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P-Adic Numbers Ultrametric Analysis and Applications

On Gibbs distribution for quantum systems

Valery Vasil'evich Kozlov

Published Jan 1, 2012• 0 citations
Publisher

Abstract

Some problems from quantum statistical mechanics concerning the regular and chaotic behavior of quantum systems are discussed.
DOI
10.1134/s2070046612010098

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Predicting the stationary behavior of observables in isolated many-body quantum systems is a central challenge in quantum statistical mechanics. While one can often use the Gibbs ensemble, which is simple to compute, there are many scenarios where this is not possible and one must instead use another ensemble, such as the diagonal, microcanonical or generalized Gibbs ensembles. However, these all require detailed information about the energy or other conserved quantities to be constructed. Here we propose a general and computationally easy approach to determine the stationary probability distribution of observables with few outcomes. Interpreting coarse measurements at equilibrium as noisy communication channels, we provide general analytical arguments in favor of the applicability of a maximum entropy principle for this class of observables. We show that the resulting theory accurately predicts stationary probability distributions without detailed microscopic information like the energy eigenstates. Extensive numerical experiments on 7 non-weakly interacting spin-1/2 Hamiltonians demonstrate the broad applicability and robustness of this framework in both quantum integrable and chaotic models.
This chapter presents an investigation of the dynamics of one anisotropic spin in an external, periodically time-dependent field. The simple quantum system displays chaotic behavior in the classical limit. Chaotic behavior in classical deterministic systems is characterized by an extremely sensitive dependence on initial conditions. The statistics of quasienergy levels are discussed for our driven spin system with a twofold purpose. The statistics of quasienergy levels is equivalent to the energy level statistics in a time-independent system and the dependence of the quasienergy level statistics on the quantum parameter is investigated. The statistical properties of the spectrum of a quantum problem in its classical limit are associated with the regularity properties of the corresponding classical system. Using the complementary approaches of phase space analysis and level statistics we have shown in both schemes that for large spins, i.e. in the semiclassical regime, stochasticity can be discussed using concepts developed for classical systems.
Similarly to the derivation of the Gibbs-Boltzmann distribution for structureless indistinguishable particles, we consider multi-particle systems some of which are contained (or delimited) inside others (Problem 1), as well as systems of particles delimited within other particles, which, in turn, are delimited yet inside another kind of particles (Problem 2). Under the natural assumptions concerning the conservation laws, such as the conservation of the total number of particles, total energy, etc, the problem of the most probable energy distributions is studied in a combinatorial formulation, with the obtained distributions treated as externally observable. For Problem 1, the particle distributions over maximal and minimal energies are also established and shown to coincide with the ones found in the framework of the original combinatorial treatment. The results for Problems 1 and 2 can be interpreted as a sorting of lower-level particles based on their statistical mechanics behavior observed in various experiments. An effective Pauli principle arises in a non-contradictory manner in one-particle observations for the complete probabilistic Problem 1 in the combinatorial formulation as well as in the problem of distributions over maximal and minimal energies. Several of the calculated distributions describe particles having no negative energy states.