Test what has actually thermalized

Any density matrix diagonal in an isolated Hamiltonian's energy basis is stationary. Its populations need not have one Gibbs temperature. Coherent superpositions can also yield apparently relaxed local observables through dephasing of many frequencies, without collisions having established Bose or Fermi occupations. Integrable or nearly integrable systems can retain additional conserved quantities and approach a constrained ensemble instead of an ordinary Gibbs ensemble. A constant observable is therefore the start of a diagnostic, not the end of a thermalization argument.

For a closed system, unitary evolution preserves the eigenvalues of its full density matrix and hence its von Neumann entropy. A reduced subsystem can nevertheless become entangled with its environment inside that closed system, changing its reduced entropy and local correlations. Coarse-grained kinetic entropy can increase under additional approximations as well. These statements address different entropies. Distinguish the full microscopic state from the selected local observables before interpreting an irreversible-looking approach as fundamental loss of quantum information.

Some observables lose sensitivity to initial details on microscopic time scales while other quantities retain nonthermal occupations. Such a prethermal plateau can arise from rapid dephasing, approximate conservation, a hierarchy between elastic and number-changing processes, or weak breaking of an integrable limit. Its duration and final state depend on the actual dynamics. A common-temperature fit to one sector can coexist with chemical imbalance or a fluctuation relation that fails elsewhere. State which observables have settled and which processes remain slow.

An illustrative perturbative mechanism has a weak symmetry-breaking parameter epsilon and a characteristic microscopic energy scale Lambda. If a golden-rule calculation gives rate $\Gamma_{\mathrm{slow}}\sim\epsilon^2\Lambda$, a much longer relaxation time follows than the microscopic inverse-Lambda time. This is a conditional scaling model, not a theorem for every prethermal system. Resonances, selection rules, many-body densities of states, or exponentially long near-integrable regimes can produce other behavior. The relevant rate should be derived or measured for the specified model.

Ordinary perturbation theory can develop secular terms that grow with time, so a small coupling may control an early interval without controlling late evolution. High occupation can also enhance nominally weak interactions. Self-consistent schemes may improve that behavior, but their truncation, conservation laws, spectral sum rules, and numerical resolution still need checks. Classical-statistical approximations are useful at high occupation yet do not reproduce every quantum low-occupation process or guarantee the correct ultraviolet thermal equilibrium. An apparent steady state can be a regulator or closure artifact.

A demanding thermalization test compares several observables and initial conditions at fixed conserved quantities, checks the relevant KMS or fluctuation relations, and verifies the assumed statistical distribution over its valid range. Finite volume and finite observation time complicate claims about strict infinite-time limits or recurrences. Transport or hydrodynamic behavior can become useful before complete thermal equilibrium; its success should be stated at that level. Report the tested time window and approximation uncertainties rather than promoting a prethermal fit into an unconditional statement about the full quantum field theory.