Unitary derivation of thermalization
For a specified translation-invariant nonintegrable local Hamiltonian and a stated low-correlation initial-state family with narrow energy density, prove local-observable convergence or time-fraction closeness to the appropriate thermal ensemble, with explicit
Problem statement
For a specified translation-invariant nonintegrable local Hamiltonian and a stated low-correlation initial-state family with narrow energy density, prove local-observable convergence or time-fraction closeness to the appropriate thermal ensemble, with explicit relaxation times, finite-size errors and ordering of limits.
Lean 4
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Background
Beyond ETH as an ansatz (B4), the goal is to prove from the microscopic unitary dynamics that a nonintegrable many-body system relaxes to thermal equilibrium, with explicit relaxation time scales, bounds on residual fluctuations, and control of finite-size effects.
What is known: Equilibration-on-average theorems use effective-dimension and energy-gap nonresonance or bounded-degeneracy assumptions to bound fluctuations around a time-averaged state; these are not assumptions of a positive ground-state spectral gap. They do not alone identify that state with a Gibbs ensemble or give physically useful relaxation times. Controlled free and selected integrable quenches have generalized-ensemble results; this is not a universal relaxation theorem for every integrable model. Finite closed systems recur, so persistent global trace-norm convergence cannot be the unrestricted target.
The residual programme combines ensemble identification, local relaxation and quantitative rates for concrete interacting models. Conserved charges and symmetry sectors must be fixed; arbitrary initial states include deliberately stationary or atypical ones.
Non-Abelian subtarget: for a model with conserved noncommuting charges, derive the appropriate local equilibrium predictions and finite-size/rate bounds under a stated charge-preparation convention. Noncommuting charges cannot simply be assigned simultaneous sharp values. This is an operational extension of thermalization, not a claim that no non-Abelian thermodynamic formalism exists. Related: B4, B6, O10.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Linden, Popescu, Short, Winter, 'Quantum mechanical evolution towards thermal equilibrium', PRE 79, 061103 (2009) link
- [2]Gogolin & Eisert, 'Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems', Rep. Prog. Phys. 79 (2016) link
- [3]Quantum thermodynamics roadmap, non-Abelian conservation-law questions (2025) link
Recorded progress
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