Periodic hard-sphere Boltzmann–Grad scaling convention

Fix an integer dimension $d\ge 2$, a period length $L>0$ metres, and an equal particle mass $m>0$ kilograms. The position space is the flat torus ${\mathbb{T}}_L^d:=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^d$. For lifts $x,y\in {\mathbb{R}}^d$ of points $[x],[y]\in {\mathbb{T}}_L^d$, define

${\mathrm{dist}}_{{\mathbb{T}}_L^d}([x],[y]):={\min }_{k\in {\mathbb{Z}}^d}\mathrm{\mid }x-y+Lk\mathrm{\mid },$

where $\mathrm{\mid }z\mathrm{\mid }=({\sum }_{\mathrm{\ell }=1}^d{z}_{\mathrm{\ell }}^2{)}^{1\mathrm{/}2}$ is the Euclidean norm. For $q_i=[x]$ and $q_j=[y]$, the vectors $x-y+Lk$, $k\in {\mathbb{Z}}^d$, are the displacement representatives from $q_j$ to $q_i$. The minimum is attained because $\mathrm{\mid }x-y+Lk\mathrm{\mid }\to \mathrm{\infty }$ as $\mathrm{\mid }k\mathrm{\mid }\to \mathrm{\infty }$, and its value is independent of the chosen lifts because changing lifts reindexes $k$. For each $N\ge 2$ under consideration, let ${\delta }_N>0$ be the physical sphere diameter in metres and put $a_N:={\delta }_N\mathrm{/}L$. An admissible configuration $(q_1,\dots ,q_N)$ has pairwise distances ${\mathrm{dist}}_{{\mathbb{T}}_L^d}(q_i,q_j)>{\delta }_N$. For a hard-sphere trajectory below, require ${\delta }_N<L\mathrm{/}2$; this restriction holds eventually in the scaling prescription. If two quotient displacement representatives of length ${\delta }_N$ existed, their difference would be a nonzero vector in $L{\mathbb{Z}}^d$, of length at least $L$, but the triangle inequality bounds its length by $2{\delta }_N<L$; thus the contact normal is unique.

A hard-sphere trajectory, when defined, is specified on one interval $I\subseteq \mathbb{R}$ with nonempty interior, with time $t\in I$ measured in seconds. For each $i=1,\dots ,N$ its maps are $q_i:I⟶{\mathbb{T}}_L^d,v_i:I⟶{\mathbb{R}}^d,$ where $v_i$ is measured in $\mathrm{m}{\mathrm{s}}^{-1}$. Require ${\mathrm{dist}}_{{\mathbb{T}}_L^d}(q_i(t),q_j(t))\ge {\delta }_N$ for every $t\in I$ and $i\mathrm{\ne }j$. The interval may be bounded or unbounded and may include either finite endpoint. Positions are continuous on $I$. Define the interior contact-time set

$C:=\{t\in I^{\circ }:{\mathrm{dist}}_{{\mathbb{T}}_L^d}(q_i(t),q_j(t))={\delta }_N\text{ for some }i<j\}\mathrm{.}$

Require $C$ to be locally finite in $I^{\circ }$: every compact interval $[a,b]\subset I^{\circ }$ contains only finitely many elements of $C$. Each $t_c\in C$ must be a binary contact as defined below. In particular, there is an $\eta >0$ with $(t_c-\eta ,t_c+\eta )\subset I^{\circ }$ and $C\cap (t_c-\eta ,t_c+\eta )=\{t_c\}$; this is the meaning of an isolated contact. All pairwise distances are strictly greater than ${\delta }_N$ at interior times outside $C$. A free-flight interval is a maximal open subinterval of $I^{\circ }$ containing no point of $C$. On each such interval require every $q_i$ and $v_i$ to be $C^1$ and to satisfy ${\dot{q}}_i=v_i$ and ${\dot{v}}_i=0$. Here ${\dot{q}}_i=v_i$ means that any local Euclidean lift of $q_i$ has derivative $v_i$; changing that lift by a constant lattice vector leaves the derivative unchanged.

At an interior contact time use both one-sided velocity limits and set $v_i(t_c)=v_i^{+}$, the outgoing value. A finite endpoint is included only when the corresponding limits from within $I$ exist, and the position and velocity maps take those values there. No collision rule involving a missing side of an endpoint is imposed; a two-sided update there would require an extension of the trajectory through that time. Contact times may accumulate only at endpoints of $I$ or at infinity; included finite endpoints still require the stated one-sided limits. These conventions specify a partial trajectory only and assert no continuation through a collision accumulation. A binary contact of particles $i,j$ at an interior time $t_c\in I^{\circ }$ means ${\mathrm{dist}}_{{\mathbb{T}}_L^d}(q_i(t_c),q_j(t_c))={\delta }_N$, while no other pair is in contact at $t_c$. Write $v_i^{-}$ and $v_i^{+}$ for the one-sided velocity limits at the contact. Let $n$ be the unique unit vector from the centre of $j$ to the centre of $i$ along the shortest quotient displacement. All particles other than $i,j$ retain their velocities at this contact. The equal-mass elastic reflection rule is

$v_i^{+}=v_i^{-}-((v_i^{-}-v_j^{-})\cdot n)n,v_j^{+}=v_j^{-}+((v_i^{-}-v_j^{-})\cdot n)n\mathrm{.}$

The Boltzmann–Grad scaling prescription is a sequence, indexed by all sufficiently large integers $N$, with $N\to \mathrm{\infty }$, $a_N\to 0$, and

$N{a}_N^{d-1}=\kappa$

for a fixed dimensionless $\kappa \in (0,\mathrm{\infty })$. Under this exact relation, $a_N=(\kappa \mathrm{/}N{)}^{1\mathrm{/}(d-1)}\to 0$, so the trajectory restriction ${\delta }_N<L\mathrm{/}2$ holds for every sufficiently large $N$. The factor $a_N$ is the diameter measured relative to the fixed box length, so the displayed parameter is dimensionless. This is a definition of a finite hard-sphere model and a scaling prescription only; it asserts neither global existence of the hard-sphere flow nor convergence to a Boltzmann equation.