty$ while keeping $k$ fixed. In order for the right hand side of each equation in the hierarchy to remain finite in that limit, one should at the same time let $r\to 0$ so that the product $Nr^2$ converges to some finite, positive quantity. Thus, the Boltzmann-Grad scaling is defined by the following prescription (Grad, 1949):
Boltzmann-Grad Scaling \begin{equation}\tag{20} N\to\infty\,,\quad \hbox{ and }r\to 0^+\,,\quad\hbox{ such that }N(2r)^2\to\lambda\in(0,+\infty)\,. \end{equation} Henceforth, the molecular radius is chosen so that $r:=r(N)\sim\tfrac12 N^{-1/2}$ in the limit as $N\to\infty$. (In the case of argon, the covalent radius and the Van der Waals radius are both of the same order of magnitude, between $10^{-10}$ and $2\cdot 10^{-10}$m; therefore, with $N$ of the order of the Avogadro number, i.e. $0.25\cdot 10^{26}$ molecules per cubic meter at the atmospheric pressure and a temperature of $300$K, one finds that $\lambda^{-1}$ is of the order of $10^{-6}$m. Thus, the Boltzmann-Grad scaling is verified in rarefied gases.)
Assuming that $F_{N:k}\to F_k$ for each $k\ge 1$ as $N\to\infty$, we pass to the limit in each equation of the BBGKY hierarchy. The equation for $F_1$ is found to be \begin{equation}\tag{21} \frac{\partial F_1}{\partial t}+v_1\cdot\nabla_{x_1}F_1=\lambda\mathcal{C}^{12}(F_2)\,, \end{equation} where $$ \mathcal{C}^{12}(F_2)(t,x_1,v_1):=\int_{\mathbf{S}^2\times\mathbf{R}^3}(F_2(t,x_1,x_1,T_{12}[n](v_1,v_2))-F_2(t,x_1,x_1,v_1,v_2))((v_2-v_1)\cdot n)_+dv_2dn\,. $$
Likewise, $F_k$ satisfies the equation \begin{equation}\tag{22} \frac{\partial F_k}{\partial t}+\sum_{j=1}^kv_j\cdot\nabla_{x_j}F_k=\lambda\sum_{j=1}^k\mathcal{C}^{j,k+1}(F_{k+1})\,, \end{equation} where $$ \mathcal{C}^{j,k+1}(F_{k+1})(t,X_k,V_k):=\int_{\mathbf{S}^2\times\mathbf{R}^3}(F_{k+1}(t,X_k,x_k,T_{j,k+1}[n]V_{k+1}) -F_{k+1}(t,X_k,x_k,V_{k+1}))((v_{k+1}-v_j)\cdot n)_+dv_{k+1}dn $$ for all $j,k$ such that $1\le j\le k$.
(Notice that the last term on the right hand side of (19) \begin{equation}\tag{23} \sum_{1\le i<j\le k}\delta_{|x_i-x_j|=2r}(v_j-v_i)\cdot n_{ij}F_{N:k}\Big|_{\partial\Gamma^r_k}\to 0 \end{equation} in the sense of distributions as $N\to\infty$, under the assumption that $$ M_k:=\sup_{1\le i\le N\atop N\ge 1}\sup_{(t,X_k,V_k)\in\mathbf{R}^{6k+1}}|v_iF_{N:k}(t,X_k,V_k)|<\infty $$ for all $k\ge 1$. Indeed \begin{equation}\tag{24} \int\left|\sum_{1\le i<j\le k}\delta_{|x_i-x_j|=2r}(v_j-v_i)\cdot n_{ij}F_{N:k}\Big|_{\partial\Gamma^r_k}\right|dX_kdV_k\le 2k(k-1)M_k\tfrac43\pi (2r)^3\to 0 \end{equation} as $N\to\infty$.)
The sequence of equations (21)-(22) for $k\ge 2$ is referred to as the Boltzmann hierarchy (see chapter 2, section 5 in (Cercignani and al., 1994), (Bouchut and al., 2000) on pp. 133-134, especially equations (3.16)-(3.18), or section 4.4 in (Gallagher and al., 2012)).
Although the BBGKY and the Boltzmann hierarchies look similar, there are important differences between them. While the BBGKY hierarchy consists of finitely many equations - in fact, of $N$ equations, where $N$ is the number of gas molecules - the Boltzmann hierarchy is infinite. As explained above, the BBGKY hierarchy is often presented as a sequence of $N$ equations satisfied by the restrictions $F_{N:k}\Big|_{\Gamma^{r(N)}_k}$, that are set on $\mathbf{R}\times\Gamma^{r(N)}_k$ for $2\le k\le N$, subject to the boundary conditions implied by (11) (see section 2.4 in (Cercignani and al., 1994), for instance). In the presentation above, the restriction to $\Gamma^{r(N)}_k$ and the corresponding boundary conditions are replaced with the surface measure $$ \sum_{1\le i<j\le k}\delta_{|x_i-x_j|=2r}(v_j-v_i)\cdot n_{ij}F_{N:k}\Big|_{\partial\Gamma^r_k} $$ on the right hand side of the $k$-th equation in the BBGKY hierarchy. On the contrary, the Boltzmann hierarchy is a sequence of equations for $F_k$ set on the whole Euclidean space $\mathbf{R}^{6k+1}$, without additional Dirac measure on their right hand sides. Finally, while the collision integrals in the BBGKY hierarchies are delocalized - meaning that the average with respect to the unit vector $n$ in $\mathcal{C}^{j,k+1}_N$ involves the dependence of $F_{N:k+1}$ in the position as well as the velocity variables - while the analogous collision integral $\mathcal{C}^{j,k+1}$ in the Boltzmann hierarchy involves only the velocity variables in $F_{k+1}$.
The Boltzmann hierarchy and the Boltzmann equation
The Boltzmann hierarchy is related to the Boltzmann equation as follows. Assume that $f$ is a solution of the Boltzmann equation (4), and set \begin{equation}\tag{25} F_k(t,X_k,V_k)=\prod_{j=1}^kf(t,x_j,v_j)\,. \end{equation} Then, for each $k\ge 1$ $$ \frac{\partial F_k}{\partial t}+\sum_{j=1}^kv_j\cdot\nabla_{x_j}F_k(t,X_k,V_k) = \sum_{j=1}^k\prod_{l=1\atop l\not=j}^kf(t,x_l,v_l)\left(\frac{\partial f}{\partial t}+v\cdot\nabla_xf\right)(t,x_j,v_j) = \sum_{j=1}^k\mathcal{C}(f)(t,x_j,v_j)\prod_{l=1\atop l\not=j}^kf(t,x_l,v_l)\,. $$ Since $$ \mathcal{C}(f)(t,x_j,v_j)\prod_{l=1\atop l\not=j}^kf(t,x_l,v_l)=\lambda\mathcal{C}^{j,k+1}(F_{k+1})(t,X_k,V_k)\,, $$ the sequence $F_k$ constructed from the solution of the Boltzmann equation by the formula (25) is a solution of the Boltzmann hierarchy (21)-(22). (See chapter 2, section 5 in (Cercignani and al., 1994).)
Therefore, if one knows (a) that the Boltzmann equation has a solution $f$ defined on $[0,T]\times\mathbf{R}^3\times\mathbf{R}^3$ and (b) that the solution $F_k$ of the Boltzmann hierarchy is determined uniquely by its restriction at $t=0$, one concludes that $$ \hbox{if }F_k(0,X_k,V_k)=\prod_{j=1}^kf(0,x_j,v_j)\,,\quad\hbox{ then }F_k(t,X_k,V_k)=\prod_{j=1}^kf(t,x_j,v_j)\hbox{ for all }t\in[0,T]. $$
With this observation, the Boltzmann-Grad limit can be formulated as follows:
Let $f^{in}\equiv f^{in}(x,v)$ defined on $\mathbf{R}^3\times\mathbf{R}^3$ be a probability density, and assume that the Boltzmann equation (4) has a solution $f$ on $[0,T]\times\mathbf{R}^3\times\mathbf{R}^3$ for some $T>0$ whose restriction at time $t=0$ is $f^{in}$. Set $r(N):=\lambda\tfrac12 N^{-1/2}$, and (for instance) \begin{equation}\tag{26} F^{in}_{N}(X_N,V_N):=\frac1{Z_N}\prod_{j=1}^Nf(t,x_j,v_j)\prod_{1\le m<n\le N}\chi\left(\frac{|x_m-x_n|}{r(N)}\right) \end{equation} with $$ Z_N:=\int\prod_{j=1}^Nf(t,x_j,v_j)\prod_{1\le m<n\le N}\chi\left(\frac{|x_m-x_n|}{r(N)}\right)dX_NdV_N\,, $$ where $\chi$ is a smooth function with values in $[0;1]$, such that $\chi(z)=1$ if $z>3$ and $\chi(z)=0$ if $z<2$.
Define $$ F_N(t,X_N,V_N)=\left\{\begin{array}{ll} F^{in}_N(S^{N,r(N)}_{-t}(X_N,V_N))&\hbox{ for }t\in[0,T]\hbox{ and }(X_N,V_N)\in\Gamma^{(r(N)}_N\,, \\ 0&\hbox{ for }t\in[0,T]\hbox{ and }(X_N,V_N)\notin\Gamma^{(r(N)}_N\,. \end{array}\right. $$ Then there exists $T'\in(0,T]$ such that $F_{N:1}\to f$ on on $[0,T']\times\mathbf{R}^3\times\mathbf{R}^3$ as $N\to\infty$ - and more generally $F_{N:k}\to F_k$ defined in terms of $f$ by the formulas (25) for all $k\ge 2$.
This is essentially Lanford's theorem, stated as Theorem 4.4.1 on p. 77 in (Cercignani and al., 1994), or as Theorem 4 on p. 19 in (Gallagher and al., 2012).
Method of proof
The argument sketched in the previous three sections, due to C. Cercignani (Cercignani, 1972), is purely formal, and significantly differs from the proof proposed by O. Lanford (Lanford, 1975) (especially in chapter 7). Lanford's idea is first to solve explicitly (18) for $F_{N:1}$ by the method of characteristics. For simplicity, set $$ \mathbf{C}^{k,k+1}_NF_{N:k+1}:=\sum_{j=1}^k\mathcal{C}^{j,k+1}_N(F_{N:k+1}) $$ for all $k\ge 1$. Thus \begin{equation}\tag{27} F_{N:1}(t,x,v)=F_{N:1}(0,x-tv,v)+(N-1)(2r)^2\int_0^t\mathbf{C}^{1,2}_NF_{N:2}(t_1,x-(t-t_1)v,v)dt_1\,. \end{equation} Then one solves (19) with $k=2$ for $F_{N:2}$ similarly: $$ F_{N:2}(t_1,X_2,V_2)=\tilde S^{2,r(N)}_{t_1}F_{N:2}(0,X_2,V_2)+(N-2)(2r)^2\int_0^{t_1}\tilde S^{2,r(N)}_{t_1-t_2}\mathbf{C}^{2,3}_NF_{N:3}(t_2,X_2,V_2)dt_2 $$ for all $(X_2,V_2)\in\Gamma^{r(N)}_2$. This expression of $F_{N:2}$ is substituted in the right hand side of (27), which involves in turn $F_{N:3}$. Iterating this procedure, one finds that \begin{equation}\tag{28} F_{N:1}(t,x,v)=F^{in}_{N:1}(x-tv,v)+\sum_{k\ge 1}(N-1)\ldots(N-k)(2r)^{2k}\int_0^t\int_0^{t_1}\ldots\int_0^{t_{k-1}}\tilde S^{1,r(N)}_{t-t_1}\mathbf{C}^{1,2}_N \tilde S^{2,r(N)}_{t_1-t_2}\ldots\mathbf{C}^{k,k+1}_N\tilde S^{k,r(N)}_{t_k}F^{in}_{N:k+1}(x,v)dt_k\ldots dt_2dt_1\,. \end{equation}
Then one passes to the limit as $N\to\infty$ in the series above.
First, one shows that \begin{equation}\tag{29} \int_0^t\int_0^{t_1}\ldots\int_0^{t_{k-1}}\tilde S^{1,r(N)}_{t-t_1}\mathbf{C}^{1,2}_N\tilde S^{2,r(N)}_{t_1-t_2}\ldots\mathbf{C}^{k,k+1}_N\tilde S^{k,r(N)}_{t_k}F^{in}_{N:k+1}(x,v) dt_k\ldots dt_2dt_1 \to \int_0^t\int_0^{t_1}\ldots\int_0^{t_{k-1}}\tilde S^{1}_{t-t_1}\mathbf{C}^{1,2}\tilde S^{2}_{t_1-t_2}\ldots\mathbf{C}^{k,k+1}\tilde S^{k}_{t_k}F^{in}_{k+1}(x,v)dt_k\ldots dt_2dt_1 \end{equation} with the following elements of notation: the initial $k$-body limiting distribution function is \begin{equation}\tag{30} F_k^{in}(X_k,V_k):=\prod_{j=1}^kf(x_j,v_j)\,, \end{equation} while $\tilde S^k_t$ designates the free transport group acting on $k$-particle densities: $$ \tilde S^k_tF^{in}_k(X_k,V_k):=F^{in}_k(X_k-tV_k,V_k)\,, $$ and $$ \mathbf{C}^{k}F_{k+1}:=\sum_{j=1}^k\mathcal{C}^{j,k+1}(F_{k+1})\,. $$
Next one shows that one can pass to the limit term by term in the series (28) by dominated convergence, on some time interval $[0,T']$ for some $T'\in(0,T]$, so that \begin{equation}\tag{31} F_{N:1}(t,x,v)\to F^{in}_1(x-tv,v)+\sum_{j\ge 1}\lambda^j\int_0^t\int_0^{t_1}\ldots\int_0^{t_{j-1}}\tilde S^{1}_{t-t_1}\mathbf{C}^{1}\tilde S^{2}_{t_1-t_2} \ldots\mathbf{C}^{j}\tilde S^{j}_{t_j}F^{in}_{j+1}(x,v)dt_j\ldots dt_2dt_1\,. \end{equation}
Likewise, one can show that $$ F_{N:k}(t,X_k,V_k)\to \tilde S^k_tF^{in}_k(X_k,V_k)+\sum_{j\ge 1}\lambda^j\int_0^t\int_0^{t_1}\ldots\int_0^{t_{j-1}}\tilde S^{k}_{t-t_1}\mathbf{C}^{k}\tilde S^{k+1}_{t_1-t_2} \ldots\mathbf{C}^{k+j}\tilde S^{k+j}_{t_k}F^{in}_{k+j+1}(X_k,V_k)dt_j\ldots dt_2dt_1\,, $$ on the same time interval $[0,T']$.
One recognizes in the series above the explicit expression of a solution $F_k$ of the Boltzmann hierarchy. After proving the uniqueness of the solution of the Boltzmann hierarchy with initial data (30), one concludes as explained above that $F_1=f$, and that $F_k$ is given by the formula (25) on the interval of time $[0,T']$. The details of this proof can be found in chapter 4, section 4 of (Cercignani and al., 1994), and occupies part II and part IV of (Gallagher and al., 2012).
Domain of validity of Lanford's theorem
The convergence stated above has been established on the short interval of time $[0,T']$, where $T'$ is a fraction of the mean free time - i.e. of the average time between two successive collisions involving one molecule. This restriction fueled some doubt at first on the physical value of this result. A global in time convergence statement (Illner and Pulvirenti, 1989) was proved subsequently, for initial distributions of molecules corresponding with very rarefied gases. This new result was based on the dispersive effect of the free transport equation (see Theorem 4.5.1 in (Cercignani and al., 1994)).
Obviously, the validity of the Boltzmann-Grad limit is limited to time intervals on which the solution of the Boltzmann equation is known to exist and is uniquely determined by its restriction at time $t=0$. While global solutions of the Boltzmann equation have been constructed for arbitrary initial data with finite total mass, energy and entropy (DiPernaLions, 1990), these solutions are not uniquely determined by their restriction at time $t=0$ in general. Moreover, this construction leads to a very weak notion of solution, whose interest in the context of the Boltzmann-Grad limit remains unclear.
In the discussion above, the Boltzmann-Grad limit has been described above in the only case where the molecular interaction corresponds with elastic hard sphere collisions. Indeed, the Boltzmann collision integral appears in the derivation outlined above as a consequence of the geometry of the phase space $\Gamma^r_N$ in the Boltzmann-Grad scaling (20). Since the assumption of hard sphere collisions is physically unrealistic, extending the Boltzmann-Grad limit to more general molecular interactions is of significant interest. This has been done in the case where the molecular interaction is described by a compactly supported, smooth, radial repulsive potential (see Theorem 5 on p. 19 in (Gallagher and al. 2013), and the references therein). It should also be mentioned that extending the Boltzmann-Grad limit to the case of any molecular interaction given by an inverse power law repulsive potential - meaning that the force exerted by the $j$-th molecule on the $i$-th molecule is of the form $a_0(x_i-x_j)/|x_i-x_j|^{s+2}$ for some $s>0$ - seems beyond reach, at least with the method of proof outlined above. Indeed, in this case, the Boltzmann collision integral should be viewed as a nonlinear singular integral - analogous in some sense to Cauchy's principal value. Therefore, the collision terms in the BBGKY hierarchy can no longer be considered as bounded perturbations of the free transport operator, as controlling them would involve derivatives of $F_{N:k}$ with respect to the velocity variables. From the physical viewpoint, this is due to the importance of grazing collisions in the case where the molecular interaction is given by a long range potential.
One limitation of the classical Boltzmann equation is that applies to ideal gases only. This results from the Boltzmann-Grad scaling (20): the total volume occupied by the gas molecules, assuming that they are concentrated in one portion of the Euclidian space like oranges in a grocery store, is $N\cdot\tfrac43\pi r^3=O(N^{-1/2})$, which vanishes as $N\to\infty$. In the case of hard spheres, this volume is referred to as the excluded volume in the context of the Van der Waals equation of state and appears as a correction to the ideal gas law.
Irreversibility and the Boltzmann-Grad limit
The kinetic theory of gases has been the subject of much controversy in the days of Boltzmann (Cercignani, 1998). Among other things, it was argued that Newton's equations for the system of gas molecules are mechanically reversible, while the Boltzmann equation is an irreversible model since Boltzmann's $H$ theorem asserts that the $H$ function associated to any solution of the Boltzmann equation is nonincreasing in time. Some scientific contemporaries of Boltzmann, including J. Loschmidt, H. Poincaré and E. Zermelo, formulated several paradoxes based on the alleged incompatibility between Boltzmann's $H$ theorem and the reversible nature of Newton's equations of classical mechanics (see chapter 3, sections 5-6 in (Cercignani and al., 1994)). All these issues are also related to the notions of time's arrow and Boltzmann's entropy.
Mechanical reversibility is the following property enjoyed by the 1-parameter group $S^{N,r}_t$ in (3): \begin{equation}\tag{32} R_NS^{N,r}_tR_NS^{N,r}_t=I\,,\hbox{ for all }t\in\mathbf{R}\,,\quad\hbox{ where }R_N(X_N,V_N)=(X_N,-V_N)\,. \end{equation} Thus, for each $N$-particle probability distribution function $F^{in}_N$, the solution $F_N(t,\cdot,\cdot)=\tilde S^{N,r}_tF^{in}$ of the Liouville equation (10) starting from $F^{in}_N$ at time $t=0$, which is defined by the formula (8), satisfies the identity \begin{equation}\tag{33} \tilde S^{N,r}_t\tilde R_N\tilde S^{N,r}_t\tilde R_NF_N^{in}=F^{in}_N\quad\hbox{ for all }t\in\mathbf{R}\,, \end{equation} where $\tilde R_N$ is defined by $$ \tilde R_NF_N(X_N,V_N):=F_N(X_N,-V_N)\,. $$
There is no analogue of this property for the Boltzmann equation. More precisely, denote by $S^B_tf^{in}:=f(t,\cdot,\cdot)$ the solution of the Boltzmann equation (4) starting from the initial condition $f\Big|_{t=0}=f^{in}$. (This definition assumes the existence and uniqueness of a solution of the Cauchy problem for the Boltzmann equation, which is known, at least in some regimes (Illner and Shinbrot, 1984).) Then $$ \tilde R_1S^B_t\tilde R_1S^B_tf^{in}\not=f^{in} $$ in general, since $$ H(\tilde R_1S^B_t\tilde R_1S^B_tf^{in})=H(S^B_t\tilde R_1S^B_tf^{in})\le H(\tilde R_1S^B_tf^{in})=H(S^B_tf^{in})<H(f^{in}) $$ unless $f^{in}$ is a local Maxwellian, i.e. is of the form $f^{in}(x,v)=\mathcal{M}_{(\rho^{in}(x)u^{in}(x),\theta^{in}(x))}(v)$. (In fact, the equality $H(S^B_tf^{in})<H(f^{in})$ is equivalent to the fact that $S^B_sf^{in}$ is a local Maxwellian for a.e. $s\in[0,t]$.) Both inequalities above are implied by Boltzmann's $H$ theorem, while the equalities follow from the obvious formula $$ \iint_{\mathbf{R}^3\times\mathbf{R}^3}\tilde R_1g(x,v)dxdv=\iint_{\mathbf{R}^3\times\mathbf{R}^3}g(x,-v)dxdv=\iint_{\mathbf{R}^3\times\mathbf{R}^3}g(x,v)dxdv $$ valid for all integrable functions $g$.
However, the mechanical reversibility of the Liouville equation and the irreversible character of the Boltzmann equation are compatible with Lanford's theorem. Indeed, the mechanical reversibility of the Liouville equation involves the transformation $R_N$ which exchanges jointly the velocities $v_1,\ldots,v_N$ of the $N$ particles into $-v_1,\ldots,-v_N$. Since the Boltzmann equation is posed in the $1$-particle phase space, it is impossible to formulate, much less to perform, such a transformation by any kind of operation on the $1$-particle distribution $f$ that is the solution of the Boltzmann equation. In particular, the operator $\tilde R_1$ corresponds to changing the velocity $v$ of one typical gas particle into $-v$, and this does not imply that the velocities of all the other gas particles are changed accordingly at the same instant of time. One of the benefits of having such a precise formulation of the Boltzmann-Grad limit as in Lanford's theorem is to avoid this type of confusion.
Boltzmann's $H$ theorem can be viewed as a special case of the second principle of thermodynamics. Indeed, whenever the distribution function $f$ is a Maxwellian, i.e. if $f(t,x,v)=\mathcal{M}_{(\rho(t,x)u(t,x),\theta(t,x))}$, then $$ H(f)(t)=\int_{\mathbf{R}^3}\rho(t,x)\left(\ln\frac{\rho(t,x)}{(2\pi\theta(t,x))^{3/2}}-\tfrac32\right)dx $$ coincides with the explicit formula for (minus) the entropy of a monatomic ideal gas (with polytropic index $\tfrac32$), up to a constant proportional to the total mass of the gas. Thus it is natural to think of $H(f)$ as a measure of the thermodynamical irreversibility of the Boltzmann equation. The notion of reversible transformations in thermodynamics differs from the notion of mechanical reversibility (32): a reversible transformation on an isolated thermodynamical system is a transformation in the course of which the entropy of that system remains constant. Therefore, it is worth discussing the compatibility of other features of the Liouville equation (10) than mechanical reversibility with Boltzmann's $H$ theorem in the Boltzmann-Grad limit.
Boltzmann's $H$ theorem and Liouville's theorem
One such feature is Liouville's theorem, i.e. the fact that the Lebesgue measure of the $N$-particle phase space is invariant under the $1$-parameter group $S^{N,r}_t$. The identity (9) is equivalent to Liouville's theorem.
Specializing (9) to the case where $\Phi(z):=z\ln z$ shows that the $N$-particle distribution function $F_N$ defined by (15) satisfies \begin{equation}\tag{34} \int_{\mathbf{R}^{6N}}F_N\ln F_N(t,X_N,V_N)dX_NdV_N=\int_{\mathbf{R}^{6N}}F^{in}_N\ln F^{in}_N(X_N,V_N)dX_NdV_N=\hbox{Const.} \end{equation} There is no contradiction between Boltzmann's $H$ theorem, and the equality deduced from (9) with $\Phi(z):=z\ln z$. Indeed, at time $t=0$ and with the initial condition (26) $$ \int_{\mathbf{R}^{6N}}F^{in}_N\ln F^{in}_N(X_N,V_N)dX_NdV_N\sim N\int_{\mathbf{R}^6}f^{in}\ln f^{in}(x,v)dxdv\,. $$ In other words \begin{equation}\tag{35} \int_{\mathbf{R}^6}F_{N:1}\ln F_{N:1}(0,x,v)dxdv\sim\frac1N\int_{\mathbf{R}^{6N}}F_N\ln F_N(0,X_N,V_N)dX_NdV_N \end{equation} as $N\to\infty$. On the other hand, by convexity of the function $z\mapsto z\ln z$, one has $$ F_N\ln F_N(t,X_N,V_N)\ge F_N(t,X_N,V_N)\ln\prod_{j=1}^NF_{N:1}(t,x_j,v_j)+F_N(t,X_N,V_N)-\prod_{j=1}^NF_{N:1}(t,x_j,v_j) $$ so that \begin{equation}\tag{36} \int_{\mathbf{R}^6}F_{N:1}\ln F_{N:1}(t,x,v)dxdv\le\frac1N\int_{\mathbf{R}^{6N}}F_N\ln F_N(t,X_N,V_N)dX_NdV_N\,. \end{equation} Putting together the convexity inequality (36), the conservation law (34) stating that the $H$ function of the $N$-particle distribution density is constant under the dynamics of the Liouville equation (10) and the asymptotic equivalent of the initial entropy per particle (35) leads to the inequality \begin{equation}\tag{37} \int_{\mathbf{R}^6}F_{N:1}\ln F_{N:1}(t,x,v)dxdv\lesssim\int_{\mathbf{R}^6}F_{N:1}\ln F_{N:1}(0,x,v)dxdv\,. \end{equation}
Since $F_{N:1}\to f$, applying Fatou's lemma to the integral on the left hand side of the inequality above, and passing to the limit by dominated convergence in the integral on the right hand side - assuming that $0\le f^{in}(x,v)\le Ce^{-\beta(|x|^2+|v|^2)}$ for some $C,\beta>0$ - shows that $$ H(f)(t)=\int_{\mathbf{R}^6}f\ln f(t,x,v)dxdv\le\varliminf_{N\to\infty}\int_{\mathbf{R}^6}F_{N:1}\ln F_{N:1}(t,x,v)dxdv \le\lim_{N\to\infty}\int_{\mathbf{R}^6}F_{N:1}\ln F_{N:1}(0,x,v)dxdv=\int_{\mathbf{R}^6}f^{in}\ln f^{in}(x,v)dxdv=H(f)(0)\,. $$ Therefore the inequality (37) is compatible with Boltzmann's H theorem when passing to the Boltzmann-Grad limit. Notice that an essential step in the proof of the inequality (37) is the conservation law (34), which is a consequence of the Liouville theorem in the formulation (9).
However the discussion above proves that $H(f)(t)\le H(f)(0)$ for each $t>0$, but not that $H(f)(s)\le H(f)(t)$ whenever $s\ge t\ge 0$. In other words, the argument above does not imply that the function $t\mapsto H(f)(t)$ is nonincreasing, as predicted by Boltzmann's $H$ theorem. Indeed, the asymptotic equivalence (35) is known to hold at time $t=0$ only, so that the argument above cannot be repeated on an arbitrary time interval $[t,s]$ with $t>0$.
In fact, although the inequality (37) is compatible in the Boltzmann-Grad limit with the inequality $H(f)(t)\le H(f)(0)$ for $t>0$ implied by Boltzmann's H theorem, both inequalities express two different physical realities. The inequality $H(f)(t)\le H(f)(0)$ in Boltzmann's H theorem measures by how much the $1$-particle distribution function $f$ differs from a Maxwellian on the time interval $[0,t]$. (Since Maxwellians are equilibrium states for the Boltzmann equation, this is consistent with the usual conception of the entropy of a closed system which increases until the system reaches reaches an equilibrium.) On the other hand, the inequality (37) measures by how much the $N$-particle probability distribution function $F_N$ differs from the factorized expression \begin{equation}\tag{38} \prod_{j=1}^NF_{N:1}(t,x_j,v_j) \end{equation} at time $t$.
As in the discussion involving mechanical reversibility, the conservation law (34) is a property of Newton's equations of mechanics in the $N$-particle phase space. The Boltzmann equation defines a dynamics in the $1$-particle phase space - acting on the first marginal of the $N$-particle distribution function in the large $N$ limit. Since some amount of information is lost when considering $F_{N:1}$ instead of $F_N$, unless $F_N$ is of the factorized form (38), there is no contradiction between the $H$ theorem and (34). The choice of the initial data (26) (approximately) in factorized form in Lanford's theorem, and the reduction to the $1$-particle phase space involved in the derivation of the Boltzmann equation partly account for the irreversible nature of the Boltzmann equation - see chapter 4, section 7 in (Cercignani and al., 1994) for a detailed discussion of these issues.
Comments on the proof
In the proof outlined above, one passes to the limit as $N\to\infty$ in the explicit expressions of the marginals of the $N$-particle distribution function - see formula (28) in the case of the first marginal. The limiting equations in the Boltzmann hierarchy are obtained as a consequence of the explicit formulas for $F_k$, which is somewhat unsatisfying. (From the mathematical viewpoint, it is more natural to deduce the solution from the equation it satisfies than the converse.) Although it may seem natural at first sight that the $k$-th equation in the Boltzmann hierarchy is the limiting form of the $k$-th equation in the BBGKY hierarchy, as argued in the argument presented above, one should be aware that this viewpoint is only formal (see the discussion on p. 75 in (Cercignani and al., 1994)).
Term by term convergence of the solution of the BBGKY hierarchy
For instance, the expression on the left hand side of (29) involves the 1-parameter group $\tilde S^{j,r(N)}_{t_{j-1}-t_j}$ corresponding with the Liouville equation (10) set on the domain $\Gamma^r_j$ with the boundary condition (11). The expression on the right hand side of (29) involves the 1-parameter group $\tilde S^{j}_{t_{j-1}-t_j}$ corresponding with the free transport equation, i.e. with (11) in the domain $\mathbf{R}^{6j}$. In other words, the expression on the right hand side of (29) represents situations where the only possible collisions result from adding a $j$-th particle to collisionless group of $j-1$ particles. The expression on the left hand side of (29) represent situations where groups of $j$ particles may be subject to binary collisions between the instants of time $t_{j-1}$ and $t_j$. Therefore, in the situation described by the right hand side of (29), the $j$-th particle added to the group of $j-1$ particles at time $t_{j-1}$ may have already collided with at least one of these $j-1$ particles before $t_{j-1}$. These events are known as recollisions, and are excluded in all the terms appearing in the series that represents the solution $F_1$ of the first equation in the Boltzmann hierarchy.
The effect of recollisions is manifested in each equation of the BBGKY hierarchy (except the first) by the Dirac terms on the right hand side of (19). Indeed, as explained above, these Dirac terms are equivalent to the boundary condition satisfied by $F_{N:k}$ on $\partial\Gamma^r_k$, which are inherited from (11) verified by $F_N$ itself. The formal argument (23) based on a bound (24) of the Dirac terms on the right hand side of (19) is not sufficient to prove that recollisions are negligible in the Boltzmann-Grad limit. Indeed , the bound (24) would also be verified in the case of particle models where the particle velocities are chosen in some finite subset of $\mathbf{R}^3$. Yet it has been observed in (Uchiyama, 1988) that the effect of recollisions is not negligible in the Boltzmann-Grad limit for a particle system leading to a discrete velocity model of the Boltzmann equation. Disposing of recollisions is a rather technical step in the proof of Lanford's theorem (see section 12 of (Gallagher and al. 2012)).
Uniqueness of the Boltzmann hierarchy
Once the term by term convergence in (28) is established, some uniform (in $N$) estimate on each term of (28) is needed in order to prove (31) by dominated convergence. This part of Lanford's argument much less technical and is best understood on a (considerably) simpler analogue of the Boltzmann hierarchy. Consider the Riccati equation \begin{equation}\tag{39} \dot{x}(t)=x(t)^2\,,\qquad x(0)=x^{in}\,, \end{equation} whose solution is known to be \begin{equation}\tag{40} x(t)=\frac{x^{in}}{1-tx^{in}}\,. \end{equation}
Obviously, one has $$ \frac{d}{dt}x(t)^k=kx(t)^{k-1}\dot{x}(t)=kx(t)^{k+1} $$ for each $k\ge 1$, which suggests considering the infinite hierarchy \begin{equation}\tag{41} \dot{y}_k(t)=ky_{k+1}(t)\,,\qquad y_k(0)=y^{in}_k\,,\qquad k\ge 1\,. \end{equation} The infinite hierarchy (41) and the Riccati equation (39) are the analogues of the Boltzmann hierarchy and the Boltzmann equation respectively.
Instead of solving the infinite hierarchy (41) by iterations (as suggested in the proof of the Boltzmann-Grad limit outlined above), one can consider instead the formal power series in the variable $z$ defined as $$ U(t,z):=\sum_{k\ge 1}y_k(t)z^{k-1}\,. $$ If the sequence of functions $t\mapsto y_k(t)$ satisfies the growth condition \begin{equation}\tag{42} \sup_{|t|\le T}\varlimsup_{k\to\infty}|y_k|^{1/k}<+\infty\,, \end{equation} it is a solution of the infinite hierarchy (41) if and only if $$ \frac{\partial U}{\partial t}(t,z)=\sum_{k\ge 1}\dot{y}_k(t)z^{k-1}=\sum_{k\ge 1}ky_{k+1}(t)z^{k-1}=\frac{\partial U}{\partial z}(t,z)\,, $$ i.e. if and only if $U$ is a solution of the partial differential equation \begin{equation}\tag{43} \frac{\partial U}{\partial t}-\frac{\partial U}{\partial z}=0\,. \end{equation}
Assume that the initial data of the hierarchy (41) satisfies $$ \varlimsup_{k\to\infty}|y_k|^{1/k}=1/R $$ for some finite $R>0$. Then the power series \begin{equation}\tag{44} U^{in}(z)=\sum_{k\ge 1}y^{in}_kz^{k-1} \end{equation} has convergence radius $R$ and defines a holomorphic function (at least) on the open disk $D(0,R)$. The partial differential equation (43) with initial data (44) has a unique solution that is real analytic on the domain $$ \{(t,z)\in\mathbf{R}^2\,|\,|z|+|t|<R\}\,, $$ and given by the formula $$ U(t,z)=U^{in}(t+z)\,. $$
Therefore, the infinite hierarchy has a unique solution $t\mapsto y_k(t)$ satisfying the growth condition (42). This solution is defined for $|t|<R$ in terms of $U$ by the formula $$ y_k(t)=\frac1{(k-1)!}\frac{\partial^{k-1}U}{\partial z^{k-1}}(t,0)=\frac1{(k-1)!}\frac{d^{k-1}}{dt^{k-1}}U^{in}(t)\,. $$
In particular, if $y^{in}_k=(x^{in})^k$ for all $k\ge 1$, one finds that $R=1/|x^{in}|$, and that $$ U^{in}(z)=\frac{x^{in}}{1-zx^{in}}\,, $$ so that $$ U(t,z)=\frac{x^{in}}{1-(z+t)x^{in}}\,,\qquad|t|+|z|<\frac1{|x^{in}|}\,. $$ One obviously recovers the solution of the Riccati equation (39) by the formula $$ x(t)=U(t,0)\qquad\hbox{ for }|t|<\frac1{|x^{in}|}\,. $$ The idea of using tools from the theory of partial differential equations in the complex plane (especially some abstract variant of the Cauchy-Kovalevskaya theorem) to handle infinite hierarchies has been proposed by S. Ukai on the example of the Boltzmann hierarchy (Ukai, 2001).
Interestingly, this procedure, which is equivalent to the one outlined above in the case of the Boltzmann-Grad limit, leads to a suboptimal existence range for the solution of the Riccati equation. (Indeed, the explicit formula for $U$ shows that $U$ is real analytic on the larger domain $$ \{(t,z)\in\mathbf{R}^2\,|\,(z+t)x^{in}<1\}\,, $$ and this leads to the maximal solution of the Riccati equation, that is defined for all $t\in\mathbf{R}$ such that $tx^{in}<1$.) Therefore, this very simple example suggests that the Boltzmann-Grad limit might in fact hold true on an interval of time larger than predicted by the proof outlined above.
The growth condition (42) ensures that the radius of convergence of $z\mapsto U(t,z)$ is positive for $t\in[0,T]$. Of course this growth condition cannot be dispensed with - otherwise, the uniqueness of the solution of the infinite hierarchy is lost. Consider for instance the function $E$ defined by $$ E(t)=\left\{\begin{array}{ll}e^{-1/t}&\quad\hbox{ if }t>0\,,\\ 0&\quad\hbox{ if }t\le 0\,,\end{array}\right. $$ which is of class $C^\infty$ on the real line and satisfies $E^{(n)}(0)=0$ for all $n\ge 0$. The infinite hierarchy (41) with $y^{in}_k=0$ for all $k\ge 0$ has a unique solution satisfying the growth condition (42), which is $y_k(t)=0$ for all $|t|\le T$ and $k\ge 1$. However the formula $$ y_k(t)=\frac1{(k-1)!}E^{k-1}(t)\,,\quad t\in\mathbf{R} $$ also defines a solution of (41). This example ((Cercignani and al., 1994) on p. 104) shows that solutions of the infinite hierarchy are not uniquely determined by their initial data without additional assumptions. (Of course, since $E$ is not real-analytic in any open neighborhood of $0$, this solution does not satisfy the growth condition (42).)
Propagation of chaos and the Boltzmann-Grad limit
While the Boltzmann equation is stated above in terms of the marginals of the $N$-particle distribution function $F_N$, it would have been perhaps more natural to study the asymptotic behavior of the empirical measure of the $N$-particle system. Given $N$ spherical particles of radius $r$, centered at the points $x_1,\ldots,x_N$ and with velocities $v_1,\ldots,v_N$ - so that $(X_N,V_N):=(x_1,\ldots,x_N,v_1,\ldots,v_N)\in\Gamma^r_N$, their empirical measure is \begin{equation}\tag{45} \mu_{(X_N,V_N)}:=\frac1N\sum_{j=1}^N\delta_{x_j,v_j}\,, \end{equation} which is a probability measure on $\mathbf{R}^6$. An important advantage in considering the empirical measure $\mu_N$ of the $N$-particle system instead of its $N$-particle distribution function $F_N$ is that the former is defined on a finite dimensional Euclidean space - $\mathbf{R}^6$ in the present case - that is independent of $N$, while the latter is defined on $\Gamma^r_N$ which depends on $r$ and $N$. How both viewpoints are related is explained by the following observation.
Let $G_N$ be sequence of probability densities on $\mathbf{R}^{6N}$ satisfying the symmetry relation (13). The two following conditions are equivalent (Lemma 4.6.3 in (Cercignani and al., 1994)):
- (i) there exists a probability density $g$ on $\mathbf{R}^6$ such that, for each $\epsilon>0$ and each continuous, compactly supported test function $\phi$ defined on $\mathbf{R}^6$
$$ G_Ndm_N-\hbox{meas}(\{(X_N,V_N)\in\mathbf{R}^{6N}\hbox{ s.t. }|\langle\mu_{(X_N,V_N)}-g,\phi\rangle|>\epsilon\})\to 0 $$ as $N\to\infty$;
- (ii) for each $k\ge 1$, the sequence of marginals of $G_N$ satisfies
$$ G_{N:k}(X_k,V_k)\to\prod_{j=1}^kg(x_j,v_j) $$ in the sense of distributions on $\mathbf{R}^{6k}$ as $N\to\infty$.
In view of this observation, the Boltzmann-Grad limit can be equivalently stated as follows (see the reformulation of Theorem 4.5.1 on p. 93 in (Cercignani and al., 1994)).
Let $f^{in}\equiv f^{in}(x,v)$ be a probability density defined on $\mathbf{R}^3\times\mathbf{R}^3$, and assume that the Boltzmann equation (4) has a solution $f$ on $[0,T]\times\mathbf{R}^3\times\mathbf{R}^3$ for some $T>0$ whose restriction at time $t=0$ is $f^{in}$. Set $r(N):=\lambda\tfrac12 N^{-1/2}$, and let $F^{in}_N$ be defined by (26). Then there exists $T'\in(0,T]$ such that $$ F^{in}_Ndm_N-\hbox{meas}(\{(X_N,V_N)\in\mathbf{R}^{6N}\hbox{ s.t. }|\langle\mu_{S^{N,r(N)}_t(X_N,V_N)}-f(t,\cdot,\cdot),\phi\rangle|>\epsilon\})\to 0 $$ as $N\to\infty$ for all $t\in[0,T']$.
In other words, the empirical measure at time $t$ of the $N$-particle system converges in the sense of distributions to the solution of the Boltzmann equation for most initial $N$-particle configurations (measured in terms of the $N$-particle initial distribution function). This convergence of the empirical measure is the basis for the Monte-Carlo method used in numerical simulations of the Boltzmann equation.
Sequences $G_N$ of probability densities satisfying the symmetry (13) and the property (ii) above are referred to as chaotic sequences. The Boltzmann-Grad limit stated above can be viewed as a statement concerning the propagation of chaos: if the $N$ particle density $F^{in}_N$ is chaotic at time $t=0$, it will also be chaotic for all subsequent instants of time $t\in[0,T']$. (Once it is known that the sequence $F_N$ is chaotic for all $t\in[0,T']$, the Boltzmann equation is deduced from the first equation in the Boltzmann hierarchy.)
The Boltzmann-Grad limit for the Lorentz gas
In the derivation of the Boltzmann equation of the kinetic theory of gases from the system of Newton's equations applied to each gas molecule, all the gas molecules are assumed to be identical.
Consider now the case of a system consisting of a very large number $N$ of heavy spherical particles of mass $M$ and radius $r$ and one light point particle of mass $m$. Assume that all the heavy particles are at rest initially, while the light particle is moving freely in the domain of the Euclidean space left empty by the $N$ heavy particles. Assume further that the collisions between the light particle and any one of the heavy particles are elastic.
This dynamical system is usually referred to as the Lorentz gas, since H.A. Lorentz proposed to apply the methods of the kinetic theory of gases to explain the motion of electrons in metals. Lorentz's idea was to view electrons as a gas of light particles colliding with the metallic atoms; neglecting collisions between electrons, Lorentz described the interaction of electrons with the metallic atoms by a collision integral analogous to Boltzmann's. At variance with the system described above, the model considered by Lorentz also included an external electric field. Being mainly interested in the collision integral, we only consider below the case where no external electric field is applied. In addition to Lorentz's investigations, one should also mention the work of P. Drude, who also proposed a kinetic theory of electrons in metals.
In a different physical context, the diffusion of a light gas mixed with a heavy gas is a classical problem in the kinetic theory of (neutral) gases, and can be described by the same type of equations.
If $m\ll M$, one can assume that each heavy particle remains at rest, while the light particle is specularly reflected when impinging on the surface of any one of the heavy particles. Observe that the speed of the light particle remains constant in this dynamics, so that it is completely determined by its value at time $t=0$, henceforth denoted $c>0$. Let $a_j$ the center of the $j$-th heavy particle - assuming that \begin{equation}\tag{46} |a_i-a_j|>2r\,,\quad i\not=j\ge 1\,. \end{equation}
Let $x(t)$ and $\omega(t)$ be respectively the position and the direction of the light particle at time $t$; they satisfy \begin{equation}\tag{47} \dot{x}(t)=c\omega(t)\,,\quad\dot\omega(t)=0\qquad\hbox{ while }\,\,|x(t)-a_j|>r\,\,\hbox{ for all }j \end{equation} and \begin{equation}\tag{48} x(t+0)=x(t-0)\,,\qquad\omega(t+0)=\mathcal{R}_{x(t\pm 0)}\omega(t-0)\,, \end{equation} if $|x(t\pm 0)-a_j|=r$ for some $j$. Here $\mathcal{R}_x$ is the specular reflection at the point $x$ belonging to the surface of one the heavy spherical particles. Explicitly $$ \mathcal{R}_x\omega=\omega-2\omega\cdot n_xn_x\qquad\hbox{ with }n_x:=\tfrac1r(x-a_j)\,. $$
Denote by $\vec{a}=\{a_j\,|\,j\ge 1\}$ the configuration of heavy particles - i.e. the countable set of the heavy particle centers, and by $S^{r,\vec{a}}_t$ the transformation defined for all $t\in\mathbf{R}$ by \begin{equation}\tag{49} S^{r,\vec{a}}_t(x(0),\omega(0)):=(x(t),\omega(t))\quad\hbox{ assuming that }|\omega(0)|=1\hbox{ and }|x(0)-a_j|>r\hbox{ for all }j\ge 1\,. \end{equation} The problem is to identify asymptotic regimes in the particle radius $r$ and the number of heavy particles per unit volume for which the dynamics defined by $S^{r,\vec{a}}_t$ can be approximated by a kinetic model.
The random case
Assume that the centers of heavy particles are independent and distributed under a Poisson law of parameter $N>0$. This means that, for each $k\ge 0$, each choice of $k$ disjoint measurable subsets $A_1,\ldots,A_k$ of the Euclidian space and each $k$-tuple $(m_1,\ldots,m_k)$ of nonnegative integers \begin{equation}\tag{50} \hbox{Prob}(\#(\vec{a}\cap A_j)=m_j\,,\,\,1\le j\le k)=\prod_{j=1}^ke^{-N|A_j|}\frac{N^{m_j}}{m_j!}|A_j|^{m_j} \end{equation} The physical meaning of $N$ is explained by the following computation: the average number of particle centers in a measurable subset $A$ of the Euclidian space is $$ \sum_{m\ge 0}m\hbox{Prob}(\#(\vec{a}\cap A)=m)=e^{-N|A|}\sum_{m\ge 1}m\frac{N^m}{m!}|A|^m=N|A| $$ so that $N$ is the average number of heavy particles per unit volume in the Euclidian space.
The following result has been established in (Gallavotti, 1969) and is discussed in detail in Appendix 1.A.2 of (Gallavotti, 1999). It is stated below in space dimension $2$.
Assume that $r$ and $N$ satisfy the Boltzmann-Grad scaling in space dimension $2$: \begin{equation}\tag{51} N\to\infty\,,\quad r\to 0^+\quad\hbox{ so that }2rN\to\lambda>0\,. \end{equation} In other words, assume that $r\to 0$ and set $N\equiv N(r):=\lambda/2r$. Let $f^{in}\equiv f^{in}(x,\omega)$ be a continuous, compactly supported c probability density on $\mathbf{R}^2\times\mathbf{S}^1$, and define $f_r$ by the formula $$ f_r(t,x,\omega,\vec{a}):=\left\{ \begin{array}{ll}f^{in}(S^{r,\vec{a}}_{-t}(x,\omega))&\quad\hbox{ if }|x-a_j|>r\hbox{ for all }j\ge 1\,,\\ 0&\quad\hbox{ if }|x-a_j|<r\hbox{ for some }j\ge 1\,.\end{array} \right. $$
Then $\langle f_r\rangle\to f$ as $r\to 0$, where $f$ is the solution of the Lorentz kinetic equation \begin{equation}\tag{52} \left(\frac{\partial f}{\partial t}+c\omega\cdot\nabla_xf\right)(t,x,\omega) =c\lambda\int_{\mathbf{S}^1}(f(t,x,\omega-2\omega\cdot nn)-f(t,x,\omega))(\omega\cdot n)_+dn\,, \end{equation} and where $\langle\cdot\rangle$ denotes the average over the configuration $\vec{a}$ of heavy particles.
This result can be viewed as a linear analogue of the Boltzmann-Grad limit from the system of Newton's equations satisfied by all the molecules of a monatomic gas to the Boltzmann equation of the kinetic theory of gases. After (Gallavotti, 1969), this result was strengthened in various ways, most notably in (Spohn, 1978) (allowing for more general distributions of heavy particles) and (Boldrighini and al. 1983) (where it is proved that $f_r\to f$ as $r\to 0$ for a.e. configuration of heavy particles).
The periodic case
Whether the assumption of a random distribution of heavy particles is essential in the derivation of the Lorentz kinetic equation (52) is a natural question. Indeed, in the original work of Lorentz, this kinetic equation was proposed to describe the motion of electrons in metals; however, assuming that the metallic atoms are distributed under a Poisson law is somewhat unrealistic.
Consider instead the case where the heavy particles are periodically distributed, i.e. centered at the vertices of a lattice in the Euclidean space. For simplicity, assume that the lattice is $\epsilon\mathbf{Z}^d$ and that the radius of the heavy particles is $r=\epsilon^{d/(d-1)}$. As $\epsilon\to 0$, the number of heavy particles per unit volume is $N\simeq1/\epsilon^d$, so that $Nr^{d-1}\to 1$.
Denote by $S^\epsilon_t$ the one-parameter group $$ S^\epsilon_t:=S^{r,\vec{a}}_t\quad\hbox{ with }r=\epsilon^{d/(d-1)}\hbox{ and }\vec{a}=\epsilon\mathbf{Z}^d $$ where $S^{r,\vec{a}}_t$ is defined in (49).
For simplicity, we discuss only the case $d=2$, so that $r=\epsilon^2\simeq 1/N$.
Let $f^{in}\equiv f^{in}(x,\omega)$ be a continuous, compactly supported probability density on $\mathbf{R}^2\times\mathbf{S}^1$, and define $f_\epsilon$ by the formula $$ f_\epsilon(t,x,\omega):=\left\{ \begin{array}{ll}f^{in}(S^\epsilon_{-t}(x,\omega))&\quad\hbox{ if }|x-\epsilon k|>r\hbox{ for all }k\in\mathbf{Z}^2 \\ 0&\quad\hbox{ if }|x-\epsilon k|<r\hbox{ for some }k\in\mathbf{Z}^2\end{array} \right. $$
At variance with the random case, the following negative result has been obtained (Golse, 2008).
There does not exist any $\kappa>0$ and $p\in C(\mathbf{S}^1\times\mathbf{S}^1)$ satisfying $$ p(\omega,\omega')=p(\omega',\omega)\ge 0\quad\hbox{ and }\int_{\mathbf{S}^1}p(\omega,\omega')d\omega'=1 $$ such that $f_\epsilon(t,x,\omega)\to f(t,x,\omega)$ in the sense of distributions on $(0,\infty)\times\mathbf{R}^2\times\mathbf{S}^1$, where $f$ is the solution of \begin{equation}\tag{53} \left(\frac{\partial f}{\partial t}+c\omega\cdot\nabla_xf\right)(t,x,\omega)=\kappa\int_{\mathbf{S}^1}p(\omega,\omega')(f(t,x,\omega')-f(t,x,\omega))d\omega' \end{equation} with initial data $$ f(0,x,\omega)=f^{in}(x,\omega)\,. $$
In particular, $f_\epsilon$ cannot converge in the sense of distributions to the solution of the Lorentz kinetic equation (52), which is a special case of the linear Boltzmann equation (53), corresponding with $\kappa=2c\lambda$ and $p(\omega,\omega')=1/(2\pi)$.
This negative result is a consequence of the following observation: when the heavy particles are periodically distributed, there are too many long straight segments in the light particle trajectory. More precisely, the lengths of straight segments included in $$ \{x\in\mathbf{R}^2\,|\,\hbox{dist}(x,\epsilon\mathbf{Z}^2)>\epsilon^2\} $$ - corresponding with collisionless parts in the trajectories of the light particle - are not exponentially distributed, which excludes the convergence of $f_\epsilon$ to the solution of any linear Boltzmann equation of the type (53) (see (Bourgain and al., 1998)).
The description of the limit of $f_\epsilon$ as $\epsilon\to 0$ involves a kinetic equation in a phase space greater than $\mathbf{R}^2\times\mathbf{S}^1$ that is the single-particle phase space for monokinetic particles. This limit can be described as follows (taking $c=1$ for simplicity). As $\epsilon\to 0$ \begin{equation}\tag{54} f_\epsilon(t,x,\omega)\to f(t,x,\omega)=\int_0^\infty\int_{-1}^1F(t,x,\omega,s,h)dsdh \end{equation} in the sense of distributions on $(0,\infty)\times\mathbf{R}^2\times\mathbf{S}^1$ where $F(t,x,\omega,s,h)$ is the solution of the following problem: \begin{equation}\tag{55} \left(\frac{\partial F}{\partial t}+\omega\cdot\nabla_xF-\frac{\partial F}{\partial s}\right)(t,x,\omega,s,h)=\int_{-1}^1P(s,h|h')F(t,x,R[h]\omega,0,h')dh'\,, \end{equation} with initial data \begin{equation}\tag{56} F(0,x,\omega,s,h)=f^{in}(x,\omega)\int_s^\infty\int_{-1}^1P(\tau,h|h')dh'd\tau\,, \end{equation} where $R[h]$ is the rotation of an angle $\pi-2\arcsin(h)$ and \begin{equation}\tag{57} P(\tau,h|h')=\tfrac{6}{\pi^2}\inf\left(1,\frac{1}{h-h'}\left(\frac{1}{\tau}-1-h'\right)_+\right)\,. \end{equation}
While the limiting distribution function $f$ in (54) cannot be the solution of any linear kinetic equation of the type (53), it is expressed by (54) in terms of the function $F$ that involves additional variables and is governed by the linear kinetic equation (55). The physical meaning of the additional variables $s$ and $h$, and of the function $P(s,h|h')$ is as follows. The function $F(t,x,\omega,s,h)$ is the density at time $t$ of particles located at the position $x$, with direction $\omega$, whose next collision will occur at time $t+s$, with impact parameter $h$. (In the event of a collision between the light particle and the surface of a heavy particle at the position $z$, the impact parameter is $\sin(n_z,\omega)$, where $n_z$ is the unit outward normal vector to the surface of the heavy particle at the point $z$.) The function $P(s,h|h')$ denotes the probability that a collision involving the light particle with the surface of a heavy particle with impact parameter $h'$ will result in a new segment of the trajectory of the light particle, where the next collision will happen after a time $s$ with impact parameter $h$.
Equation (55)-(56) was proposed in (Caglioti and Golse, 2008), based on a decorrelation assumption left unverified (see also (Bykovskii and Ustinov, 2009) for (57)). (Marklof and Strömbergsson, 2008) provides explicit formulas for $P(s,h|h')$ in the case of particle interactions defined by scattering maps more general than hard sphere collisions. The first complete proof of the Boltzmann-Grad limit of the periodic Lorentz gas, valid for all lattices and in all space dimensions, can be found in (Marklof and Strömbergsson, 2011). The mathematical properties of equation (55) are analyzed in (Caglioti and Golse, 2010).
References
Alexander, R. (1976). Time Evolution for Infinitely Many Hard Spheres. Commun. Math. Phys. 49:217-232.
Bardos, C. (1970). Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport. "Ann. Sci. Ecole Norm. Sup.", Sér. 4, 3:185-233.
Bardos, C., Erdös, L., Golse, F., Mauser, N., Yau, H.-T. (2002). Derivation of the Schrödinger-Poisson equation from the quantum $N$-body problem. C. R. Math. Acad. Sci. Paris 334:515–520.
Boldrighini, C., Bunimovich, L.A., Sinai, Ya.G. (1983). On the Boltzmann equation for the Lorentz gas. J. Stat. Phys. 32:477–501.
Bouchut, F., Golse, F., Pulvirenti, M. (2000). Kinetic Equations and Asymptotic Theory, L. Desvillettes and B. Perthame eds, Gauthier-Villars, Paris. ISBN: 2-84299-110-9.
Bourgain, J., Golse, F., Wennberg, B. (1998). On the distribution of free path lengths for the periodic Lorentz gas. Commun. Math. Phys. 190:491–508.
Bykovskii, V.A., Ustinov, A.V. (2009). The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice. Izv. Math. 73:669–688.
Caglioti, E., Golse, F. (2008). The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions. C. R. Math. Acad. Sci. Paris 346:477–482.
Caglioti, E., Golse, F. (2010). On the Boltzmann-Grad Limit for the Two Dimensional Periodic Lorentz Gas. J. Stat. Phys. 141:264–317.
Cercignani, C. (1972). On the Boltzmann equation for rigid spheres. Transport Theory Statist. Phys. 2:211–225.
Cercignani, C. (1998). Ludwig Boltzmann. The Man Who Trusted Atoms. Oxford University Press Inc., New York. ISBN 0-19-850154-4 978-0-19-850154-1.
Gallavotti, G. (1969). Divergences and approach to equilibrium in the Lorentz and the wind–tree–models. Phys. Rev. (2) 185:308–322.
Gallagher, I., Saint-Raymond, L., Texier, B. (2012). From Newton to Boltzmann: hard spheres and short-range potentials. Preprint arXiv:1208.5753. To appear in Zürich Lectures in Advanced Mathematics, European Math. Soc. Publishing House.
Golse, F. (2008). On the periodic Lorentz gas in the Boltzmann-Grad scaling. Ann. Fac. Sci. Toulouse 17:735–749.
Grad, H. (1949): On the kinetic theory of rarefied gases. Comm. on Pure and Appl. Math. 2:331-407.
Illner, R., Shinbrot, M. (1984). The Boltzmann Equation: Global Existence for a Rare Gas in an Infinite Vacuum. Comm. Math. Phys. 95:217-226.
Illner, R., Pulvirenti, M. (1989). Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Comm. Math. Phys. 121:143–146.
Lanford, O.E. III (1975). The evolution of large classical system, in: Dynamical Systems, theory and applications, J. Moser, ed., Lecture Notes in Physics 38:1– 111, Springer-Verlag, Heidelberg. ISBN: 978-3-540-07171-6.
Lax, P.D. (2002). Functional Analysis. J. Wiley and Sons Inc. ISBN: 0-47L-55604-L.
Marklof, J., Strömbergsson, A. (2008): Kinetic transport in the two-dimensional periodic Lorentz gas. Nonlinearity 21:1413–1422.
Marklof, J., Strömbergsson, A. (2011). The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. of Math. (2) 174:225–298.
Sone, Y. (2007). Molecular Gas Dynamics. Theory, Techniques, and Applications. Birkhäuser, Boston. ISBN: 978-0-8176-4345-4
Spohn, H. (1978). The Lorentz process converges to a random flight process. Commun. Math. Phys. 60:277–290.
Uchiyama, K. (1988). On the Boltzmann-Grad limit for the Broadwell model of the Boltzmann equation. J. Statist. Phys. 52:331–355.
Ukai, S. (2001). The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem. Japan J. Indust. Appl. Math. 18: 383–392.
Recommended reading
Cercignani, C., Illner, R., Pulvirenti, M. (1994). The Mathematical Theory of Dilute Gases, Springer-Verlag, New York. ISBN: 0-387-94294-7.
Gallavotti, G. (1999). Statistical Mechanics. A Short Treatise. Springer Verlag, Berlin Heidelberg. ISBN: 3-540-64883-6.
Spohn, H. (1980). Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52:600–640.
