Dependencies

The entropy dissipation functional is

For \(\rho _{\mathrm{ion}} {\gt} 0\) and \(T {\gt} 0\), the equilibrium Maxwellian is

For \(v \in \mathbb {R}^3\), define \(|v| = \sqrt{|v|^2}\).

A flat 3-torus is a compact, nonempty, first-countable measure space \(X\) equipped with spatial gradient, divergence, and curl operators, a differentiability predicate IsSpatiallyDiff, and axioms encoding: operator linearity, closure of IsSpatiallyDiff under const/add/smul/log/grad, integration by parts, curl-integral vanishing, harmonic \(\Rightarrow \) constant, maximum principle, Killing/curl-free/div-free \(\Rightarrow \) harmonic, and Fubini for spatial–velocity double integrals.

A function \(f : \mathbb {R}^3 \to \mathbb {R}\) is Maxwellian if there exist \(a_0 \in \mathbb {R}\), \(b \in \mathbb {R}^3\), and \(c_0 {\lt} 0\) such that

for all \(v \in \mathbb {R}^3\).

Given a kernel function \(\Psi : \mathbb {R}\to \mathbb {R}\), the Landau matrix \(A(z)\) is the \(3 \times 3\) matrix with entries

The Landau collision operator is

For \(v \in \mathbb {R}^3\), define \(|v|^2 = \sum _{i=1}^{3} v_i^2\).

A concrete VelocityDecayConditions instance for the Coulomb kernel \(\Psi (r) = r^{-3}\) with Schwartz-class \(f\) and nonzero \(E\), \(B\). All 17 integrability/Fubini/IBP conditions verified.

A concrete FlatTorus3 instance on \(\mathbb {T}^3 = (\mathbb {R}/\mathbb {Z})^3\), realized as \(\mathrm{Fin}\, 3 \to \mathrm{AddCircle}\, 1\). All 23 fields proved: IBP via \(1\)D FTC + Fubini + periodicity, harmonic \(\Rightarrow \) constant via the energy method, Laplacian maximum principle via the second derivative test, etc.

The VelocityDecayConditions structure bundles 17 integrability/Fubini/IBP conditions on \(f\), \(E\), \(B\) that justify the analytical manipulations (integration by parts on \(\mathbb {R}^3\), Fubini exchange, Leibniz differentiation under the integral, entropy dissipation continuity, etc.).

If \(\Psi {\gt} 0\), \(f {\gt} 0\), \(f\) smooth and integrable, and \(D(f) = 0\), then \(f\) is Maxwellian. This is the core analytic step: \(D = 0\) forces the PSD integrand to vanish, which by the nullspace characterization of \(A(z)\) forces \(\nabla \log f\) to be affine, hence \(f = \exp (\text{quadratic})\).

For any Lorentz force \((E_0 + v \times B_0)\), velocity-space IBP combined with \(\nabla \cdot _v(E_0 + v \times B_0) = 0\) gives \(\int (E_0 + v \times B_0) \cdot \nabla _v f \cdot \log f\, dv = 0\).

The double integral from Landau IBP can be symmetrized by swapping \(v \leftrightarrow w\) and using \(A(v-w) = A(w-v)\), yielding the PSD integrand form with \((\nabla \log f(v) - \nabla \log f(w))\).

Integration by parts applied to \(\int Q(f,f) \log f\), rewriting the collision operator in divergence form and integrating by parts in \(v\):

If \(\Psi (|z|) \ge 0\), then the Landau matrix \(A(z)\) is positive semi-definite: \(Y^T A(z) Y \ge 0\) for all \(Y \in \mathbb {R}^3\).

If \(\Psi (|z|) {\gt} 0\) and \(z \ne 0\), then \(Y^T A(z) Y = 0\) iff \(Y \parallel z\).

For any \(E_0, B_0 \in \mathbb {R}^3\), \(\nabla \cdot _v (E_0 + v \times B_0) = 0\).

Integration by parts on \(\mathbb {R}^3\) in velocity space: \(\int \partial _i g \cdot h = -\int g \cdot \partial _i h\) under suitable decay.

In steady state with constant temperature and Ampère’s law, the bulk velocity is zero: \(b_0 = 0\).

Specialization of Theorem 42 to the Coulomb kernel \(\Psi (r) = r^{-3}\) on the concrete torus \(\mathbb {T}^3 = (\mathbb {R}/ \mathbb {Z})^3\), with Schwartz-class decay assumptions on \(f\). The conclusion is the same: \(f\) is an equilibrium Maxwellian with an injective parameter \(T_{\mathrm{eq}}\), \(E = 0\), and \(B\) is constant.

By the maximum principle on the compact torus, the density \(\rho \) is constant and \(E(x) = 0\) for all \(x\).

If \(\rho {\gt} 0\), \(T_1 {\gt} 0\), \(T_2 {\gt} 0\), and \(M_{\rho , T_1} = M_{\rho , T_2}\), then \(T_1 = T_2\).

If \(\Psi \ge 0\) and \(f {\gt} 0\) is smooth, then the entropy dissipation is non-positive: \(D(f) \le 0\).

With \(J = 0\) (since \(b_0 = 0\)), \(\nabla \times B = 0\). Combined with \(\nabla \cdot B = 0\) and the torus axiom that curl-free div-free fields are harmonic (hence constant): \(\exists B_0,\, \forall x,\, B(x) = B_0\).

Given a VMLInput (which encodes \(D(f(x,\cdot )) = 0\) for all \(x\)), constructs a VMLSteadyState and delegates to main_steady_state.

Given a VMLSteadyState, there exist \(T_{\mathrm{eq}} {\gt} 0\) and \(B_0\) such that \(f = M_{\rho ,T_{\mathrm{eq}}}\), \(E = 0\), \(B = B_0\).

If \(\log f(v) = a_0 + b \cdot v + c_0 |v|^2\) with \(c_0 {\lt} 0\), then \(Q(f,f)(v) = 0\) for all \(v\).

When \(b_0 = 0\) (isotropic Maxwellian), substituting \(Q(f,f) = 0\) into the Vlasov equation and combining with Gauss’s law yields: \(T_\infty \Delta (\log \rho ) = \rho - \rho _{\mathrm{ion}}\).

If \(f\) is a local Maxwellian satisfying the Vlasov equation, then substituting \(Q(f,f) = 0\) (by nullspace sufficiency) and matching polynomial coefficients in \(v\) yields: \(\nabla c = 0\) (constant temperature) and transport identities for \(a\), \(b\).

If \(\Psi {\gt} 0\), \(f {\gt} 0\), \(f\) smooth, and \(D(f(x, \cdot )) = 0\) for all \(x\), then for every \(x\), \(f(x, \cdot )\) is Maxwellian.

Let \(X\) be a flat 3-torus, \(f {\gt} 0\) smooth and integrable, \(\Psi {\gt} 0\). From the Vlasov equation + Maxwell equations + velocity decay conditions:

1. Landau IBP + Fubini symmetrization \(\Rightarrow \) score form of \(D(f)\)

2. \(H\)-theorem: \(D(f(x,\cdot )) \le 0\) for each \(x\)

3. Transport entropy identity: \(\int _X D = 0\)

4. Combining: \(D(f(x,\cdot )) = 0\) for all \(x\)

5. Polynomial identity + Poisson–Boltzmann from the Vlasov equation

6. Main from physics: conclude \(f = M_{\rho ,T}\), \(E = 0\), \(B = B_0\)

Under the Vlasov equation, combined with spatial IBP (on the torus) and the force transport lemma: \(\int _X D(f(x, \cdot ))\, dx = 0\).