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Quantum Statistics and Coherent Access Hypothesis

N. G. de Almeida

Published Feb 22, 2012• 0 citations
Publisher

Abstract

The properties of quantum systems at equilibrium are mainly displayed by the very nature ofthe particles involved. As for example, distinguishable particles behave in a quite differentmanner of indistinguishable bosons, which in turns behave very different from the alsoindistinguishable fermions. Among the standard methods of statistical mechanics to handlewithquantumsystemsatequilibriumisincludedthewayofcountingtheaccessiblestatestoagivenparticle(1). Thus,thethermodynamicalprobability Ω comprisingthesetofpossibilitiesfor a system of N particles to populate the available states will depend on the type of particlebeing considered, as shown in Table 1. The method for calculating Ω consists in supposingthe existence of a number of energy levels, labeled by and index

DOI
10.5772/34197

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³ÑÆÇÓÉÂÐËÇ 1. £ÄÇÆÇÐËÇ (747).2. ¬ËÐÇÕËÚÇÔÍÑÇ ÖÓÂÄÐÇÐËÇ ÆÎâ ÔÓÇÆÐËØ ÚËÔÇÎ ÊÂÒÑÎÐÇÐËâ Ä ÃÑÎßÛÑÏ ÍÂÐÑÐËÚÇÔÍÑÏ ÂÐÔÂÏÃÎÇ (747).3. ¥ÑÒÎÇÓÑÄÔÍÑÇ ÑØÎÂÉÆÇÐËÇ (749).4. ¢ÑÎßÛÑÌ ÏËÍÓÑÍÂÐÑÐËÚÇÔÍËÌ ÂÐÔÂÏÃÎß (749). 5. ©ÂÍÎáÚÇÐËÇ (750).³ÒËÔÑÍ ÎËÕÇÓÂÕÖÓÞ (751).1. £ÄÇÆÇÐËÇ £ ÒÑÔÎÇÆÐÇÇ ÄÓÇÏâ Ä ÔÄâÊË Ô àÍÔÒÇÓËÏÇÐÕÂÎßÐÞÏË ÆÑÔÕË-ÉÇÐËâÏË Ä ÑÃÎÂÔÕË ÅÎÖÃÑÍÑÅÑ ÑØÎÂÉÆÇÐËâ ÓÂÊÓÇÉÇÐÐÞØ ÅÂÊÑÄ [1 ë 3] Ë ÑÔÖÜÇÔÕÄÎÇÐËÇÏ ÃÑÊÇ-àÌÐÛÕÇÌÐÑÄÔÍÑÅÑ ÍÑÐÆÇÐÔÂÕ [4, 5] ÄÑÊÐËÍΠÐÇÑÃØÑÆËÏÑÔÕß ÒÓËÏÇÐÇÐËâ ÍÄÂÐÕÑÄÞØ ÔÕÂÕËÔÕËÍ Í ÂÐÔÂÏÃÎâÏ, ÔÑÔÕÑâÜËÏ ËÊ ÔÓÂÄ-ÐËÕÇÎßÐÑ ÐÇÃÑÎßÛÑÅÑ ÚËÔΠÚÂÔÕËÙ (ÒÑÓâÆÍ 1000 ë 1000000), ÍÑÅÆÂ ËÔÒÑÎßÊÑÄÂÐËÇ ÕÇÓÏÑÆËÐÂÏËÚÇÔÍÑÅÑ ÒÓÇÆÇΠÑÍÂÊÞÄÂÇÕÔâ ÐÇÑÒÓÂÄÆÂÐÐÞÏ Ë Í ÕÑÏÖ ÉÇ ÒÓËØÑ-ÆËÕÔâ ÖÚËÕÞÄÂÕß ÆËÔÍÓÇÕÐÖá ÔÕÓÖÍÕÖÓÖ àÐÇÓÅÇÕËÚÇÔÍËØ ÖÓÑÄÐÇÌ [6 ë 31].

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The properties of quantum systems at equilibrium are mainly displayed by the very nature ofthe particles involved. As for example, distinguishable particles behave in a quite differentmanner of indistinguishable bosons, which in turns behave very different from the alsoindistinguishable fermions. Among the standard methods of statistical mechanics to handlewithquantumsystemsatequilibriumisincludedthewayofcountingtheaccessiblestatestoagivenparticle(1). Thus,thethermodynamicalprobability Ω comprisingthesetofpossibilitiesfor a system of N particles to populate the available states will depend on the type of particlebeing considered, as shown in Table 1. The method for calculating Ω consists in supposingthe existence of a number of energy levels, labeled by and index
In the previous chapters, the focus was on systems of particles following the law of classical mechanics. Of course, this was mainly done within the framework of classical mechanics but a quantum description of individual (discretised) states was also used, while assuming distinguishability of the particles (for instance, particles located at the nodes of a lattice (spin system, quantum harmonic oscillators,) ). The present chapter as well as Chapters 7 and 8 are dedicated to the study of statistical physics in a fully quantum framework. The first obstacle encountered by a quantum approach is the impossibility to define a quantum phase space (in which the ergodic hypothesis would hold for instance). This problem is inherent to quantum mechanics: because of Heisenberg’s uncertainty principle, one cannot define both the momentum and the position of a particle. Besides, statistical physics considers systems made of a large number of particles, a priori in mutual interaction. In classical physics, model situations for which the interactions between the constituents of the system are neglected can be studied. In quantum mechanics, the indistinguishability of identical particles makes the study of the problem in the absence of interaction already difficult: even for non-interacting particles, the symmetrisation (or antisymmetrisation) of the wave function of https://www.w3.org/1998/Math/MathML" display="inline"> N https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_25.tif "/> identical particles induces correlations which have non-trivial thermodynamical effects such as the existence of a “Fermi sea” for fermions (see Section 8.1 ) or the Bose–Einstein condensation phenomenon for bosons (see Section 7.4 ). The present chapter addresses these two points consecutively.
In quantum mechanics, elementary particles of the same type are considered identical. For example, an electron in an atom could be replaced by another electron and the atom would behave in exactly the same way. Nature has taken particular advantage of this, allowing only two types of elementary particles: bosons and fermions. They differ fundamentally in how a collection of identical particles interact. Identical fermions cannot occupy the same quantum state (only one per state!). Identical bosons can occupy the same quantum state. The statistics of identical particles (how they act in groups) has profound affects. For example, the rich and complex structure of the periodic table arises from the principle that only one electron can occupy each quantum state. The spin and the statistics of particles are closely connected: (fermions) bosons have (half-)integer spin. (Spin is an internal angular momentum measured in units of Planck’s constant.)
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.
According to the quantum chaos paradigm, the nature of a system's classical dynamics, whether integrable or chaotic, is universally reflected in the fluctuations of its quantum spectrum. However, since many-body spectra in the mean-field limit are composed of independent single-particle energy levels, their spectral fluctuations always display Poissonian behavior and hence cannot be used to distinguish underlying chaotic from integrable single-particle dynamics. We demonstrate that this distinction can, instead, be revealed from the mean many-body level density (at fixed energy) and its variance after averaging over ensembles representing different types of single-particle dynamics. This is in strong contrast to the energy-averaged mean level density (of a given system) that is assumed not to carry such information and is routinely removed to focus on universal signatures. To support our claim we systematically analyze the role of single-particle level correlations, that enter through Poisson and random matrix statistics (of various symmetry classes) into the ensemble-averaged density of states and its variance, contrasting bosonic and fermionic many-body systems. Our analytical study, together with extensive numerical simulations for systems with N≥5 particles consistently reveal significant differences (up to an order of magnitude for fermions and even larger for bosons) in the mean many-body level densities, depending on the nature of the underlying dynamics. Notably, in the fermionic case Poisson-type single-particle level fluctuations precisely cancel contributions from indistinguishability, such that the average many-body spectral density equals the (Thomas-Fermi) volume term. We further highlight the difference between the mean level density and its variance as functions of the total energy E and the excitation energy Q.
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.
³ÑÆÇÓÉÂÐËÇ 1. £ÄÇÆÇÐËÇ (747).2. ¬ËÐÇÕËÚÇÔÍÑÇ ÖÓÂÄÐÇÐËÇ ÆÎâ ÔÓÇÆÐËØ ÚËÔÇÎ ÊÂÒÑÎÐÇÐËâ Ä ÃÑÎßÛÑÏ ÍÂÐÑÐËÚÇÔÍÑÏ ÂÐÔÂÏÃÎÇ (747).3. ¥ÑÒÎÇÓÑÄÔÍÑÇ ÑØÎÂÉÆÇÐËÇ (749).4. ¢ÑÎßÛÑÌ ÏËÍÓÑÍÂÐÑÐËÚÇÔÍËÌ ÂÐÔÂÏÃÎß (749). 5. ©ÂÍÎáÚÇÐËÇ (750).³ÒËÔÑÍ ÎËÕÇÓÂÕÖÓÞ (751).1. £ÄÇÆÇÐËÇ £ ÒÑÔÎÇÆÐÇÇ ÄÓÇÏâ Ä ÔÄâÊË Ô àÍÔÒÇÓËÏÇÐÕÂÎßÐÞÏË ÆÑÔÕË-ÉÇÐËâÏË Ä ÑÃÎÂÔÕË ÅÎÖÃÑÍÑÅÑ ÑØÎÂÉÆÇÐËâ ÓÂÊÓÇÉÇÐÐÞØ ÅÂÊÑÄ [1 ë 3] Ë ÑÔÖÜÇÔÕÄÎÇÐËÇÏ ÃÑÊÇ-àÌÐÛÕÇÌÐÑÄÔÍÑÅÑ ÍÑÐÆÇÐÔÂÕ [4, 5] ÄÑÊÐËÍΠÐÇÑÃØÑÆËÏÑÔÕß ÒÓËÏÇÐÇÐËâ ÍÄÂÐÕÑÄÞØ ÔÕÂÕËÔÕËÍ Í ÂÐÔÂÏÃÎâÏ, ÔÑÔÕÑâÜËÏ ËÊ ÔÓÂÄ-ÐËÕÇÎßÐÑ ÐÇÃÑÎßÛÑÅÑ ÚËÔΠÚÂÔÕËÙ (ÒÑÓâÆÍ 1000 ë 1000000), ÍÑÅÆÂ ËÔÒÑÎßÊÑÄÂÐËÇ ÕÇÓÏÑÆËÐÂÏËÚÇÔÍÑÅÑ ÒÓÇÆÇΠÑÍÂÊÞÄÂÇÕÔâ ÐÇÑÒÓÂÄÆÂÐÐÞÏ Ë Í ÕÑÏÖ ÉÇ ÒÓËØÑ-ÆËÕÔâ ÖÚËÕÞÄÂÕß ÆËÔÍÓÇÕÐÖá ÔÕÓÖÍÕÖÓÖ àÐÇÓÅÇÕËÚÇÔÍËØ ÖÓÑÄÐÇÌ [6 ë 31].