[Paper Review] Ergodicity, eigenstate thermalization, and the foundations of statistical mechanics in quantum and classical systems

Lorenzo Campos Venuti, Lawrence Liu|arXiv (Cornell University)|Apr 4, 2019

Advanced Thermodynamics and Statistical Mechanics2 references4 citations

TL;DR

This paper establishes a rigorous equivalence between quantum ergodicity, eigenstate thermalization hypothesis (ETH), and thermalization in isolated quantum systems. By defining quantum ergodicity via metric indecomposability of energy shells, it proves that ETH's diagonal matrix element condition is both necessary and sufficient for thermalization, resolving a long-standing conjecture and revealing ETH as the quantum foundation of statistical mechanics.

ABSTRACT

Boltzmann's ergodic hypothesis furnishes a possible explanation for the emergence of statistical mechanics in the framework of classical physics. In quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead generally considered as a possible route to thermalization. This is because the notion of ergodicity itself is vague in the quantum world and it is often simply taken as a synonym for thermalization. Here we show, in an elementary way, that when quantum ergodicity is properly defined, it is, in fact, equivalent to ETH. In turn, ergodicity is equivalent to thermalization, thus implying the equivalence of thermalization and ETH. This result previously appeared in [De Palma et al., Phys. Rev. Lett. 115, 220401 (2015)], but becomes particularly clear in the present context. We also show that it is possible to define a classical analogue of ETH which is implicitly assumed to be satisfied when constructing classical statistical mechanics. Classical and quantum statistical mechanics are built according to the familiar standard prescription. This prescription, however, is ontologically justified only in the quantum world.

Motivation & Objective

• To clarify the foundational role of ergodicity in quantum statistical mechanics, where it is often conflated with thermalization.
• To define quantum ergodicity rigorously in finite-dimensional Hilbert spaces using metric indecomposability of energy shells.
• To establish the equivalence between quantum ergodicity, the Eigenstate Thermalization Hypothesis (ETH), and thermalization.
• To show that ETH is not only sufficient but also necessary for thermalization in isolated quantum systems.
• To demonstrate that a classical analogue of ETH underlies the construction of classical statistical mechanics, though with fundamental differences in energy precision.

Proposed method

• Define quantum ergodicity via the metric indecomposability of energy shells, ensuring no invariant subsets of non-zero measure exist within the shell.
• Use the time-averaged expectation value of observables as the criterion for thermalization: ⟨A⟩_V = lim_{T→∞} T⁻¹∫₀ᵀ tr(A(t)ρ₀) dt.
• Establish equivalence between ergodicity and thermalization by showing that phase space averages over energy shells equal time-averaged observables for all initial states in the shell.
• Prove that the diagonal matrix elements of observables in energy eigenstates must satisfy the ETH condition: ⟨E_n|A|E_n⟩ ≈ ⟨A⟩_V for all n in the shell.
• Introduce a proxy for ETH-C via the energy scale ε_f = ⟨f⟩_Ē / |⟨f⟩′_Ē|, showing that Δ ≪ ε_f ensures approximate thermalization.
• Translate classical ergodicity concepts (e.g., Birkhoff's theorem) into quantum language to derive the quantum-ETH equivalence.

Experimental results

Research questions

• RQ1 Is there a precise, non-circular definition of ergodicity in quantum mechanics that does not equate it with thermalization?
• RQ2 Does the Eigenstate Thermalization Hypothesis (ETH) fully characterize thermalization in isolated quantum systems?
• RQ3 Can quantum ergodicity be shown to be equivalent to both ETH and thermalization?
• RQ4 What is the classical analogue of ETH, and is it implicitly assumed in classical statistical mechanics?
• RQ5 Why is exact energy eigenstate preparation impossible in quantum mechanics, and how does this affect the equivalence between ergodicity and thermalization?

Key findings

• Quantum ergodicity, defined via metric indecomposability of energy shells, is mathematically equivalent to thermalization for all initial states in the shell.
• The diagonal part of the ETH condition is both necessary and sufficient for thermalization, confirming ETH as the fundamental mechanism behind quantum thermalization.
• The equivalence between ergodicity and ETH resolves a long-standing conjecture, with the proof being conceptually elementary and transparent.
• For observables to thermalize, the energy shell width Δ must be much smaller than the energy scale ε_f = ⟨f⟩_Ē / |⟨f⟩′_Ē|, ensuring small relative error in the shell average.
• The classical limit of ETH is implicitly assumed in classical statistical mechanics, though the classical framework lacks the quantum constraint of infinite energy precision.
• The paper shows that the standard prescription of statistical mechanics is ontologically justified only in the quantum world, where ETH provides the foundational mechanism.