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Molecular Physics

Equilibrium and ergodicity in small spin systems

John S. Waugh

Published Dec 10, 1998• 10 citations
Publisher

Abstract

The rearrangement of initial magnetic polarizations in a small isolated spin system does not lead to a uniform distribution as required by the microcanonical ensemble, but to a distribution of intermediate entropy as required by the initial conditions and auxiliary constants of the motion. The final equilibriumstate and the time to reach it are defined uniquely because of the finite number of discrete quantum states in such a system. Is such a system ergodic?

DOI
10.1080/00268979809483208

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The rearrangement of initial magnetic polarizations in a small isolated spin system does not lead to a uniform distribution as required by the microcanonical ensemble, but to a distribution of intermediate entropy as required by the initial conditions and auxiliary constants of the motion. The final equilibriumstate and the time to reach it are defined uniquely because of the finite number of discrete quantum states in such a system. Is such a system ergodic?
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
We introduce a unitary dynamics for quantum spins which is an extension of a model introduced by Mark Kac to clarify the phenomenon of relaxation to equilibrium. When the number of spins becomes very large, the magnetization satisfies an autonomous equation as a function of time with exponentially fast relaxation to the equilibrium magnetization as determined by the microcanonical ensemble. This is proved as a law of large numbers with respect to a class of initial data. The corresponding Gibbs–von Neumann entropy is also computed and its monotonicity in time discussed.
A stringent analysis of nonequilibrium dynamics and thermalization in a complex interacting spin system, which transits from integrability to chaos, is performed. The applicability of both standard thermodynamical ensembles—canonical and microcanonical—and the generalized Gibbs ensemble is tested. The role of constants of motion is studied, and a critical analysis of the usual definition of quantum integrability is performed. Different possible thermalization mechanisms are also analyzed. We find significant discrepancies between the time averages of some observables and the prediction from the thermodynamical ensembles within the ergodic region, so we conclude that quantum chaos does not guarantee thermalization. We also find that quantum integrable dynamics keeps more memory concerning the initial state, which is in part stored in a set of constants of motion that can be explicitly obtained.
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrodinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.
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.