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arXiv:chao-dyn/9911018 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Classical statistical mechanics of a few-body interacting spin model

F. Borgonovi, Felix M. Izrailev

Published Nov 1, 2000• 18 citations
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Abstract

We study the emergence of Boltzmann's law for the "single-particle energy distribution" in a closed system of interacting classical spins. It is shown that for a large number of particles Boltzmann's law may occur, even if the interaction is very strong. Specific attention is paid to classical analogs of the average shape of quantum eigenstates and "local density of states," which are very important in quantum chaology. Analytical predictions are then compared with numerical data.

DOI
10.1103/physreve.62.6475

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We study the emergence of Boltzmann's law for the "single-particle energy distribution" in a closed system of interacting classical spins. It is shown that for a large number of particles Boltzmann's law may occur, even if the interaction is very strong. Specific attention is paid to classical analogs of the average shape of quantum eigenstates and "local density of states," which are very important in quantum chaology. Analytical predictions are then compared with numerical data.
It is shown that statistical mechanics is applicable to quantum systems with finite numbers of particles, such as complex atoms, atomic clusters, etc., where the residual two-body interaction is sufficiently strong. This interaction mixes the unperturbed shell-model basis states and produces ``chaotic'' many-body eigenstates. As a result, an interaction-induced equilibrium emerges in the system, and temperature can be introduced. However, the interaction between the particles and their finite number can lead to prominent deviations of the equilibrium occupation numbers distribution from the Fermi-Dirac shape. For example, this takes place in the cerium atom with four valence electrons, which was used to compare the theory with realistic numerical calculations.
The onset of thermalization in a closed system of randomly interacting bosons at the level of a single eigenstate is discussed. We focus on the emergence of Bose-Einstein distribution of single-particle occupation numbers, and we give a local criterion for thermalization dependent on the eigenstate energy. We show how to define the temperature of an eigenstate, provided that it has a chaotic structure in the basis defined by the single-particle states. The analytical expression for the eigenstate temperature as a function of both interparticle interaction and energy is complemented by numerical data.
Abstract A study of quantum-classical correspondence for the onset of thermalization in an isolated system of four interacting particles is presented. Provided that classical dynamics of the system is chaotic, thermalization is present and fits the expected statistical mechanics results. Quantum classical correspondence is established for the classical analogue of the single particle distributions of occupation numbers. The latter quantity is useful to test thermalization and it allows to obtain the microcanonical Boltzmann factor in the quantum and semi-classical cases. The correspondence between them and the statistical mechanics results are surprisingly good considering that we have only four interacting particles. The resulting Boltzmann factor for the occupation number distribution depends on the eigenfunction width, which is a semi-classical parameter explicitly calculated and consistent with the quantum one.
We study the model of two interacting particles moving in a 1D box, paying main attention to the quantum-classical correspondence for the average shape of quantum eigenstates and for the local density of states (LDOS). We show that if the classical motion is chaotic, in a deep semi-classical region of a quantum system, both the shape of eigenstates and of the LDOS coincide with their classical analogs, on average. However, individual eigenstates exhibit quite large fluctuations which may not be treated as statistical ones. Thus, comparison of quantum quantities to the classical ones allows one to detect quantum effects of localization which for conservative systems emerge in the energy space.
We demonstrate analytically and numerically that in isolated quantum systems of many interacting particles, the number of many-body states participating in the evolution after a quench increases exponentially in time, provided the eigenstates are delocalized in the energy shell. The rate of the exponential growth is defined by the width Γ of the local density of states and is associated with the Kolmogorov-Sinai entropy for systems with a well-defined classical limit. In a finite system, the exponential growth eventually saturates due to the finite volume of the energy shell. We estimate the timescale for the saturation and show that it is much larger than ℏ/Γ. Numerical data obtained for a two-body random interaction model of bosons and for a dynamical model of interacting spin-1/2 particles show excellent agreement with the analytical predictions.
It is believed that thermalization in closed systems of interacting particles can occur only when the eigenstates are fully delocalized and chaotic in the preferential (unperturbed) basis of the total Hamiltonian. Here we demonstrate that at variance with this common belief the typical situation in the systems with two-body inter-particle interaction is much more complicated and allows to treat as thermal even eigenstates that are not fully delocalized. Using a semi-analytical approach we establish the conditions for the emergence of such thermal states in a model of randomly interacting bosons. Our numerical data show an excellent correspondence with the predicted properties of localized thermal eigenstates.