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

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

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 \(f = M_{\rho ,T}\), \(E = 0\), \(B = B_0\)

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.

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