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Cutoff elastic-reflection collision measure and gain

1 · Prerequisites

2 · Summary

This page fixes a homogeneous three-dimensional kinetic model with a finite normalized angular law and a two-particle elastic reflection. The mathematical items establish the gain map, its collision identities, global weighted-L1 evolution, a regular Gaussian-mixture branch, exact unclosed moment balances, and the H-theorem equality case. The physical postulate adopts the closed equation as a restricted model assumption. No hard-sphere Boltzmann–Grad derivation, general hydrodynamic limit, or empirical result is asserted.

The published mathematics page td-math-mixture-dynamics-and-shells provides related context. Its conditional collision-entropy theorem Conditional Boltzmann entropy inequality and equality form assumes a symmetric weak collision identity. The reflection map and parameter measure used here have incoming/outgoing and particle-exchange symmetries, and Well-defined elastic reflection gain and collision identities proves the corresponding weak identity for this gain. The local entropy proof uses that exact identity and its own regularity estimates; it does not take the separately formulated theorem as a premise.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Cutoff elastic-reflection collision measure and gain

Definition

Assume the Axiom of Countable Choice, written ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Under this assumption Lebesgue measure is the completion of Borel measure ( $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets), and every Lebesgue-measurable integrable class has a Borel representative (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra). Write $λ_{3}$ for Lebesgue measure on $R^{3}$ (Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ ). Fix a positive reference speed $v_{0} > 0$ , and put $w_{0} (v) = 1 + ∣ v ∣^{2} / v_{0}^{2}$ . Let $X$ be the real vector space of equivalence classes modulo $λ_{3}$ -almost-everywhere equality of measurable functions $f : R^{3} \to R$ with

$∥ f ∥_{1, w_{0}} : = \int_{R^{3}} w_{0} (v) ∣ f (v) ∣ d λ_{3} (v) < \infty .$

Let $S^{2} = {n \in R^{3} : ∣ n ∣ = 1}$ . For a Borel set $E \subseteq S^{2}$ , define the normalized finite angular measure

$σ (E) : = \frac{1}{4 π} \int_{0}^{π} \int_{0}^{2 π} 1_{E} (\sin θ \cos ϕ, \sin θ \sin ϕ, \cos θ) \sin θ d ϕ d θ .$

For $v, w \in R^{3}$ and $n \in S^{2}$ , define the postcollision velocities

$v^{'} = v - ((v - w) \cdot n) n, w^{'} = w + ((v - w) \cdot n) n .$

The collision-parameter measure is $d μ = d λ_{3} (v) d λ_{3} (w) d σ (n)$ on $R^{3} \times R^{3} \times S^{2}$ . Its angular factor is finite and normalized; there is no singular angular weight. For $f, g \in X$ , choose Borel representatives and define the gain $G (f, g)$ by the almost-everywhere integral

$G (f, g) (v) : = \int_{R^{3} \times S^{2}} f (v^{'}) g (w^{'}) d λ_{3} (w) d σ (n) .$

The value on any exceptional null set is immaterial because the output is an equivalence class. The next item proves that this definition is independent of the representatives, maps $X \times X$ to $X$ , and has the stated collision and moment identities.

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Well-defined elastic reflection gain and collision identities

Statement

Let $G$ , $v^{'}$ , $w^{'}$ , $X$ , $w_{0}$ , and $μ$ be as in Cutoff elastic-reflection collision measure and gain. For $f, g \in X$ , the gain is independent of the chosen Borel representatives, is bilinear, and satisfies

$∥ G (f, g) ∥_{1, w_{0}} \leq 2 ∥ f ∥_{1, w_{0}} ∥ g ∥_{1, w_{0}} .$

For $f \geq 0$ with finite mass $M = \int f d v$ and finite second moment, it also satisfies

$\int G (f, f) d v = M^{2}, \int v G (f, f) d v = M \int v f d v, \int ∣ v ∣^{2} G (f, f) d v = M \int ∣ v ∣^{2} f d v .$

If $M = 1$ , set $A = f (v) f (w)$ , $B = f (v^{'}) f (w^{'})$ , and $Δ φ = φ (v) + φ (w) - φ (v^{'}) - φ (w^{'})$ . For every real-valued measurable test $φ$ such that

$\int_{R^{3}} ∣ φ (v) ∣ G (f, f) (v) d v < \infty, \int_{R^{3}} ∣ φ (v) ∣ f (v) d v < \infty, \int ∣ A Δ φ ∣ d μ < \infty, \int ∣ B Δ φ ∣ d μ < \infty,$

the following identity holds:

$\int_{R^{3}} φ (v) (G (f, f) (v) - f (v)) d v = \frac{1}{4} \int (B - A) Δ φ d μ .$

For each fixed $n$ , the map $(v, w) \mapsto (v^{'}, w^{'})$ is a linear orthogonal involution, preserves $v + w$ and $∣ v ∣^{2} + ∣ w ∣^{2}$ , and has Jacobian determinant of absolute value one. The angular measure has full support, hence the product parameter measure $μ$ has full support on $R^{3} \times R^{3} \times S^{2}$ . Every pair $(v, w)$ and $(\tilde{v}, \tilde{w})$ with the same momentum and squared-speed sum is related by one such reflection.

Facts & Assumptions

Given: ${A C}_{ω}$ , a reference speed $v_{0} > 0$ , elements of the weighted space $X$ , and the collision parameterization of Cutoff elastic-reflection collision measure and gain.

[A1]

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), as required by the completed Lebesgue product and measurable change-of-variables suppliers used below.

[F1]

Cutoff elastic-reflection collision measure and gain: $v^{'} = v - ((v - w) \cdot n) n$ , $w^{'} = w + ((v - w) \cdot n) n$ , $d μ = d v d w d σ (n)$ , and $G (f, g) (v) = \int f (v^{'}) g (w^{'}) d w d σ (n)$ .

[F2]

$L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets and A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra: under ${A C}_{ω}$ , Lebesgue measurable sets are the completion of Borel sets, and each completed-measurable function is almost-everywhere equal to a Borel-measurable representative.

[F3]

Tonelli and Fubini for the completed product, with only almost-everywhere section measurability: for sigma-finite factors, nonnegative measurable functions and integrable functions on the completed product have measurable sections and equal iterated integrals outside the stated null exceptional sections.

[F4]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: under ${A C}_{ω}$ , a $C^{1}$ diffeomorphism $T : U \to V$ satisfies $\int_{V} h (y) d y = \int_{U} h (T (x)) ∣ \det D T (x) ∣ d x$ for every nonnegative Lebesgue measurable $h$ .

Proof

Given: [A1], $v_{0} > 0$ , and [F1]–[F4].

Proof technique: direct.

1.1F1algebra

Put $c = (v + w) / 2$ and $r = (v - w) / 2$ . The formulas in [F1] give $v^{'} = c + r - 2 (r \cdot n) n$ and $w^{'} = c - r + 2 (r \cdot n) n$ . The map $r \mapsto r - 2 (r \cdot n) n$ is reflection across the plane perpendicular to $n$ : applying it twice returns $r$ , and its squared norm is $∣ r ∣^{2}$ . The center $c$ is fixed. Hence the pair map is an orthogonal involution on $R^{6}$ , its determinant has absolute value one, $v^{'} + w^{'} = v + w$ , and $∣ v^{'} ∣^{2} + ∣ w^{'} ∣^{2} = ∣ v ∣^{2} + ∣ w ∣^{2}$ .

1.2F1constructalgebra

The defining integral of $σ$ has total mass $(4 π)^{- 1} (2) (2 π) = 1$ . Every nonempty relative-open subset of $S^{2}$ contains a patch away from a coordinate pole and seam; its preimage contains an open rectangle on which $\sin θ > 0$ , so it has positive $σ$ -measure. Every nonempty open subset of the full parameter space contains a product of two Euclidean balls and such a sphere patch; each factor has positive measure, so $μ$ has full support. For an elastic pair, set $c = (v + w) / 2$ , $r = (v - w) / 2$ , and $\tilde{r} = (\tilde{v} - \tilde{w}) / 2$ . Momentum and squared-speed conservation give the same center and $∣ \tilde{r} ∣ = ∣ r ∣$ . If $\tilde{r} \neq r$ , choose $n = (r - \tilde{r}) / ∣ r - \tilde{r} ∣$ ; the equal-norm identity gives $2 (r \cdot n) n = r - \tilde{r}$ , hence the reflection sends $r$ to $\tilde{r}$ . If $\tilde{r} = r$ , choose any unit $n$ perpendicular to $r$ when $r \neq 0$ , and any unit $n$ when $r = 0$ . Thus every elastic pair is attained.

2.1F1F2F3F4step 1.1algebra

For each fixed $n$ , let $T_{n} (v, w) = (v^{'}, w^{'})$ . If $h \in L^{1} (R^{6})$ is real-valued, write $h = h^{+} - h^{-}$ , where $h^{+} = \max (h, 0)$ and $h^{-} = \max (- h, 0)$ . Applying [F4] to $∣ h ∣$ shows that $h \circ T_{n}$ is absolutely integrable because $T_{n}$ has unit absolute Jacobian. Applying [F4] separately to $h^{+}$ and $h^{-}$ , whose integrals are finite, and subtracting gives $\int_{R^{6}} h (T_{n} (v, w)) d v d w = \int_{R^{6}} h (v, w) d v d w .$ By [F2], choose a finite Borel representative $h_{0}$ equal to $h$ almost everywhere. For every fixed $n$ , [F4] gives the same change-of-variables equality for $h_{0}$ , hence for $h$ . The map $(v, w, n) \mapsto h_{0} (T_{n} (v, w))$ is Borel, and [F4] gives $\int_{R^{6}} ∣ h_{0} (T_{n} (v, w)) ∣ d v d w = ∥ h_{0} ∥_{1}$ for every $n$ . Since $σ (S^{2}) = 1$ , [F3] permits integration of this absolutely integrable identity in $n$ . Thus the cited nonnegative change-of-variables result has the signed $L^{1}$ consequence used below.

2.2A1F2F3F4step 1.1algebra

Replace the representatives of $f$ and $g$ by Borel representatives using [F2]. Tonelli [F3] makes the sections measurable and permits integration in any order. For each fixed $n$ , the pair reflection in step 1.1 is a $C^{1}$ diffeomorphism with unit absolute Jacobian, so [F4] gives $\int_{R^{6}} h (v^{'}, w^{'}) d v d w = \int_{R^{6}} h (v, w) d v d w .$ It follows that $\int ∣ G (f, g) ∣ d v \leq ∥ f ∥_{1} ∥ g ∥_{1}$ . The same change of variables shows that altering either representative on a Lebesgue-null set changes the integrand only on a $μ$ -null set; therefore the output class is independent of representatives. Bilinearity follows from linearity of the integral. Since $∣ v ∣^{2} \leq ∣ v^{'} ∣^{2} + ∣ w^{'} ∣^{2}$ , one has $w_{0} (v) \leq w_{0} (v^{'}) + w_{0} (w^{'})$ . Applying Tonelli and [F4] again gives $∥ G (f, g) ∥_{1, w_{0}} \leq ∥ f ∥_{1, w_{0}} ∥ g ∥_{1} + ∥ f ∥_{1} ∥ g ∥_{1, w_{0}} \leq 2 ∥ f ∥_{1, w_{0}} ∥ g ∥_{1, w_{0}} .$ The same bounds applied to $∣ f ∣, ∣ g ∣$ define the signed bilinear extension.

3.1F1F3F4step 1.1step 2.1algebra

For $f \geq 0$ , use Tonelli and the unit-Jacobian involution to obtain $\int G (f, f) d v = \int f (v) f (w) d v d w = M^{2}$ . For the first moment, the same change of variables gives $\int v G (f, f) d v = \int v^{'} f (v) f (w) d μ$ . The simultaneous exchange $(v, w, n) \mapsto (w, v, n)$ preserves $μ$ and swaps $v^{'}$ with $w^{'}$ , while $f (v) f (w)$ is unchanged. Averaging the two expressions replaces $v^{'}$ by $(v^{'} + w^{'}) / 2 = (v + w) / 2$ , yielding $M \int v f d v$ . Applying the same averaging to the second moment uses $(∣ v^{'} ∣^{2} + ∣ w^{'} ∣^{2}) / 2 = (∣ v ∣^{2} + ∣ w ∣^{2}) / 2$ and gives $M \int ∣ v ∣^{2} f d v$ . The first-moment change of variables is applied coordinatewise to signed integrable functions; finiteness follows from the finite mass and second moment.

4.1F1F2F3F4step 1.1step 2.1algebra∎

Let $M = 1$ and let $φ$ satisfy the four absolute-integrability conditions in the Statement. By [F2], choose a finite Borel representative $φ_{0}$ equal to $φ$ almost everywhere. The Lebesgue integrals weighted by $f$ or $G (f, f)$ are unchanged, and the products with $A$ or $B$ agree $μ$ -almost everywhere: the unprimed exceptional sets are null by Tonelli, while for each fixed $n$ the primed exceptional sets are null by the measure-preserving involution in step 1.1, so Tonelli makes their union $μ$ -null. Thus it suffices to prove the identity for $φ_{0}$ . Tonelli and the gain definition give $\int φ_{0} (v) G (f, f) (v) d v = \int φ_{0} (v) B d μ$ . By Tonelli, $h_{n} (v, w) = φ_{0} (v) f (v^{'}) f (w^{'})$ belongs to $L^{1} (R^{6})$ for $σ$ -almost every $n$ , since its absolute integral over $n, v, w$ is $\int ∣ φ_{0} ∣ G (f, f) d v < \infty$ . For those $n$ , step 2.1 applied to $h_{n}$ gives $\int φ_{0} (v) f (v^{'}) f (w^{'}) d v d w = \int φ_{0} (v^{'}) f (v) f (w) d v d w$ ; the fixed- $n$ change of variables also shows the right side is absolutely integrable. Fubini then gives $\int φ_{0} (v) B d μ = \int φ_{0} (v^{'}) A d μ$ . The loss term is $\int φ_{0} (v) A d μ$ because $\int f = 1$ and $σ (S^{2}) = 1$ . The stated integrability conditions make these terms absolutely integrable; particle exchange gives the same for the $w^{'}$ and $w$ terms, and the conditions on $A Δ φ_{0}$ and $B Δ φ_{0}$ make the incoming/outgoing change of variables integrable. Averaging the difference with its image under simultaneous particle exchange yields $\frac{1}{2} \int A [φ_{0} (v^{'}) + φ_{0} (w^{'}) - φ_{0} (v) - φ_{0} (w)] d μ = - \frac{1}{2} \int A Δ φ_{0} d μ$ . The incoming/outgoing involution sends $A$ to $B$ and $Δ φ_{0}$ to $- Δ φ_{0}$ , so $\int (B - A) Δ φ_{0} d μ = - 2 \int A Δ φ_{0} d μ$ . Combining these equalities gives the claimed factor $1 / 4$ identity.

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Kinetic density, velocity, covariance stress and heat flux

Definition

Use the velocity space $R^{3}$ and Lebesgue measure $d v$ fixed by Cutoff elastic-reflection collision measure and gain. Let $Ω = \prod_{j = 1}^{3} (R / L_{j} Z)$ be a flat periodic spatial domain with periods $L_{j} > 0$ (in metres in the physical interpretation), and let $I \subseteq R$ be an open time interval. Let $F : Ω \times R^{3} \times I \to [0, \infty)$ be a measurable number density in position and velocity. At each $(r, t)$ where the integrals below are finite, assume $0 < n (r, t) : = \int_{R^{3}} F (r, v, t) d v < \infty$ and $\int ∣ v ∣^{3} F (r, v, t) d v < \infty$ . Fix a particle mass $m > 0$ in kilograms, with number count dimensionless. In SI, $F$ has units $m^{- 3} (m s^{- 1})^{- 3}$ .

Define $(x \otimes y)_{i j} = x_{i} y_{j}$ , $\nabla_{r} \cdot T$ by $(\nabla_{r} \cdot T)_{i} = \sum_{j} \partial_{r_{j}} T_{i j}$ , and $A : B = \sum_{i, j} A_{i j} B_{i j}$ . Define the number density $n$ , mass density $ρ = m n$ , mean velocity $u = n^{- 1} \int v F d v$ , kinetic covariance momentum flux

$Π = m \int_{R^{3}} (v - u) \otimes (v - u) F d v,$

internal kinetic-energy density $e = \frac{1}{2} tr Π$ , and heat-flux vector

$q = \frac{m}{2} \int_{R^{3}} ∣ v - u ∣^{2} (v - u) F d v .$

Their SI units are respectively $m^{- 3}$ , $k g m^{- 3}$ , $m s^{- 1}$ , pascals, $J m^{- 3}$ , and $J m^{- 2} s^{- 1}$ . The tensor $Π$ is symmetric because each integrand $(v - u) \otimes (v - u)$ is symmetric, and it is positive semidefinite because for every $ξ \in R^{3}$ , $ξ^{T} Π ξ = m \int [ξ \cdot (v - u)]^{2} F d v \geq 0$ . It is the kinetic momentum-flux covariance; in the balance law below it enters with a plus sign, while a continuum tensile-stress variable defined from it would be $- Π$ .

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Selected phenomenological cutoff collision model

Physical scope

An adopted spatially homogeneous one-particle velocity-density model for identical nonrelativistic particles with a finite normalized angular reflection law and prescribed collision rate. It is not derived from finite-particle mechanics, a hard-sphere Boltzmann–Grad limit, or a general hydrodynamic limit.

Postulate

For this selected model, an identical classical particle is nonrelativistic in the explicit sense that its velocity is a vector $v \in R^{3}$ and its kinetic energy is the Newtonian scalar $m ∣ v ∣^{2} / 2$ . The mass $m > 0$ is a primitive model parameter in kilograms. At time $t \geq 0$ , $f_{t} (v)$ is the probability density of velocity for a randomly selected particle with respect to Lebesgue measure $d v$ ; it is nonnegative, normalized by $\int f_{t} d v = 1$ , and therefore has SI units $(m s^{- 1})^{- 3}$ . Assume ${A C}_{ω}$ for the completed Lebesgue measure used by the mathematical gain definition (The Axiom of Countable Choice ( ${A C}_{ω}$ )). A positive speed $v_{0}$ in $m s^{- 1}$ is a reference scale used both in the dimensionless weight $w_{0} (v) = 1 + ∣ v ∣^{2} / v_{0}^{2}$ 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 $J K^{- 1}$ is a primitive conversion factor when that functional is assigned entropy units.

Let $X$ and $w_{0} (v) = 1 + ∣ v ∣^{2} / v_{0}^{2}$ be as defined in Cutoff elastic-reflection collision measure and gain. On the interior of a time interval $J \subseteq [0, \infty)$ , 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 $f_{t} \geq 0$ almost everywhere and $\int f_{t} d v = 1$ 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 $\tilde{f} : J \times R^{3} \to [0, \infty)$ that is continuously differentiable in $t$ and for which the gain integral is finite and continuous in $(t, v)$ , the pointwise form $\partial_{t} \tilde{f} (t, v) = λ (G ({\tilde{f}}_{t}, {\tilde{f}}_{t}) (v) - \tilde{f} (t, v))$ 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

$\frac{d}{d t} f_{t} = λ (G (f_{t}, f_{t}) - f_{t}) in X, λ > 0,$

where $G$ and the elastic reflection $(v, w) \mapsto (v^{'}, w^{'})$ 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 $λ$ is a prescribed rate in $s^{- 1}$ , independent of relative speed; it is not a measured cross-section or a derived microscopic frequency. The selected incoming pair law is $f (v) f (w) d v d w$ , and the gain uses the corresponding product $f (v^{'}) f (w^{'})$ 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.

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Regularity and invariance of the cutoff Gaussian-mixture class

Statement

Fix $u \in R^{3}$ and positive velocity-squared parameters $0 < a < b$ (in a physical interpretation, both have units $(m s^{- 1})^{2}$ ). Let

$K_{a, b} : = {Σ \in R^{3 \times 3} : Σ = Σ^{T}, a I \leq Σ \leq b I},$

where the matrix inequalities mean $a ∣ z ∣^{2} \leq z^{T} Σ z \leq b ∣ z ∣^{2}$ for every $z \in R^{3}$ . For $Σ \in K_{a, b}$ , define the normalized Gaussian density

$γ_{Σ, u} (v) : = (2 π)^{- 3 / 2} (\det Σ)^{- 1 / 2} \exp ⁣ (- \frac{1}{2} (v - u)^{T} Σ^{- 1} (v - u)) .$

Let $M_{u, a, b}$ be the closure in the weighted space $X$ of all finite convex mixtures $\sum_{j = 1}^{N} α_{j} γ_{Σ_{j}, u}$ with $Σ_{j} \in K_{a, b}$ , $α_{j} \geq 0$ , and $\sum_{j} α_{j} = 1$ . Then:

Every $f \in M_{u, a, b}$ has a $C^{\infty}$ representative satisfying

$(2 π b)^{- 3 / 2} e^{- ∣ v - u ∣^{2} / (2 a)} \leq f (v) \leq (2 π a)^{- 3 / 2} e^{- ∣ v - u ∣^{2} / (2 b)} .$

For every multi-index $α$ , there is a finite constant $C_{α, a, b}$ , depending only on $α, a, b$ and not on $f$ , such that

$∣ \partial^{α} f (v) ∣ \leq C_{α, a, b} (1 + ∣ v - u ∣^{∣ α ∣}) e^{- ∣ v - u ∣^{2} / (2 b)} .$

The common derivative bounds imply that every sequence in $M_{u, a, b}$ has a subsequence converging locally uniformly together with all velocity derivatives.

The gain in Cutoff elastic-reflection collision measure and gain maps this class into itself: $G (f, g) \in M_{u, a, b}$ whenever $f, g \in M_{u, a, b}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $u \in R^{3}$ , positive velocity-squared parameters $0 < a < b$ , and the definitions in the Statement.

[A1]

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )); it is the only choice principle used in the measurable representatives and countable diagonal subsequence arguments.

[F1]

Cutoff elastic-reflection collision measure and gain: $G (f, g) (v) = \int f (v^{'}) g (w^{'}) d w d σ (n)$ , with $v^{'} = v - ((v - w) \cdot n) n$ and $w^{'} = w + ((v - w) \cdot n) n$ .

[F2]

Well-defined elastic reflection gain and collision identities: the pair reflection is orthogonal with unit absolute Jacobian, and $G$ is bilinear with $∥ G (f, g) ∥_{1, w_{0}} \leq 2 ∥ f ∥_{1, w_{0}} ∥ g ∥_{1, w_{0}}$ .

[F3]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: a real self-adjoint endomorphism of a finite-dimensional real inner-product space has an orthonormal eigenbasis.

[F4]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ : $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ .

[F5]

Dominated convergence: almost-everywhere convergence dominated by a single integrable function permits passage of limits through the Lebesgue integral.

[F6]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: under ${A C}_{ω}$ , a $C^{1}$ diffeomorphism changes Lebesgue integrals by the absolute Jacobian determinant.

[F7]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for sigma-finite factors, Tonelli permits iterated integration of nonnegative measurable functions.

Proof

Given: [A1], $u$ , $a$ , $b$ , and [F1]–[F7].

Proof technique: direct.

1.1F3F4algebra

By [F3], diagonalize $Σ$ in an orthonormal basis with eigenvalues $s_{1}^{2}, s_{2}^{2}, s_{3}^{2} \in [a, b]$ . The linear substitution $y_{i} = (v - u)_{i} / s_{i}$ has Jacobian $\det (Σ)^{- 1 / 2}$ ; [F4] in each coordinate gives $\int γ_{Σ, u} (v) d v = 1$ . The same eigenvalue bounds give $(\det Σ)^{- 1 / 2} \in [b^{- 3 / 2}, a^{- 3 / 2}]$ and $∣ v - u ∣^{2} / b \leq (v - u)^{T} Σ^{- 1} (v - u) \leq ∣ v - u ∣^{2} / a$ , proving the two Gaussian envelopes for each component and hence for every finite mixture.

1.2F2F3F4F6F7algebra

Let $O_{n} : R^{6} \to R^{6}$ be the orthogonal pair map from [F2], and let $P : R^{6} \to R^{3}$ project onto the first component. For component densities $γ_{Σ, u}$ and $γ_{T, u}$ , the joint centered covariance is $S = diag (Σ, T)$ . Using [F6] under $O_{n}$ , the density of the pair after reflection is the centered Gaussian with covariance $S_{n} = O_{n} S O_{n}^{T}$ and mean $(u, u)$ . Write $S_{n}$ in blocks as $(\begin{array}{cc} C & D \\ D^{T} & E \end{array})$ . Its first marginal is Gaussian with covariance $C$ : indeed $C$ is positive definite, the Schur complement $H = E - D^{T} C^{- 1} D$ is positive definite, and block elimination gives $\det S_{n} = \det C \det H$ and $(x, y)^{T} S_{n}^{- 1} (x, y) = x^{T} C^{- 1} x + (y - D^{T} C^{- 1} x)^{T} H^{- 1} (y - D^{T} C^{- 1} x) .$ Integrating in $y$ using [F3], [F4], and [F7] gives the normalized marginal $γ_{C, u}$ . Also $C = P S_{n} P^{T}$ satisfies $a I \leq C \leq b I$ : for every $z$ , $z^{T} C z = (z, 0)^{T} O_{n} S O_{n}^{T} (z, 0)$ lies between $a ∣ z ∣^{2}$ and $b ∣ z ∣^{2}$ .

2.1F3step 1.1algebra

Differentiating a component $γ_{Σ, u}$ in $v$ multiplies it by a polynomial in $Σ^{- 1} (v - u)$ ; each coefficient is uniformly bounded because $∥ Σ^{- 1} ∥ \leq a^{- 1}$ . The exponent has the upper bound from step 1.1, so every derivative of a finite mixture has the stated polynomial-Gaussian envelope, uniformly over all finite mixtures in this class.

2.2F1F5F7step 1.1step 1.2algebra

Therefore $G (γ_{Σ, u}, γ_{T, u})$ is the pointwise angular average $\int_{S^{2}} γ_{C_{n}, u} d σ (n)$ with $C_{n} \in K_{a, b}$ . The map $n \mapsto C_{n}$ is continuous, and $C \mapsto γ_{C, u}$ is continuous in $X$ : pointwise convergence follows from the formula, while the weighted upper Gaussian envelope from step 1.1 is an integrable dominator and [F5] gives weighted- $L^{1}$ convergence. For each $n$ , continuity supplies a relative-open neighborhood on which the $X$ -distance from $γ_{C_{n}, u}$ is less than $ε$ ; compactness of $S^{2}$ gives a finite subcover. Successively subtracting earlier open sets makes this a finite Borel partition $E_{j}$ , and choosing one $n_{j}$ in each nonempty piece gives variation at most $2 ε$ on that piece. Tonelli [F7] and the triangle inequality bound the weighted- $L^{1}$ distance from the angular average to $\sum_{j} σ (E_{j}) γ_{C_{n_{j}}, u}$ by $2 ε \sum_{j} σ (E_{j}) = 2 ε$ . Since $ε > 0$ is arbitrary, the angular average belongs to $M_{u, a, b}$ . Bilinear expansion proves the same for finite convex mixtures in both inputs.

3.1F2step 2.2algebra

For general $f, g \in M_{u, a, b}$ , choose finite-mixture sequences $f_{k} \to f$ and $g_{k} \to g$ in $X$ . The bound [F2] gives $∥ G (f_{k}, g_{k}) - G (f, g) ∥_{1, w_{0}} \leq 2 ∥ f_{k} - f ∥_{1, w_{0}} ∥ g_{k} ∥_{1, w_{0}} + 2 ∥ f ∥_{1, w_{0}} ∥ g_{k} - g ∥_{1, w_{0}} ⟶ 0.$ Each $G (f_{k}, g_{k})$ is in $M_{u, a, b}$ by step 2.2, so closedness proves $G (f, g) \in M_{u, a, b}$ .

4.1A1F5step 1.1step 2.1algebra∎

Let $f_{k}$ be finite mixtures converging in $X$ to $f \in M_{u, a, b}$ . By step 2.1, every derivative sequence $(\partial^{α} f_{k})_{k}$ is uniformly bounded and equicontinuous on every closed ball, since the next derivatives have a common bound there. On the countable dense rational grid in each ball, bounded real sequences admit convergent subsequences: repeatedly bisect a bounded interval, retain a half containing infinitely many terms, and use the nested intervals of lengths tending to zero to select a convergent subsequence. A countable diagonal selection over grid points, integer-radius balls, and multi-indices makes every derivative converge on every grid. Equicontinuity and a finite rational $δ$ -net of each ball then make these subsequences uniformly Cauchy there, hence locally uniformly convergent. The fundamental theorem of calculus along coordinate segments identifies the limits as derivatives of a smooth function $\tilde{f}$ . Weighted- $L^{1}$ convergence identifies $\tilde{f} = f$ almost everywhere on every ball, so $\tilde{f}$ represents $f$ . The pointwise envelopes and derivative bounds pass to the local uniform limit. This proves part 1 for every class element.

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Global finite-energy cutoff collision evolution

Statement

Assume ${A C}_{ω}$ . Fix a reference speed $v_{0} > 0$ , a rate $λ > 0$ , and the weighted space $X = L^{1} (R^{3}, w_{0} d v)$ with $w_{0} (v) = 1 + ∣ v ∣^{2} / v_{0}^{2}$ , as in Cutoff elastic-reflection collision measure and gain. Let $f_{0} \in X$ be nonnegative with $\int f_{0} d v = 1$ , first moment $m_{1} = \int v f_{0} (v) d v$ , and finite second moment $E_{2} = \int ∣ v ∣^{2} f_{0} (v) d v$ . Put

$K : = {g \in X : g \geq 0 a.e., \int g d v = 1, \int v g d v = m_{1}, \int ∣ v ∣^{2} g d v = E_{2}} .$

Then there is a unique $f \in C ([0, \infty), K)$ , with continuity in the $X$ norm, satisfying

$f_{t} = e^{- λ t} f_{0} + λ \int_{0}^{t} e^{- λ (t - s)} G (f_{s}, f_{s}) d s,$

where the time integral is the norm limit of Riemann sums in $X$ . For all $t \geq 0$ , $f_{t} \geq 0$ , $\int f_{t} d v = 1$ , $\int v f_{t} d v = \int v f_{0} d v$ , and $\int ∣ v ∣^{2} f_{t} d v = E_{2}$ . If $f_{0} > 0$ almost everywhere, then $f_{t} \geq e^{- λ t} f_{0}$ almost everywhere.

If in addition $f_{0} \in M_{u, a, b}$ for some $u \in R^{3}$ and $0 < a < b$ , with $M_{u, a, b}$ as in Regularity and invariance of the cutoff Gaussian-mixture class, then each $f_{t}$ has a positive $C^{\infty}$ velocity representative with the Gaussian bounds in that lemma; these representatives depend continuously on $t$ locally uniformly in velocity, are $C^{1}$ in $t$ , and satisfy $\partial_{t} f_{t} = λ (G (f_{t}, f_{t}) - f_{t})$ pointwise.

Facts & Assumptions

Given: ${A C}_{ω}$ , $v_{0} > 0$ , $λ > 0$ , and $f_{0} \in X$ with the nonnegativity, normalization, and finite-second-moment hypotheses of the Statement.

[A1]

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the completed Lebesgue measure suppliers in the collision-operator construction.

[F1]

Cutoff elastic-reflection collision measure and gain: $X$ has norm $∥ f ∥_{1, w_{0}} = \int (1 + ∣ v ∣^{2} / v_{0}^{2}) ∣ f (v) ∣ d v$ , and $G$ is the elastic-reflection gain.

[F2]

Well-defined elastic reflection gain and collision identities: $G$ maps $X \times X$ bilinearly to $X$ with $∥ G (f, g) ∥_{1, w_{0}} \leq 2 ∥ f ∥_{1, w_{0}} ∥ g ∥_{1, w_{0}}$ , and for mass-one $f$ its gain has the same mass, first moment, and second moment as $f$ .

[F3]

Regularity and invariance of the cutoff Gaussian-mixture class: the closed convex class $M_{u, a, b}$ is invariant under $G$ , and its elements have common positive Gaussian lower bounds and smooth Gaussian derivative envelopes.

[F4]

A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point: a contraction of a nonempty complete metric space into itself has a unique fixed point.

[F5]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative measurable functions on a sigma-finite product have measurable sections and equal iterated integrals.

[F6]

Dominated convergence: pointwise almost-everywhere convergence dominated by one integrable function permits passage of limits through the Lebesgue integral.

Proof

Given: [A1], $v_{0} > 0$ , $λ > 0$ , $f_{0} \in X$ , and [F1]–[F6].

Proof technique: direct.

1.1A1F5construct

The weighted measure $w_{0} d v$ is sigma-finite because $w_{0}$ is bounded on every bounded ball. To prove $X$ complete, let $(g_{n})$ be Cauchy in its norm and choose a subsequence $(g_{n_{j}})$ with $∥ g_{n_{j + 1}} - g_{n_{j}} ∥_{1, w_{0}} \leq 2^{- j}$ . By [F5], the nonnegative series $\sum_{j} w_{0} ∣ g_{n_{j + 1}} - g_{n_{j}} ∣$ is finite almost everywhere and has finite integral. Hence $g_{n_{j}}$ converges pointwise almost everywhere to a measurable $g$ , and the integral of each tail is bounded by $\sum_{j \geq J} 2^{- j}$ , so $g_{n_{j}} \to g$ in $X$ . The same estimates give $∥ g ∥_{1, w_{0}} < \infty$ . Since the original sequence was Cauchy, it converges to the same $g$ . Thus $X$ is complete.

2.1F1step 1.1algebra

Put $m_{1} = \int v f_{0} d v$ and $E_{2} = \int ∣ v ∣^{2} f_{0} d v$ , and let $K$ be the subset of $X$ of nonnegative functions with mass $1$ , first moment $m_{1}$ , and second moment $E_{2}$ . It is nonempty because $f_{0} \in K$ . It is closed: if $g_{n} \to g$ in $X$ , then $\int ∣ g_{n} - g ∣ \leq ∥ g_{n} - g ∥_{1, w_{0}}$ , $\int ∣ v ∣ ∣ g_{n} - g ∣ \leq (v_{0} / 2) ∥ g_{n} - g ∥_{1, w_{0}}$ , and $\int ∣ v ∣^{2} ∣ g_{n} - g ∣ \leq v_{0}^{2} ∥ g_{n} - g ∥_{1, w_{0}}$ ; also the negative part of $g$ is bounded by $∣ g - g_{n} ∣$ whenever $g_{n} \geq 0$ . Thus all constraints pass to the limit, and $K$ is complete. The continuous path space $C ([0, T], K)$ with the supremum metric is complete: a uniformly Cauchy sequence has pointwise limits in $K$ , converges uniformly, and the triangle inequality with one continuous approximant proves continuity of the limit.

2.2F1F2step 1.1algebra

Let $M_{0} = 1 + E_{2} / v_{0}^{2}$ ; every $g \in K$ has $∥ g ∥_{1, w_{0}} = M_{0}$ . On $C ([0, T], K)$ define $Φ (g)_{t} = e^{- λ t} f_{0} + λ \int_{0}^{t} e^{- λ (t - s)} G (g_{s}, g_{s}) d s .$ The integrand is norm-continuous because $s \mapsto g_{s}$ is continuous and [F2] bounds the bilinear map. Its Riemann sums are Cauchy in $X$ by uniform continuity on $[0, T]$ and completeness from step 1.1. The gain identities in [F2] show that $Φ (g)_{t}$ has mass $1$ , first moment $m_{1}$ , and second moment $E_{2}$ , since $e^{- λ t} + λ \int_{0}^{t} e^{- λ (t - s)} d s = 1$ ; nonnegativity is preserved as well. Therefore $Φ$ maps this path space into itself.

3.1F2F4step 2.1step 2.2algebra

Bilinearity and [F2] give $∥ G (g_{s}, g_{s}) - G (h_{s}, h_{s}) ∥_{1, w_{0}} \leq 2 ∥ g_{s} - h_{s} ∥_{1, w_{0}} (∥ g_{s} ∥_{1, w_{0}} + ∥ h_{s} ∥_{1, w_{0}}) = 4 M_{0} ∥ g_{s} - h_{s} ∥_{1, w_{0}} .$ Hence $∥ Φ (g) - Φ (h) ∥_{C ([0, T], X)} \leq 4 λ M_{0} T ∥ g - h ∥_{C ([0, T], X)}$ . Choose $T = (8 λ M_{0})^{- 1}$ , so this constant is $1 / 2$ . The nonempty complete path space from step 2.1 and [F4] yield a unique fixed point on $[0, T]$ . Differentiating its norm-continuous mild integral gives $f_{t}^{'} = λ (G (f_{t}, f_{t}) - f_{t})$ in $X$ .

3.2F3F4step 2.2

Suppose $f_{0} \in M_{u, a, b}$ . The class is closed and convex by definition. If a path takes values in this class, [F3] puts every gain value in the class. For a partition $0 = s_{0} < \dots < s_{N} = t$ , assign each subinterval $I_{j} = [s_{j - 1}, s_{j}]$ the exact nonnegative weight $λ \int_{I_{j}} e^{- λ (t - s)} d s$ and evaluate the gain at one point of that interval; these weights sum to $1 - e^{- λ t}$ . Adding the weight $e^{- λ t}$ on $f_{0}$ gives convex combinations in the class. As the mesh tends to zero, the Riemann sums converge in $X$ to $Φ (g)_{t}$ , so closedness puts $Φ (g)_{t}$ in the class. Thus Picard iteration of $Φ$ starting at the constant path $f_{0}$ remains in $M_{u, a, b}$ , and [F4] makes its uniform limit the fixed point. Therefore every $f_{t}$ lies in this class and has the smooth positive representative and Gaussian envelopes from [F3].

4.1F2step 3.1algebra

At $t = T$ the solution is again in $K$ . Restart the same construction with initial value $f_{T}$ ; the same $M_{0}$ gives the same positive interval length. Iterating constructs a solution on every $[0, k T]$ , and uniqueness on overlaps makes the concatenation a unique global mild solution. Substituting the restarted mild formula into the preceding interval gives the displayed global formula. Since the gain is nonnegative, the formula also gives $f_{t} \geq e^{- λ t} f_{0}$ almost everywhere.

5.1F3F6step 1.1step 4.1algebra

On any compact time interval, the uniform spatial derivative bounds from [F3] make the family of representatives locally relatively compact in $C^{\infty}$ . If $t_{j} \to t$ , continuity in $X$ gives $f_{t_{j}} \to f_{t}$ in weighted $L^{1}$ . Every subsequence has a further locally uniformly convergent subsequence by [F3]; its limit equals almost everywhere the continuous representative of $f_{t}$ , hence equals it everywhere. A subsequence on which local uniform convergence failed would contradict this property, so $t \mapsto f_{t}$ is locally uniformly continuous in velocity. For $v$ in a compact velocity set, the gain integrand $f_{s} (v^{'}) f_{s} (w^{'})$ is continuous in $(s, v)$ , and pair-energy preservation gives the integrable majorant $C e^{- ∣ w - u ∣^{2} / (2 b)}$ , uniform on that compact set and compact time intervals. Dominated convergence [F6] therefore makes the pointwise gain integral finite and jointly continuous in $(s, v)$ ; it is a continuous representative of the $X$ -class gain. In particular, for fixed $v$ this supplies the integrable majorant $C (v) e^{- ∣ w - u ∣^{2} / (2 b)}$ uniformly on compact time intervals.

6.1F3F5F6step 1.1step 3.2step 5.1algebra∎

For each fixed $t$ , the Riemann sums defining the mild integral converge in $X$ ; their pointwise limits are the ordinary time integral because the gain is continuous in time by step 5.1. Choose a subsequence whose successive $X$ -norm differences are summable; the Tonelli argument in step 1.1, using [F5], shows it converges almost everywhere to the $X$ -class limit. Hence the pointwise time integral represents the mild right-hand side. It is continuous in velocity by the same compact-velocity Gaussian majorant in step 5.1. The left side has the smooth representative from step 3.2. Since these continuous representatives agree as $X$ classes, they agree everywhere; a continuous function vanishing almost everywhere vanishes everywhere. The ordinary fundamental theorem of calculus applied to the pointwise mild integral gives $\partial_{t} f_{t} (v) = λ (G (f_{t}, f_{t}) (v) - f_{t} (v))$ . This proves the stated pointwise $C^{1}$ time regularity.

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Exact kinetic moment balances and unclosed fluxes

Statement

Let $Ω = \prod_{j = 1}^{3} (R / L_{j} Z)$ with $L_{j} > 0$ , let $I \subset R$ be open, and fix $m > 0$ . Let $F : Ω \times R^{3} \times I \to [0, \infty)$ be $C^{1}$ , with $n (r, t) = \int F (r, v, t) d v > 0$ . For every compact $J \subset I$ , assume some $C_{J} < \infty$ and $N > 6$ satisfy

$∣ F ∣ + ∣ \partial_{t} F ∣ + ∣ \nabla_{r} F ∣ + ∣ \nabla_{v} F ∣ \leq C_{J} (1 + ∣ v ∣)^{- N} (t \in J) .$

Let $a : Ω \times I \to R^{3}$ be $C^{1}$ and independent of $v$ . Let $Q (r, \cdot, t)$ be measurable with $\int (1 + ∣ v ∣^{2}) ∣ Q ∣ d v < \infty$ and

$\int Q d v = 0, \int v Q d v = 0, \int ∣ v ∣^{2} Q d v = 0$

at every $(r, t)$ . Suppose pointwise that

$\partial_{t} F + v \cdot \nabla_{r} F + a \cdot \nabla_{v} F = Q .$

Define $ρ, u, Π, e, q$ as in Kinetic density, velocity, covariance stress and heat flux. Then the exact balances are

$\partial_{t} ρ + \nabla_{r} \cdot (ρ u) = 0,$

$\partial_{t} (ρ u) + \nabla_{r} \cdot (ρ u \otimes u + Π) = ρ a,$

$\partial_{t} ⁣ (\frac{1}{2} ρ ∣ u ∣^{2} + e) + \nabla_{r} \cdot ⁣ [(\frac{1}{2} ρ ∣ u ∣^{2} + e) u + Π u + q] = ρ a \cdot u,$

and

$\partial_{t} e + \nabla_{r} \cdot (e u + q) + Π : \nabla_{r} u = 0.$

These identities are exact moment equations under the stated generic conservation assumptions on $Q$ . They do not supply a constitutive relation for $Π$ or $q$ , a closure, or a hydrodynamic limit.

In the SI physical interpretation, $t$ is measured in seconds, $a$ is acceleration in $m s^{- 2}$ , and $Q$ has the units of $F$ per second.

Facts & Assumptions

Given: $Ω$ , $I$ , $m$ , $F$ , $a$ , and $Q$ satisfying the Statement.

[A1]

$F$ is periodic and $C^{1}$ in $(r, v, t)$ with the stated uniform velocity-decay bounds on compact time slabs; $a$ is $C^{1}$ and independent of velocity.

[A2]

The source $Q$ has finite absolute moments through order two and its number, momentum and squared-speed moments vanish.

[F1]

Kinetic density, velocity, covariance stress and heat flux: $ρ = m n$ , $u = n^{- 1} \int v F d v$ , $Π = m \int (v - u) \otimes (v - u) F d v$ , $e = \frac{1}{2} tr Π$ , and $q = \frac{m}{2} \int ∣ v - u ∣^{2} (v - u) F d v$ .

[F2]

Dominated convergence: an integrable majorant on a measure space permits differentiation or limits under an integral when the corresponding difference quotients converge pointwise and share that majorant.

[F3]

Fubini's theorem for L^1 functions on a sigma-finite product: absolutely integrable functions on sigma-finite products may be integrated in either order.

Proof

Given: [A1], [A2], the displayed kinetic equation, and [F1]–[F3].

Proof technique: direct.

1.1A1A2F2F3algebra

Fix $(r, t)$ and multiply the equation by one of $ψ (v) = 1$ , $ψ (v) = m v_{i}$ , or $ψ (v) = m ∣ v ∣^{2} / 2$ . Since $N > 6$ , polar integration gives $\int_{∣ v ∣ > 1} ∣ v ∣^{3} (1 + ∣ v ∣)^{- N} d v < \infty$ , so the moments and fluxes through order three are absolutely integrable; the energy integration-by-parts boundary term on $∣ v ∣ = R$ is $O (R^{4 - N}) \to 0$ . The decay in [A1] and [F2] then justify differentiating each velocity moment and commuting spatial derivatives with its integral; [F3] justifies the absolutely convergent rearrangements. Integrating the transport term gives $\nabla_{r} \cdot \int v ψ F d v$ . Integrating the acceleration term by parts in velocity gives $- \int F a \cdot \nabla_{v} ψ d v$ ; its boundary term vanishes by the preceding estimate. The $Q$ integral is zero for each selected $ψ$ by [A2]. The resulting raw identities are $\partial_{t} n + \nabla_{r} \cdot \int v F d v = 0$ , $\partial_{t} (m \int v F d v) + \nabla_{r} \cdot (m \int v \otimes v F d v) = ρ a$ , and $\partial_{t} (\frac{m}{2} \int ∣ v ∣^{2} F d v) + \nabla_{r} \cdot (\frac{m}{2} \int ∣ v ∣^{2} v F d v) = ρ a \cdot u$ .

2.1F1step 1.1algebra

Write $c = v - u$ , so $\int c F d v = 0$ . Expanding $v = u + c$ gives $m \int v \otimes v F d v = ρ u \otimes u + Π$ . Expanding $∣ v ∣^{2} v = (∣ u ∣^{2} + 2 u \cdot c + ∣ c ∣^{2}) (u + c)$ and discarding only terms whose centered first moment is zero gives $\frac{m}{2} \int ∣ v ∣^{2} v F d v = (\frac{1}{2} ρ ∣ u ∣^{2} + e) u + Π u + q .$ Substitution into the raw moment equations of step 1.1 yields the first three displayed balances.

3.1F1step 2.1algebra∎

Dot the momentum equation from step 2.1 with $u$ . The continuity equation gives $u \cdot [\partial_{t} (ρ u) + \nabla_{r} \cdot (ρ u \otimes u)] = \partial_{t} (\frac{1}{2} ρ ∣ u ∣^{2}) + \nabla_{r} \cdot (\frac{1}{2} ρ ∣ u ∣^{2} u) .$ Since $Π$ is symmetric by [F1], the product rule gives $u \cdot (\nabla_{r} \cdot Π) = \nabla_{r} \cdot (Π u) - Π : \nabla_{r} u$ . Subtracting this kinetic-energy balance from the total-energy balance in step 2.1 cancels $ρ a \cdot u$ and leaves $\partial_{t} e + \nabla_{r} \cdot (e u + q) + Π : \nabla_{r} u = 0$ .

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H theorem and Maxwellian equality for the regular cutoff model

Statement

Assume ${A C}_{ω}$ , $v_{0} > 0$ , $λ > 0$ , $u \in R^{3}$ , and positive velocity-squared parameters $0 < a < b$ . Let

$f_{0} (v) = \frac{1}{2} γ_{a I, u} (v) + \frac{1}{2} γ_{b I, u} (v),$

and let $f_{t}$ be its global solution from Global finite-energy cutoff collision evolution. Define the dimensionless one-particle functional

$H (t) : = \int_{R^{3}} f_{t} (v) \log (v_{0}^{3} f_{t} (v)) d v .$

For every $t \geq 0$ , $H$ is finite and continuously differentiable, and with $A = f_{t} (v) f_{t} (w)$ and $B = f_{t} (v^{'}) f_{t} (w^{'})$ ,

$H^{'} (t) = - \frac{λ}{4} \int_{R^{3} \times R^{3} \times S^{2}} (B - A) \log (B / A) d μ \leq 0.$

The integrand is nonnegative. Equality holds at a time $t$ exactly when $f_{t}$ is a normalized isotropic Gaussian

$f_{t} (v) = (2 π s^{2})^{- 3 / 2} \exp ⁣ (- \frac{∣ v - \overset{ˉ}{u} ∣^{2}}{2 s^{2}})$

for some $\overset{ˉ}{u} \in R^{3}$ and $s > 0$ ; every such Gaussian is stationary. The specified initial mixture is not a single Gaussian, so $H^{'} (0) < 0$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $v_{0} > 0$ , $λ > 0$ , $u \in R^{3}$ , $0 < a < b$ , and the initial density and collision map of the Statement.

[A1]

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the completed Lebesgue measure and change-of-variables suppliers.

[F1]

Cutoff elastic-reflection collision measure and gain: the gain uses the velocities $v^{'}, w^{'}$ and parameter measure $d μ = d v d w d σ (n)$ , with $v_{0} > 0$ in the weighted space.

[F2]

Well-defined elastic reflection gain and collision identities: for normalized $f$ , the weak identity applies when each of $\int ∣ φ ∣ G (f, f) d v$ , $\int ∣ φ ∣ f d v$ , $\int ∣ A Δ φ ∣ d μ$ , and $\int ∣ B Δ φ ∣ d μ$ is finite; the reflection is an involution and the parameter measure has full support.

[F3]

Regularity and invariance of the cutoff Gaussian-mixture class: the mixture class is preserved by gain and has uniform positive lower, upper, and derivative Gaussian bounds.

[F4]

Global finite-energy cutoff collision evolution: every stated mass-one finite-energy initial density has a unique global mild solution; it conserves mass, first moment and second moment, and for data in $M_{u, a, b}$ it remains in that class with pointwise $C^{1}$ evolution satisfying $\partial_{t} f = λ (G (f, f) - f)$ .

[F5]

Dominated convergence: an integrable majorant permits passage of limits through the Lebesgue integral.

[F6]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ : $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ .

[F7]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: nonnegative measurable functions on sigma-finite products may be integrated iteratively.

Proof

Given: [A1], $v_{0} > 0$ , $λ > 0$ , $0 < a < b$ , the specified $f_{0}$ , and [F1]–[F7].

Proof technique: direct.

1.1F3F4algebra

The bounds in [F3] give constants $c, C > 0$ such that $c e^{- ∣ v - u ∣^{2} / (2 a)} \leq f_{t} (v) \leq C e^{- ∣ v - u ∣^{2} / (2 b)}$ for all $t, v$ . The same upper Gaussian envelope holds for $G (f_{t}, f_{t})$ because [F3] preserves the mixture class; [F4] therefore bounds $∣ \partial_{t} f_{t} ∣$ by a constant times that envelope. The lower and upper bounds imply $∣ \log (v_{0}^{3} f_{t} (v)) ∣ \leq C_{1} (1 + ∣ v - u ∣^{2})$ , after changing the finite constant $C_{1}$ . Thus $f_{t} ∣ \log (v_{0}^{3} f_{t}) ∣$ is integrable and the entropy derivative integrand has an integrable polynomial-Gaussian majorant on every compact time interval.

2.1F4F5step 1.1algebra

For each fixed $v$ , the ordinary chain rule gives $\partial_{t} [f_{t} \log (v_{0}^{3} f_{t})] = (1 + \log (v_{0}^{3} f_{t})) \partial_{t} f_{t}$ . The uniform majorant from step 1.1 also bounds the difference quotients by the scalar mean-value theorem. Dominated convergence [F5] therefore gives $H^{'} (t) = \int (1 + \log (v_{0}^{3} f_{t})) \partial_{t} f_{t} d v$ . The mass of $f_{t}$ is constant by [F4], so $\int \partial_{t} f_{t} d v = 0$ and the constant term drops. The integrand in the resulting formula is pointwise continuous in time by [F4] and has the same local integrable majorant from step 1.1; a second application of [F5] shows that $H^{'} (t)$ is continuous.

3.1A1F1F2F3F4step 2.1algebra

Fix $t$ and write $φ (v) = \log (v_{0}^{3} f_{t} (v))$ . The lower Gaussian bound makes $φ$ finite. The mean of $f_{t}$ is still $u$ by the moment conservation in [F4]. Also $A, B$ are bounded above by a constant times $e^{- (∣ v - u ∣^{2} + ∣ w - u ∣^{2}) / (2 b)}$ , because the pair reflection fixes the common mean $(u, u)$ and preserves the sum of the two centered squared speeds. The Gaussian lower bound gives $∣ \log (A / B) ∣ \leq C_{2} (1 + ∣ v - u ∣^{2} + ∣ w - u ∣^{2})$ . The same Gaussian upper bound holds for $G (f_{t}, f_{t})$ by [F3], so $\int ∣ φ ∣ f_{t} d v$ and $\int ∣ φ ∣ G (f_{t}, f_{t}) d v$ are finite; the pair bound and polynomial bound on $Δ φ$ give $\int ∣ A Δ φ ∣ d μ$ and $\int ∣ B Δ φ ∣ d μ$ finite. Thus the four conditions in [F2] hold. By [F2] and [F4], $H^{'} (t) = \frac{λ}{4} \int (B - A) Δ φ d μ .$ The reference factors cancel in the logarithm: $Δ φ = \log (A / B)$ . Hence $H^{'} (t) = - \frac{λ}{4} \int (B - A) \log (B / A) d μ$ . Since $\log$ is strictly increasing, $(B - A) \log (B / A) \geq 0$ , proving the inequality.

4.1A1F2step 3.1algebra

If $H^{'} (t) = 0$ , the continuous nonnegative integrand in step 3.1 vanishes $μ$ -almost everywhere. The product measure has full support by [F2]; a continuous function nonzero at one point is bounded away from zero on a nonempty open neighborhood of positive measure. Thus $A = B$ for every $(v, w, n)$ . Conversely, this pointwise identity makes the production integral zero.

5.1A1F2F3step 4.1algebra

Put $g (v) = \log (v_{0}^{3} f_{t} (v))$ , a $C^{2}$ function by [F3]. By the reflection coverage in [F2], for every center $c \in R^{3}$ and every pair $r, r^{'} \in R^{3}$ with $∣ r ∣ = ∣ r^{'} ∣$ , the equality in step 4.1 gives $g (c + r) + g (c - r) = g (c + r^{'}) + g (c - r^{'})$ . For unit vectors $θ, η$ , take $r = s θ$ and $r^{'} = s η$ . Taylor expansion at $c$ yields $g (c + s θ) + g (c - s θ) = 2 g (c) + s^{2} θ^{T} D^{2} g (c) θ + o (s^{2})$ , with the same expression for $η$ . Dividing by $s^{2}$ and taking $s \to 0$ shows that the quadratic form of $D^{2} g (c)$ is constant on the unit sphere. Testing $e_{i}$ and $(e_{i} + e_{j}) / \sqrt{2}$ shows $D^{2} g (c) = α (c) I$ .

6.1step 5.1algebra

The off-diagonal Hessian entries vanish, so for each $i$ , $\partial_{i} g$ depends only on its $i$ th coordinate: changing any other coordinate leaves it fixed by the fundamental theorem of calculus along that coordinate segment. Write $\partial_{i} g (v) = G_{i} (v_{i})$ . The equal diagonal entries give $G_{i}^{'} (v_{i}) = G_{j}^{'} (v_{j})$ for independently variable coordinates $v_{i}, v_{j}$ ; hence every such derivative is one constant $α$ . Integrating gives $g (v) = c_{0} + β \cdot v + \frac{α}{2} ∣ v ∣^{2}$ .

7.1A1F1F2F3F4F6F7step 6.1algebra

Integrability of $f_{t} = e^{g} / v_{0}^{3}$ forces $α < 0$ : if $α > 0$ the density grows in every sufficiently large direction, and if $α = 0$ an exponential of an affine function is nonintegrable on $R^{3}$ . Completing the square writes $f_{t} = C e^{- ∣ v - \overset{ˉ}{u} ∣^{2} / (2 s^{2})}$ with $s^{2} = - 1 / α > 0$ ; Tonelli [F7] and the one-dimensional Gaussian integral [F6] give $\int e^{- ∣ v - \overset{ˉ}{u} ∣^{2} / (2 s^{2})} d v = (2 π s^{2})^{3 / 2}$ , so unit mass fixes the prefactor in the Statement. Conversely, for that Gaussian $\log f_{t}$ is a constant plus a linear function of $v$ and a scalar multiple of $∣ v ∣^{2}$ ; pair momentum and squared-speed conservation in [F2] give $A = B$ at every collision. Define $sgn (x) = 1, 0, - 1$ according as $x > 0, x = 0, x < 0$ , and set $φ = sgn (G (f_{t}, f_{t}) - f_{t})$ . This test is bounded and measurable; since $\int A d μ = \int B d μ = 1$ , both sides of the weak identity are absolutely integrable. Applying [F2] gives $∥ G (f_{t}, f_{t}) - f_{t} ∥_{1} = 0$ . The constant curve satisfies the mild equation because $e^{- λ t} + λ \int_{0}^{t} e^{- λ (t - s)} d s = 1$ , and uniqueness in [F4] makes this Gaussian stationary.

8.1F3F6F7step 4.1step 7.1algebra∎

The density formula in [F3] factors for $a I$ and $b I$ into products of one-dimensional Gaussians; Tonelli [F7] and normalization [F6] give the first-coordinate marginal as their equal mixture. For a centered one-dimensional Gaussian of variance $s^{2}$ , integration of the derivative of $x^{2 k - 1} e^{- x^{2} / (2 s^{2})}$ over $R$ gives $\int x^{2 k} e^{- x^{2} / (2 s^{2})} d x = (2 k - 1) s^{2} \int x^{2 k - 2} e^{- x^{2} / (2 s^{2})} d x$ for $k = 1, 2$ ; the boundary terms vanish by Gaussian decay. Thus the mixture's centered second moment is $(a + b) / 2$ and its fourth moment is $3 (a^{2} + b^{2}) / 2$ . A single Gaussian with the same second moment has fourth moment $3 ((a + b) / 2)^{2}$ , smaller by $3 (a - b)^{2} / 4 > 0$ . Thus $f_{0}$ is not a single Gaussian. By the equality characterization in steps 4.1–7.1, the production at $t = 0$ is strictly positive and $H^{'} (0) < 0$ .

Physics theoremStatement: AI-adaptedProof: AI-generatedOpen item page →

Conditional one-particle kinetic entropy production

Physical scope

Conditional prediction of nondecrease of the declared one-particle velocity-reference information entropy for the adopted homogeneous cutoff reflection model and the specified positive regular Gaussian-mixture preparation. It is not a statement about fine-grained N-particle Gibbs entropy, generic gases, a hard-sphere Boltzmann–Grad limit, hydrodynamic limits, experiments, or all physical realizations.

Statement

Assume the physical adoption Selected phenomenological cutoff collision model with a positive collision rate $λ$ , particle mass $m > 0$ , reference speed $v_{0} > 0$ , and $k_{B} > 0$ . Prepare the normalized one-particle velocity density by

$f_{0} (v) = \frac{1}{2} γ_{a I, u} (v) + \frac{1}{2} γ_{b I, u} (v), u \in R^{3}, 0 < a < b,$

where $u \in R^{3}$ is a mean velocity measured in $m s^{- 1}$ , $a, b$ are covariance variances in $(m s^{- 1})^{2}$ , and $γ_{Σ, u}$ is the normalized Gaussian defined in Regularity and invariance of the cutoff Gaussian-mixture class. For the resulting global regular model solution $f_{t}$ , define the one-particle reference information entropy

$S_{B} (t) : = - k_{B} \int_{R^{3}} f_{t} (v) \log (v_{0}^{3} f_{t} (v)) d v .$

Since $v_{0}^{3} f_{t} (v)$ is dimensionless, $S_{B}$ has SI units $J K^{- 1}$ .

Then the model deduction is

$\frac{d S_{B}}{d t} = \frac{k_{B} λ}{4} \int_{R^{3} \times R^{3} \times S^{2}} (B - A) \log (B / A) d μ \geq 0,$

where $A = f_{t} (v) f_{t} (w)$ and $B = f_{t} (v^{'}) f_{t} (w^{'})$ use the exact reflection variables of Selected phenomenological cutoff collision model. The derivative is strictly positive at $t = 0$ . Equality at any time holds exactly when, for some mean velocity $\overset{ˉ}{u} \in R^{3}$ and speed parameter $s > 0$ (in $m s^{- 1}$ ),

$f_{t} (v) = (2 π s^{2})^{- 3 / 2} \exp ⁣ (- \frac{∣ v - \overset{ˉ}{u} ∣^{2}}{2 s^{2}}) .$

This conclusion is conditional on the adopted closed homogeneous model and regular preparation. It identifies a one-particle information functional with entropy units; it does not assert the fine-grained entropy of an $N$ -particle ensemble, a general thermodynamic second law, or empirical entropy increase.

Facts & Assumptions

Given: The postulated kinetic model, ${A C}_{ω}$ , $m > 0$ , $λ > 0$ , $v_{0} > 0$ , $k_{B} > 0$ , and the stated Gaussian-mixture initial preparation.

[A1]

The selected model equation is an equality in $X$ for norm- $C^{1}$ trajectories in the stated admissible domain, with normalized velocity density and the finite angular reflection law (Selected phenomenological cutoff collision model). The Gaussian-mixture branch also supplies the specified pointwise regular representatives by [F1].

[A2]

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) as required by the mathematical measure and change-of-variables suppliers.

[F1]

H theorem and Maxwellian equality for the regular cutoff model: the specified mixture has a global positive smooth solution with $H^{'} (t) = - \frac{λ}{4} \int (B - A) \log (B / A) d μ \leq 0$ , equality exactly for a normalized isotropic Gaussian, and strict production at the non-Gaussian initial mixture.

Proof

Given: [A1], [A2], and [F1].

Proof technique: direct.

1.1A1A2F1algebra

The physical postulate [A1] supplies the trajectory and the selected one-particle entropy convention; the mathematical theorem [F1] applies to this exact reflection measure, rate, reference speed, and initial mixture. Since $S_{B} = - k_{B} H$ and $k_{B} > 0$ , differentiating gives $S_{B}^{'} = - k_{B} H^{'} = \frac{k_{B} λ}{4} \int (B - A) \log (B / A) d μ \geq 0$ .

2.1A1F1step 1.1∎

The strict-production and equality clauses in [F1] transfer with the same positive factor $k_{B}$ : the initial mixture has $S_{B}^{'} (0) > 0$ , and at any later time $S_{B}^{'} = 0$ exactly when the velocity density has the normalized isotropic Gaussian form in the Statement. This is a deduction inside the adopted model and supplies no empirical premise.

5 · Examples, experiments and counterexamples

None yet.

Cutoff elastic-reflection collision measure and gain

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Cutoff elastic-reflection collision measure and gain

Definition

Well-defined elastic reflection gain and collision identities

Statement

Facts & Assumptions

Proof

Kinetic density, velocity, covariance stress and heat flux

Definition

Selected phenomenological cutoff collision model

Postulate

Regularity and invariance of the cutoff Gaussian-mixture class

Statement

Facts & Assumptions

Proof

Global finite-energy cutoff collision evolution

Statement

Facts & Assumptions

Proof

Exact kinetic moment balances and unclosed fluxes

Statement

Facts & Assumptions

Proof

H theorem and Maxwellian equality for the regular cutoff model

Statement

Facts & Assumptions

Proof

Conditional one-particle kinetic entropy production

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources