Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Scope of the selected finite stochastic and kinetic models
Records the boundaries of the selected smooth periodic mean-field, finite angular-reflection, and finite reversible-rate formulations. It asserts no empirical applicability, microscopic rate derivation, hydrodynamic limit, or generic Hamiltonian ergodicity.
Remarks
The selected mean-field definition Smooth periodic finite-N and scaled mean-field phase data fixes a smooth periodic pair force and interaction scaling. The collision definition Cutoff elastic-reflection collision measure and gain fixes a homogeneous finite normalized angular reflection measure. The finite-rate definition Finite reversible stochastic rate model fixes a connected finite state graph and reversible rates. These formulations provide the precise regimes used by the results on this page.
No result here derives the finite-state rates from deterministic particle mechanics, proves a limit from particle dynamics to continuum fluid equations, or shows that long-time averages equal invariant-measure averages for a general Hamiltonian system. Such conclusions require their own stated microscopic model, limiting assumptions and mathematical proof. The Langevin model here is exactly Adopted inertial velocity Langevin bath: a scalar velocity process with positive constant mass, damping rate and bath temperature, additive Brownian forcing, the adopted thermal noise normalization, stationary Gaussian initial velocity, and prescribed deterministic continuous force, satisfying its displayed integral equation. The physical rate and Langevin models in this pair are adopted assumptions; their theorems are conditional predictions, and this pair contains no reported observations or empirical inference.