Thermalization or resonant-relaxation conjecture for tagged particles

Consider a conservative long-range interacting particle system at thermal equilibrium with a tagged particle whose mean-field evolution is trivial. Let the linearized mean-field operators at mean-field equilibrium determine the spectral behavior near $0$. Thermalization or resonant-relaxation conjecture. In the non-degenerate case, when these operators have purely absolutely continuous spectrum close to $0$, orthogonally to a possibly nontrivial kernel, thermalization of the tagged particle should occur on the slow timescale $t=O(N)$ and be described by a Fokker--Planck type equation. In the degenerate case, when the operators have eigenvalues accumulating at $0$, for example when they are compact, the slow dynamics should instead occur on the shorter timescale $t=O(N^{1/2})$, and thermalization should fail. More precisely, the dynamics should be given by a well-posed conservative hierarchical evolution for the tagged-particle density coupled to the infinite collection of limiting correlation functions, implying weak relaxation to equilibrium for $t\gg N^{1/2}$. The conjecture proposes a general dichotomy between Fokker--Planck thermalization and resonant relaxation, while the precise behavior depends on the spectral degeneracy of the linearized mean-field operators.