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Physica Scripta

Deviation from equilibrium in macroscopic quantum systems

Takaaki Monnai

Published Nov 1, 2012• 0 citations
Publisher

Abstract

The irreversible relaxation to equilibrium is explained for macroscopic quantum systems with an emphasis on the behavior of expectation values and extremely high dimensionality of the Hilbert space. We consider a large but isolated system that is initially out of equilibrium and eventually relaxes to equilibrium. The relaxation is described by the deviation of the expectation value of the quantity of interest from the long-time average. After relaxation, the amount of deviation from equilibrium is discussed based on probabilistic arguments, which are available for nonintegrable systems. We also evaluate how long the system stays near equilibrium.

DOI
10.1088/0031-8949/2012/t151/014043

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The irreversible relaxation to equilibrium is explained for macroscopic quantum systems with an emphasis on the behavior of expectation values and extremely high dimensionality of the Hilbert space. We consider a large but isolated system that is initially out of equilibrium and eventually relaxes to equilibrium. The relaxation is described by the deviation of the expectation value of the quantity of interest from the long-time average. After relaxation, the amount of deviation from equilibrium is discussed based on probabilistic arguments, which are available for nonintegrable systems. We also evaluate how long the system stays near equilibrium.
A quantum mechanical explanation of the relaxation to equilibrium is shown for macroscopic systems for nonintegrable cases and numerically verified. The macroscopic system is initially in an equilibrium state, subsequently externally perturbed during a finite time, and then isolated for a sufficiently long time. We show a quantitative explanation that the initial microcanonical state typically reaches a state whose expectation values are well approximated by the average over another microcanonical ensemble.
We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems, in particular, for chaotic systems, we argue that the expectation value of a local operator averaged over a time interval of length TTT is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium of all states with a given value of energy fluctuations. This result applies to any initial state with a well-defined effective temperature. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at a macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with the transport and the properties of microscopic matrix elements. The form of the deviation function indicates that among the ``slowest'' states which saturate the bound there are also those with arbitrarily long equilibration times.
General evaluation of the relaxation time to equilibrium remains as a subtle unsolved problem, partially because of its possible dependence on the specific features of models. We evaluate the relaxation time of expectation values for generic isolated quantum many-body systems, and especially show that it at most polynomially depends on the system size. This provides a first model-independent quantitative analysis of the relaxation time. The estimation holds when the Hamiltonian is non-integrable, the initial deviation of the quantity of interest is order of its spectral norm defined on a finite space, and the relaxation process is essentially monotonic.
In conclusion, by assuming that the initial state is prepared as a microcanonical ensemble, the actual state at time t=T can be replaced by another microcanonical ensemble in the evaluation of macroscopic quantities. The derivation is based on the high dimensionality of the Hilbert space, and restrictions for the system size dependence of observables as well as the perturbative treatment of the unitary transformation of the energy eigenstates. The validity of the uniformly random sampling assumption for [numerical formula] is numerically verified for a nonintegrable spin chain. It is also remarked that the initial state can be out of equilibrium, i.e. the initial density matrix is [numerical formula] with a unitary transformation U_0, since this can be regarded as the state evolved from an actual initial state [numerical formula].
We consider isolated many-body quantum systems which do not thermalize; i.e., expectation values approach an (approximately) steady longtime limit which disagrees with the microcanonical prediction of equilibrium statistical mechanics. A general analytical theory is worked out for the typical temporal relaxation behavior in such cases. The main prerequisites are initial conditions which appreciably populate many energy levels and do not give rise to significant spatial inhomogeneities on macroscopic scales. The theory explains very well the experimental and numerical findings in a trapped-ion quantum simulator exhibiting many-body localization, in ultracold atomic gases, and in integrable hard-core boson and XXZ models.