Computability of long-time averages and thermalization
For specified one-dimensional translation-invariant nearest-neighbour Hamiltonian families and product-state inputs, determine when a local-observable long-time-average threshold problem is decidable. Specify a finite input representation, a separated threshol
Problem statement
For specified one-dimensional translation-invariant nearest-neighbour Hamiltonian families and product-state inputs, determine when a local-observable long-time-average threshold problem is decidable. Specify a finite input representation, a separated threshold promise, boundary conditions, and the order of the time and thermodynamic limits. Thermalization additionally requires an explicitly defined equilibrium reference.
Lean 4
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Background
Equilibration on average, pointwise convergence and agreement with a thermal ensemble are different properties. A representative average is first the finite-chain Cesàro time average, then its infinite-chain limit when it exists; changing the limit order defines another problem. The observable may also be spatially averaged, which must be stated.
What is known: Shiraishi–Matsumoto prove an undecidability theorem in a one-dimensional shift-invariant nearest-neighbour setting with a fixed product initial state. The Hamiltonian family encodes computation; this does not mean that every particular physical model has an undecidable thermalization question.
Matsumoto's July 2025 preprint refines the long-time-average setting to product inputs that are identical except at one site. Its finite-lattice versions have PSPACE/EXPSPACE classifications depending on the input-size convention: these are complexity results, not undecidability of every finite matrix calculation.
Free models and other explicitly solvable families supply positive cases, but “integrable” alone is not a blanket effective algorithm. Equilibration bounds need their nondegeneracy and finite-size hypotheses and do not automatically provide computable thermalization times or thermodynamic-limit error bounds. Related: B4, B5, M4, U2.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
published: Undecidability in a specified 1D nearest-neighbour thermalization model with fixed product-state input. Evidence
preprint: Almost-i.i.d. input refinement and separate finite-lattice PSPACE/EXPSPACE formulations. Evidence
Recorded progress
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