Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems (1503.07538v5)

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.

• The paper demonstrates that closed quantum systems typically equilibrate and thermalise by rigorously analyzing unitary dynamics, ETH, and decoherence.
• The study shows that in high-dimensional Hilbert spaces, quantum observables reach near-equilibrium values for most times despite recurrence.
• The work explores practical implications for quantum technology, highlighting how many-body localisation and non-integrability affect energy and information flow.

Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems: An Overview

The paper by Christian Gogolin and Jens Eisert provides a comprehensive review of theoretical advancements concerning equilibration and thermalisation in quantum many-body systems, particularly focusing on how statistical mechanics emerge in closed quantum systems. The phenomenon is examined through a series of interrelated concepts that bridge the gap between micro-level quantum mechanics and macro-level statistical physics.

The paper begins by contextualizing the challenge of describing the thermodynamic behavior of quantum systems, which evolve unitarily according to the Schrödinger equation. This sets the stage for exploring the conditions under which these systems exhibit equilibration—whereby certain properties approach steady-state behaviors over time—and thermalisation, where subsystems appear to take on thermal characteristics as if in equilibrium, characterized by canonical or micro-canonical ensembles.

Key Themes and Concepts

1. Equilibration: The paper addresses the conditions under which quantum systems equilibrate, a notion compatible with recurrent and time-reversible unitary dynamics. Two scenarios are covered: equilibration on average, where properties are near their equilibrium values for most times; and equilibration during intervals, where properties stay close to equilibrium for defined periods. The work dives into rigorous results, suggesting that under reasonable conditions and given a high-dimensional Hilbert space, equilibration is a typical feature of quantum many-body systems.
2. Eigenstate Thermalisation Hypothesis (ETH): A central theme is the eigenstate thermalisation hypothesis, which posits that individual energy eigenstates of a non-integrable system should resemble micro-canonical ensemble averages. If a system satisfies the ETH, the expectation values of few-body observables should not only equilibrate but should also mimic thermal states, explaining why thermalisation is observed in many interacting systems.
3. Decoherence and Generalised Gibbs Ensembles: Decoherence, particularly in the energy eigenbasis, is highlighted as an intrinsic mechanism through which systems lose memory of their initial conditions, leading small subsystems to equilibrate to thermal states described by Generalised Gibbs Ensembles (GGE), incorporating the infinite set of integrals of motion often present in integrable systems.
4. Many-Body Localisation and Absence of Thermalisation: The paper also covers scenarios such as many-body localisation, where disorder and interactions inhibit thermalisation. Here, subsystems retain memory of initial conditions, defying conventional thermal behaviour and suggesting that entanglement and the propagation of quantum information are crucial in understanding the failure of thermalisation.
5. Practical and Theoretical Implications: By demonstrating that thermalisation and equilibration can be understood through quantum mechanics without additional postulates, the review reinforces foundational aspects of statistical mechanics. It opens avenues for understanding quantum thermodynamic processes and quantum statistical mechanics using pure state approaches.

Implications and Speculation on Future Directions

The paper suggests various future directions in both theoretical studies and practical experiments. It posits significant implications for the design of quantum technologies, such as quantum simulators or devices where control over quantum state thermalisation and decoherence is necessary. Moreover, understanding these processes could impact fields ranging from condensed matter physics to quantum computing and information theory.

Given the intricate relationship between quantum integrability, ETH, and many-body localisation, future research could explore classifying systems based on these criteria, potentially yielding new insights into how information and energy flow in quantum systems. Additionally, the practical aspects, particularly concerning systems out of equilibrium, offer a rich terrain for experimental physicists to test and explore these theoretical findings.

In summary, Gogolin and Eisert's work systematically unpackages the fundamental processes leading to equilibration and thermalisation in closed quantum systems. By doing so, it establishes a clear linkage between dynamic quantum processes and the statistical descriptions central to our understanding of macroscopic physical laws.