Selected phenomenological cutoff collision model

For this selected model, an identical classical particle is nonrelativistic in the explicit sense that its velocity is a vector $v\in {\mathbb{R}}^3$ and its kinetic energy is the Newtonian scalar $m\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}2$. The mass $m>0$ is a primitive model parameter in kilograms. At time $t\ge 0$, $f_t(v)$ is the probability density of velocity for a randomly selected particle with respect to Lebesgue measure $dv$; it is nonnegative, normalized by $\int f_{t}dv=1$, and therefore has SI units $(\mathrm{m}{\mathrm{s}}^{-1}{)}^{-3}$. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the completed Lebesgue measure used by the mathematical gain definition (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A positive speed $v_0$ in $\mathrm{m}{\mathrm{s}}^{-1}$ is a reference scale used both in the dimensionless weight $w_0(v)=1+\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{v}_0^2$ defining the norm on $X$ below and to make the logarithm argument $v_0^3{f}_t(v)$ in the one-particle information functional dimensionless. The Boltzmann constant $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$ is a primitive conversion factor when that functional is assigned entropy units.

Let $X$ and $w_0(v)=1+\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{v}_0^2$ be as defined in Cutoff elastic-reflection collision measure and gain. On the interior of a time interval $J\subseteq [0,\mathrm{\infty })$, an admissible classical trajectory is a map $t\mapsto f_t\in X$ continuously differentiable in the $X$ norm, with its derivative extending continuously to included endpoints by one-sided derivatives, and with $f_t\ge 0$ almost everywhere and $\int f_{t}dv=1$ for every $t\in J$. Thus each admissible density has finite second velocity moment. The differential equation below is an equality in the normed space $X$, hence an equality of equivalence classes of functions of velocity. When a trajectory has a chosen representative $\tilde{f}:J\times {\mathbb{R}}^3\to [0,\mathrm{\infty })$ that is continuously differentiable in $t$ and for which the gain integral is finite and continuous in $(t,v)$, the pointwise form ${\mathrm{\partial }}_t\tilde{f}(t,v)=\lambda (\mathcal{G}({\tilde{f}}_t,{\tilde{f}}_t)(v)-\tilde{f}(t,v))$ is required for every $(t,v)$ on that representative; this pointwise conclusion is not asserted for arbitrary representatives of an $X$ class.

Adopt the spatially homogeneous kinetic equation

$\frac{d}{dt}{f}_t=\lambda (\mathcal{G}(f_t,f_t)-f_t)\text{in }X,\lambda >0,$

where $\mathcal{G}$ and the elastic reflection $(v,w)\mapsto (v^{\mathrm{\prime }},w^{\mathrm{\prime }})$ are exactly those of Cutoff elastic-reflection collision measure and gain and Well-defined elastic reflection gain and collision identities. The angular measure is the finite normalized area law specified there, and $\lambda$ is a prescribed rate in ${\mathrm{s}}^{-1}$, independent of relative speed; it is not a measured cross-section or a derived microscopic frequency. The selected incoming pair law is $f(v)f(w)dvdw$, and the gain uses the corresponding product $f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }})$ after reflection. This factorized independent-pair closure is adopted here; it is not deduced from reversible many-particle mechanics or from a molecular-chaos theorem. The mathematical existence and regularity claims are separate from this physical adoption.

The model's global mild evolution and its finite-energy hypotheses are mathematical claims treated separately from this physical adoption. This postulate makes no claim of a hard-sphere Boltzmann–Grad limit, generic gas applicability, spatially inhomogeneous PDE well-posedness, a hydrodynamic limit, or observed entropy increase.