Fig. 1. The problem of thermalization concerns the long-time average of the observable.

a We consider whether a nonequilibrium initial state relaxes to the equilibrium or not. b More precisely, we decide whether the long-time average of $⟨\psi \left(t\right)\mid {\mathcal{A}}_L\mid \psi \left(t\right)⟩$ converges to the value A^* with precision ε₁, or deviates from A^* at least ε₂ > ε₁, in the thermodynamic limit (If the long-time average settles between ε₁ and ε₂, we do not have to answer). This problem is shown to be an undecidable problem.