From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics (1509.06411v3)

Abstract: This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

• The paper establishes that the eigenstate thermalization hypothesis (ETH) underpins thermalization in non-integrable quantum systems by aligning observable expectations with ensemble predictions.
• It rigorously derives thermodynamic laws, including the second law and fluctuation theorems, from the principles of quantum chaos and statistical mechanics.
• The study highlights practical implications for quantum computation and simulation by explaining nonequilibrium dynamics in isolated quantum systems.

From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics

The paper "From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics" by Luca D'Alessio et al. provides a comprehensive review of the interplay between quantum chaos, eigenstate thermalization, statistical mechanics, and thermodynamics. This essay offers an expert overview of the key concepts, findings, and implications of the research discussed in the paper.

Overview of Quantum Chaos and Eigenstate Thermalization Hypothesis (ETH)

The foundation of the review is the eigenstate thermalization hypothesis (ETH), which builds on notions from quantum chaos and random matrix theory. Quantum chaos is characterized by the exponential sensitivity of a system's dynamics to small perturbations and is often described statistically by random matrix theory. The ETH posits that in chaotic quantum systems, the eigenstates of the Hamiltonian are such that the expectation values of observables coincide with those predicted by statistical mechanics ensembles, like the microcanonical ensemble.

The review highlights how ETH allows for understanding thermalization in isolated systems by bridging the gap between quantum dynamics and classical statistical mechanics. It asserts that ETH is a key mechanism that ensures the expectation values of observables equilibrate to the values predicted by thermodynamic ensembles in non-integrable systems.

Implications to Statistical Mechanics and Thermodynamics

The review further explores the consequences of quantum chaos for thermodynamics. Through ETH, it is possible to derive thermodynamic relations for individual eigenstates, extending them beyond traditional equilibrium assumptions. The paper rigorously derives the second law of thermodynamics, fluctuation theorems, and the fluctuation-dissipation relation based on quantum chaos, asserting that these fundamental relations hold even for isolated quantum systems.

One significant result is the demonstration that under doubly stochastic evolution—a natural outcome of closed-system dynamics conditioned by ETH—the diagonal entropy of the system increases, thereby ensuring the irreversibility associated with the second law of thermodynamics. This aligns with the Kelvin formulation of the second law, precluding net work extraction in a cyclic process starting from a thermal state.

Speculative Future Developments and Practical Implications

The review also speculates on the future development of these ideas within the context of generalized thermodynamic frameworks that consider nonequilibrium quantum systems. It suggests that insights from quantum chaos and ETH could lead to a deeper understanding of quantum statistical mechanics, particularly in exploring how systems thermalize after out-of-equilibrium perturbations.

From a practical standpoint, the paper's insights may offer new ways to manipulate nonequilibrium systems, which can be of interest in fields such as quantum computation, where controlling coherence and thermalization is crucial. Additionally, understanding the thermalization process can impact simulations in statistical physics, enabling more accurate descriptions of quantum many-body systems.

Conclusion

The work reviewed provides a robust framework connecting quantum chaos and ETH to thermodynamic behavior in isolated systems, thereby broadening traditional statistical mechanics. It elucidates the intricate mechanism through which quantum chaotic systems achieve equilibrium and underscores the vital role of ETH in explaining the thermalization phenomena without recourse to external baths. This paper is a foundational contribution to the paper of quantum statistical mechanics, offering a comprehensive view of how non-integrable quantum systems behave and evolve in isolation.