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arXiv:1111.7074

Generic evaluation of the relaxation time to equilibrium

Takaaki Monnai

Published Nov 30, 2011• 0 citations
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Abstract

We evaluate the relaxation time to equilibrium, and especially show that it is almost independent from the system size for macroscopic isolated quantum systems. It at most polynomially depends on the system size. This estimation holds when the Hamiltonian is non-integrable, the initial deviation of the quantity of interest is of order its spectral norm, and the relaxation process is monotonic.

DOI
10.48550/arxiv.1111.7074

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We evaluate the relaxation time to equilibrium, and especially show that it is almost independent from the system size for macroscopic isolated quantum systems. It at most polynomially depends on the system size. This estimation holds when the Hamiltonian is non-integrable, the initial deviation of the quantity of interest is of order its spectral norm, and the relaxation process is monotonic.
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.
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General evaluation of the relaxation time to equilibrium is usually considered as difficult, since it would strongly depend on the model of interest. In this paper, we provide a generic initial relaxation time of the fidelity for the isolated large systems. The decay of the fidelity is a combination of the Lorentzian and a sinusoidal oscillation. We calculate the relaxation time of the Lorentzian envelop, and the period of the oscillation. Remarkably, these two time scales are the same order when the energy range of the microcanonical state is larger than the thermal fluctuation. Also, the power law decay generally exists for long time regime.
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].
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 provide an overview of our numerical and analytical studies of isolated interacting quantum systems that are abruptly quenched out of equilibrium instantaneously. We describe the relaxation process to a new equilibrium and obtain lower bounds for the relaxation time of full random matrices and realistic systems with two-body interactions. We show that the size of the time fluctuations after relaxation decays exponentially with system size for systems without too many degeneracies. We also discuss the conditions for thermalization and demonstrate that it can happen after local and global quenches in space. The analyses are developed for systems, initial states, and few-body observables accessible to experiments with optical lattices.