Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems

TL;DR

This paper surveys rigorous theoretical advances on how closed quantum many-body systems equilibrate and thermalise, linking unitary pure-state dynamics to emergent statistical mechanics. It develops a unified framework based on dephasing, typicality, and maximum entropy principles, and connects these to both general results (equilibration on average, Lieb–Robinson bounds, and GGEs) and model-specific insights (quenches, entanglement dynamics, and ramps). It also discusses conditions under which thermalisation holds (ETH, initial-state constraints, ensemble equivalence) and mechanisms of absence of thermalisation, such as Anderson and many-body localisation, as well as the ongoing debates around quantum integrability. The review highlights the central role of locality, information propagation, and entanglement in shaping time scales and the stability of thermal states, while pointing to open problems, experimental relevance, and the need for further rigorous results on timescales and non-equilibrium phenomena.

Abstract

We review selected advances in the theoretical understanding of complex quantum many-body systems with regard to emergent notions of quantum statistical mechanics. We cover topics such as equilibration and thermalisation in pure state statistical mechanics, the eigenstate thermalisation hypothesis, the equivalence of ensembles, non-equilibration dynamics following global and local quenches as well as ramps. We also address initial state independence, absence of thermalisation, and many-body localisation. We elucidate the role played by key concepts for these phenomena, such as Lieb-Robinson bounds, entanglement growth, typicality arguments, quantum maximum entropy principles and the generalised Gibbs ensembles, and quantum (non-)integrability. We put emphasis on rigorous approaches and present the most important results in a unified language.

Figures (6)

• Figure 1: (Reproduction from Ref. Gogolin2014) Illustration of the non-degenerate energy gaps condition. No gap between two energy levels may occur more than once in the spectrum, but the individual levels may well be degenerate.
• Figure 2: (Reproduced from Ref. Gogolin2014) Equilibration on average is compatible with the time reversal invariant and recurrent nature of the time evolution of finite dimensional quantum systems. The figure shows a prototypical example of equilibration on average. Started in a non-equilibrium initial condition at time $0$ the expectation value of some observable $A$ quickly relaxes towards the equilibrium value $\langle A \rangle_{\omega}$ and then fluctuates around it, with far excursions from equilibrium being rare. After very long times the system returns (close to) its initial state and so does the expectation value of the observable. A similar behaviour is observed when the initial state is evolved backwards in time.
• Figure 3: Schematic depiction of the Lieb-Robinson "light" cones in clean systems (a) and the more stringent bounds that can be derived in disordered systems (b). Outside the shaded area causal influences are exponentially suppressed.
• Figure 4: (Reproduction from Ref. Gogolin2014) Dephasing implies a maximum entropy principle. A quantum system started in an initial state $\rho(0)$ represented in panel (a) in an eigenbasis of its Hamiltonian $H$ with degenerate subspaces corresponding to the squares, evolves (b) in a way such that time averaging its evolution (c) has the same effect as de-phasing the initial state with respect to $H$. The time averaged state $\overline{\rho}$ is the state that maximises the von Neumann entropy under the constraint that all conserved quantities give the same expectation value as in the initial state $\rho(0)$.
• Figure 5: (Reproduction from Ref. Gogolin2014) Structure of the proof of thermalisation from Ref. Riera2012.

Theorems & Definitions (29)

• Theorem 1: Equilibration on average
• proof
• Theorem 2: Equilibration during intervals
• proof
• Theorem 3: Lieb-Robinson bound (corollary of Theorem 1 from Kliesch2013)
• Theorem 4: Lieb-Robinson bounds for quadratic bosonic systems HarmonicLiebRobinson
• Theorem 5: Equilibration under Haar random Hamiltonians 1108.0374
• Theorem 6: Equilibration under random circuit Hamiltonians 1108.0374
• Theorem 7: Fast equilibration of low rank observables Malabarba2014
• Theorem 8: Maximum entropy principle PhysRevLett.10-6