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Thermalization and its mechanism for generic isolated quantum systems

An Erratum to this article was published on 11 January 2012

Abstract

An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies1,2 of the problem have become possible, stimulating theoretical interest3,4,5,6,7. In generic isolated systems, non-equilibrium dynamics is expected8,9 to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible10. For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete11. Some recent studies4,5 even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch12 and Srednicki13. A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages—any eigenstate in the microcanonical energy window will do, because they all give the same result.

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Figure 1: Relaxation dynamics.
Figure 2: Thermalization in classical versus quantum mechanics.
Figure 3: Eigenstate thermalization hypothesis.
Figure 4: Temporal versus quantum fluctuations.

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Acknowledgements

We thank A. C. Cassidy, K. Jacobs, A. P. Young, and E. J. Heller for their comments. We acknowledge financial support from the National Science Foundation and the Office of Naval Research. We are grateful to the USC HPCC centre, where all our numerical computations were performed.

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Correspondence to Maxim Olshanii.

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Supplementary Discussion

This file contains Supplementary Discussion of: 1. some details of the model and numerical calculations; 2. the considerations necessary when considering the microcanonical ensemble for a small system; and 3. the properties and the role of the width of the energy distribution. It contains Supplementary Figure 1-2 with Legends. (PDF 225 kb)

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Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008). https://doi.org/10.1038/nature06838

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