Emergence of a second law of thermodynamics in isolated quantum systems

TL;DR

This paper demonstrates how a form of the second law of thermodynamics emerges in isolated quantum systems by showing that observable-based entropy tends to equilibrium, reconciling quantum reversibility with thermodynamic irreversibility.

Contribution

It introduces novel bounds linking observable equilibration to entropy increase, providing a quantitative connection between quantum dynamics and thermodynamic behavior.

Findings

Observable entropy approaches equilibrium values over time.

Numerical simulations confirm entropy increase and fluctuations in a quantum Ising model.

Results support the idea that non-integrability is not necessary for quantum equilibration.

Abstract

The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schr\"odinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value -- their equilibrium value. This ultimately raises the question: in what sense does the entropy of an isolated quantum system increase over time? For classical systems, one introduces the assumption of a low entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the equilibration on average of observables. While the statistical behavior of observable expectation values is well-established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends towards its equilibrium value in the course of the system's unitary evolution. These results also support recent findings which question the necessity of non-integrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations which expose the underlying reversible evolution in accordance with the derived bounds.