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arXiv:cond-mat/0310377

Exclusion Statistics Transformation and Ensemble Equivalence Tested From a Different Perspective

Dragoş-Victor Anghel

Published Oct 16, 2003• 0 citations
arXivHTMLPDF

Abstract

We generalize the method introduced in J. Phys. A: Math. Gen. 35, 7255 (2002) based on the concept of thermodynamic equivalence and we transform a Fermi system of general density of states into a thermodynamically equivalent Bose system. This consists of mapping configurations of fermions from the original system onto configurations of bosons, the initial and final configurations having the same energy above the many-body ground state energy. In this way we obtain two systems of particles of different exclusion statistics, but which have the same entropies--and therefore identical canonical thermodynamic properties. This method enables one in general to calculate the system properties in either of the bosonic and fermionic representations. We check the method here in microscopic detail by calculating the equilibrium particle distributions in the two representations using the entropy maximization at fixed particle number and fixed ``fermionic'' and ``bosonic'' energies, respectively. Analytical calculations seem difficult to do, but we check the results numerically and we find them equal within the numerical accuracy.

DOI
10.48550/arxiv.cond-mat/0310377

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We generalize the method introduced in J. Phys. A: Math. Gen. 35, 7255 (2002) based on the concept of thermodynamic equivalence and we transform a Fermi system of general density of states into a thermodynamically equivalent Bose system. This consists of mapping configurations of fermions from the original system onto configurations of bosons, the initial and final configurations having the same energy above the many-body ground state energy. In this way we obtain two systems of particles of different exclusion statistics, but which have the same entropies--and therefore identical canonical thermodynamic properties. This method enables one in general to calculate the system properties in either of the bosonic and fermionic representations. We check the method here in microscopic detail by calculating the equilibrium particle distributions in the two representations using the entropy maximization at fixed particle number and fixed fermionic'' and bosonic'' energies, respectively. Analytical calculations seem difficult to do, but we check the results numerically and we find them equal within the numerical accuracy.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.
It is demonstrated in many thermodynamic textbooks that the equivalence of the different ensembles is achieved in the thermodynamic limit. In this present work we discuss the inequivalence of microcanonical and canonical ensembles in a finite ultracold system at low energies. We calculate the microcanonical momentum distribution function (MDF) in a system of identical fermions (bosons). We find that the microcanonical MDF deviates from the canonical one, which is the Fermi-Dirac (Bose-Einstein) function, in a finite system at low energies where the single-particle density of states and its inverse are finite.
We investigate the thermodynamic properties of a bosonic Josephson junction in the full quantum approach and, in particular, we concentrate in studying the thermal averages of one- and two-body properties below and above the transition from delocalized to self-trapped regimes. This temperature dependence is determined by using the fact that at equilibrium the microcanonical and canonical ensembles should be equivalent. To establish the robustness of the equilibrium state, we first study a one body property and show numerically that any arbitrary state localized in energy, when evolved, reaches a stationary or equilibrium state. Comparison among averages of one- and two-body properties in the microcanonical and canonical ensembles reveals discrepances, thus leading to non-equivalence among these ensembles. Such averages differences can be attributed to the fact that the Hilbert space of the system scales as its size N, and consequently, the entropy does not scale as N. We further find as a natural consequence of studying the finite bosonic Josephson junction in the two-mode Bose Hubbard context, that positive and negative temperatures are obtained. This result can be generalized for any finite optical lattice.
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.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
The finite-temperature thermodynamics of a unitary Fermi gas is studied in detail. The chemical potential, energy density, and entropy are given analytically with the quasilinear approximation. The ground-state energy agrees with previous theoretical and experimental results. Recently, the generalized exclusion statistics is applied to the discussion of the finite-temperature unitary Fermi-gas thermodynamics. A concrete comparison between the two different approaches is performed. Emphasis is made on the behavior of the entropy per particle. In physics, the slope of entropy gives the information on the effective fermion mass ${m}^{\ensuremath{\ast}}/m$ in the low-temperature strongly degenerate region. Compared with ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}0.70<1$ given in terms of the generalized exclusion statistics, our quasilinear approximation determines that ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1.11>1$.