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arXiv:2407.11711 · Physical review. B./Physical review. B

Onsager's regression hypothesis adjusted to quantum systems

Peter Reimann, Christian Eidecker-Dunkel

Published Jul 15, 2024• 5 citations
arXivHTMLPDF

Abstract

Some 93 years ago, Onsager postulated his so-called regression hypothesis, which relates the time-dependent relaxation of a close-to-equilibrium system to its temporal correlations at thermal equilibrium. As far as quantum systems are concerned, it is long-known that this hypothesis cannot be quantitatively correct. Here, the authors analytically derive a suitably adjusted version of Onsager's original formula, provide a wide variety of numerical illustrations, and compare it with some pertinent previous works.

DOI
10.1103/physrevb.110.014306

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Some 93 years ago, Onsager postulated his so-called regression hypothesis, which relates the time-dependent relaxation of a close-to-equilibrium system to its temporal correlations at thermal equilibrium. As far as quantum systems are concerned, it is long-known that this hypothesis cannot be quantitatively correct. Here, the authors analytically derive a suitably adjusted version of Onsager's original formula, provide a wide variety of numerical illustrations, and compare it with some pertinent previous works.
We obtain a modified version of the Onsager regression relation for the expectation values of quantum-mechanical operators in pure quantum states of isolated many-body quantum systems. We use the insights gained from this relation to show that high-temperature time correlation functions in many-body quantum systems can be controllably computed without complete diagonalization of the Hamiltonians, using instead the direct integration of the Schrödinger equation for randomly sampled pure states. This method is also applicable to quantum quenches and other situations describable by time-dependent many-body Hamiltonians. The method implies exponential reduction of the computer memory requirement in comparison with the complete diagonalization. We illustrate the method by numerically computing infinite-temperature correlation functions for translationally invariant Heisenberg chains of up to 29 spins 1/2. Thereby, we also test the spin diffusion hypothesis and find it in a satisfactory agreement with the numerical results. Both the derivation of the modified regression relation and the justification of the computational method are based on the notion of quantum typicality.
The lack of knowledge about the detailed many-particle motion on the microscopic scale is a key issue in any theoretical description of a macroscopic experiment. For systems at or close to thermal equilibrium, statistical mechanics provides a very successful general framework to cope with this problem. However, far from equilibrium, only very few quantitative and comparably universal results are known. Here a quantum mechanical prediction of this type is derived and verified against various experimental and numerical data from the literature. It quantitatively describes the entire temporal relaxation towards thermal equilibrium for a large class (in a mathematically precisely defined sense) of closed many-body systems, whose initial state may be arbitrarily far from equilibrium.
We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory . The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.
For strongly driven systems where the coupling is weak and for frequencies near resonance, the Lax formula has proved to be an invaluable tool for calculating two-time correlation functions, as we have recently discussed in detail [1]. Nevertheless, we also stressed that this does not imply a quantum generalization of the Onsager hypothesis but in fact that the correct quantum generalization of this hypothesis is the fluctuation dissipation theorem. Here we discuss the fluctuation-dissipation theorem in more detail (by expressing a general correlation function in terms of both a relaxation function and a response function) and the conditions under which it leads to the Onsager hypothesis.
Abstract We have previously developed this framework in the context of static response, here we will generalize this perspective to dynamic response, and the evolution between distinct thermal equilibrium states. Of particular importance will be the derivation of the fluctuation-dissipation theorem, which relates the response of a system to its time dependent fluctuations, clarifying an equivalence between how a system responds to a small perturbation and how it regresses back to equilibrium in the absence of a perturbation. With this framework we can develop linear models of relaxation, classically and quantum mechanically, and derive equa- tions of motion suitable for describing the dynamics of thermal ensembles held at fixed temperature rather than fixed energy. These models are Gaussian, and approximate the dynamical correlations of interacting systems. These effective equations of motion are rendered stochastic due to the incomplete description of the microscopic degrees of freedom that evolve the system.