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Distinct kinetic scaling prescriptions

Example

Fix an integer dimension $d \geq 2$ and a spatial side length $L > 0$ in metres; compare three model families. In the smooth periodic mean-field family, $N \geq 2$ equal masses $m > 0$ kilograms have positions $q_{i} \in T_{L}^{d}$ and canonical momenta $p_{i} \in R^{d}$ measured in $k g m s^{- 1}$ , and move under the Hamiltonian

$H_{N} = \sum_{i = 1}^{N} \frac{∣ p_{i} ∣^{2}}{2 m} + \frac{1}{N} \sum_{1 \leq i < j \leq N} W (q_{i} - q_{j}),$

where $W \in C^{\infty} (T_{L}^{d})$ is even and measured in joules. In the periodic hard-sphere family, the dimensionless diameter $a_{N} = δ_{N} / L$ obeys $N a_{N}^{d - 1} = κ$ for a fixed $κ > 0$ . In the two-velocity family on $T_{L}$ , take $c, λ > 0$ , $ϵ ↓ 0$ , and well-prepared data $f_{+, 0}^{ϵ} = f_{-, 0}^{ϵ} = ρ_{0} / 2$ with a nonnegative $ρ_{0} \in C^{\infty} (T_{L})$ . Under ${A C}_{ω}$ , the two densities have speeds $\pm c / ϵ$ and switch to the opposite state at rate $λ / ϵ^{2}$ , and their total density has the heat limit with $D = c^{2} / (2 λ)$ .

These prescriptions use different model variables and interaction rules: respectively a smooth pair force with coefficient $1 / N$ , hard-core collisions with $N a_{N}^{d - 1}$ fixed, and a two-state linear transport equation with $ϵ ↓ 0$ . This comparison makes no cross-family implication claim and gives no derivation of the nonlinear Boltzmann equation.

Example references

Smooth periodic finite-N and scaled mean-field phase data

Periodic hard-sphere Boltzmann–Grad scaling convention

Two-velocity periodic diffusive-scaling model

Periodic two-velocity heat limit for well-prepared data

Facts & Assumptions

Given: ${A C}_{ω}$ , the three model specifications in the Example, $d \geq 2$ , $L > 0$ , $m > 0$ , $N \geq 2$ , and admissible hard-sphere trajectories when discussing collision invariants.

[A1]

Assume the Axiom of Countable Choice, ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), as required by the two-velocity heat-limit theorem.

[F1]

Smooth periodic finite-N and scaled mean-field phase data: $W$ is a smooth even periodic pair potential in joules, $H_{N}$ contains the dimensionless coefficient $1 / N$ , the Hamiltonian convention is $ι_{X_{H_{N}}} ω_{N} = d H_{N}$ , and the physical force law is ${\dot{p}}_{i} = N^{- 1} \sum_{j \neq i} K (q_{i} - q_{j})$ with $K = - \nabla W$ odd. In dimensionless variables the force law is $x_{i}^{'} = v_{i}$ and $v_{i}^{'} = N^{- 1} \sum_{j \neq i} k (x_{i} - x_{j})$ with odd $k$ .

[F2]

Periodic hard-sphere Boltzmann–Grad scaling convention: $a_{N} = δ_{N} / L$ is dimensionless, $N a_{N}^{d - 1} = κ$ is fixed, $δ_{N} < L / 2$ makes the torus contact normal unique, and an equal-mass hard-sphere collision reverses the relative velocity component normal to the contact surface.

[F3]

Two-velocity periodic diffusive-scaling model: $d m_{L} = d x / L$ , $\hat{g} (n) = \int g \overline{e_{n}} d m_{L}$ , $c / ϵ$ has units $m s^{- 1}$ , $λ / ϵ^{2}$ has units $s^{- 1}$ , and the total density is $ρ^{ϵ} = f_{+}^{ϵ} + f_{-}^{ϵ}$ .

[F4]

Periodic two-velocity heat limit for well-prepared data: for well-prepared smooth data, $ρ^{ϵ}$ converges in $L^{2} (d m_{L})$ uniformly on each bounded time interval to $u (t, x) = \sum_{n} \hat{ρ_{0}} (n) e^{- D κ_{n}^{2} t} e_{n} (x)$ , where $D = c^{2} / (2 λ)$ ; its zero Fourier coefficient is the constant $\hat{ρ_{0}} (0)$ .

[F5]

Well-posedness, positivity and contraction for two-velocity transport: the periodic two-velocity evolution conserves $\int_{0}^{L} ρ^{ϵ} (x, t) d x$ .

Verification

technique · direct

1.1F1givenalgebra

The mean-field coefficient $1 / N$ and particle number $N$ are dimensionless; $q_{i}$ has units m, $p_{i}$ has units $k g m s^{- 1}$ , $W$ and $H_{N}$ have units J, and $K = - \nabla W$ has units N by [F1]. The total momentum $P_{N} = \sum_{i} p_{i}$ is conserved because ${\dot{P}}_{N} = N^{- 1} \sum_{i < j} (K (q_{i} - q_{j}) + K (q_{j} - q_{i})) = 0$ by oddness of $K$ . Along any differentiable Hamiltonian trajectory, $H_{N}$ is constant: the defining convention $ι_{X_{H_{N}}} ω_{N} = d H_{N}$ gives $d H_{N} (X_{H_{N}}) = ω_{N} (X_{H_{N}}, X_{H_{N}}) = 0$ since a two-form is antisymmetric. The model keeps the finite particle count $N$ fixed; its separate mean-field scaling varies $N$ while retaining the smooth all-to-all interaction coefficient $1 / N$ .

1.2F2givenalgebra

In the hard-sphere scaling, $N$ and $a_{N} = δ_{N} / L$ are dimensionless and $a_{N} = (κ / N)^{1 / (d - 1)} \to 0$ . Between collisions every velocity is constant. At a binary collision set $U = (v_{i}^{-} + v_{j}^{-}) / 2$ and $g = v_{i}^{-} - v_{j}^{-}$ . The rule in [F2] leaves $U$ unchanged and sends $g$ to $g^{'} = g - 2 (g \cdot n) n$ , so $∣ g^{'} ∣^{2} = ∣ g ∣^{2}$ because $∣ n ∣ = 1$ . Therefore the pair momentum $m (v_{i} + v_{j}) = 2 m U$ and pair kinetic energy $\frac{m}{2} (∣ v_{i} ∣^{2} + ∣ v_{j} ∣^{2}) = m ∣ U ∣^{2} + \frac{m}{4} ∣ g ∣^{2}$ are unchanged. Other particles are unaffected at this isolated binary collision. Thus momentum and kinetic energy are unchanged between any two times across which only finitely many of these collision updates occur, by composing the free-flight and single-collision equalities. This comparison uses only the specified collision rule; it does not assert existence of a global hard-sphere flow or address accumulation of collisions.

1.3A1F3F4F5givenalgebra

For the two-velocity model, $[c / ϵ] = m s^{- 1}$ and $[λ / ϵ^{2}] = s^{- 1}$ by [F3], so $D = c^{2} / (2 λ)$ has units $m^{2} s^{- 1}$ . Its conserved quantity is the total density mass $\int_{0}^{L} ρ^{ϵ} (x, t) d x$ by [F5]; the limiting heat equation preserves the same mass because its zero Fourier mode is constant. It has no particle-count parameter $N$ , pair potential $W$ , hard-sphere diameter $δ_{N}$ , or collision events.

2.1step 1.1step 1.2step 1.3algebra∎

The three prescriptions constrain different variables and specify different state spaces: a smooth Hamiltonian $N$ -body system with $1 / N$ pair coupling, a hard-sphere system with $N a_{N}^{d - 1} = κ$ , and a two-component periodic density equation with $ϵ ↓ 0$ . The conservation checks in steps 1.1–1.3 concern their own equations and collision rule. They supply no cross-family implication or molecular-chaos or hard-sphere derivation of the two-velocity heat equation, so none is claimed here.

Distinct kinetic scaling prescriptions

Example

Example references

Facts & Assumptions

Verification

Depends on

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