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Jilin Normal University Journal

Study on Statistic Means of Non-interaction among Particles

Ju Zhang

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

In this paper,the authors suggest a new means to simplify the imference of quantum statistical distributions of uninferacting particles.

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Note on a recent proposal concerning statistical inference in quantum theory

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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.
A novel perspective on relativistic transformation recently-proposed provides an insight into the very meaning of the principle of relativity. With this novel perspective and Bell's theorem, we argue that special relativity, instead of quantum theory, should be radically reformulated to resolve inconsistencies between the two theories. A new theory of relativistic quantum mechanics is formulated upon this novel perspective. This new relativistic quantum mechanics is free from such anomalies as the negative probability density, the negative-energy states, Zitterbewegung, and the Klein paradox deep-rooted in the current relativistic quantum mechanics. Moreover, a remarkable result is found that a particle can not be confined within an infinite square well of width less than half of the Compton wavelength. As implications in nuclear physics, there is a lower bound on the size of atomic nucleus. Neither an electron, nor a positron, can be confined inside the nucleus by whatever interaction. Furthermore, we argue that the postulates of non-relativistic quantum statistics fulfill the principle of relativity, as extended to the relativistic realm. A new theory of relativistic quantum statistics is formulated such that the probability distribution functions are the same as the three well-known probability distribution functions in non-relativistic quantum statistics. A relativistic speed distribution of a dilute gas is then derived by the new relativistic quantum statistics and the new relativistic quantum mechanics. This relativistic speed distribution reduces to Maxwell speed distribution in the low temperature region. Yet, this relativistic speed distribution differs remarkably from Jüttner speed distribution in the high temperature region. Also, thermal properties of a dilute gas are studied by the new relativistic quantum statistics.
We investigate the mean field regime of the dynamics of a tracer particle in a homogenous quantum gas. For a bosonic gas, we show that this regime is constrained by the well known requirement of an appropriate mean field scaling of the interaction. For fermions, however, we find an important qualitative difference. Not only are fermions much more homogeneously distributed than bosons but also deviations from the mean are due only to fast degrees of freedom in the gas. This observation leads to an explanation of why a tracer particle behaves freely in the dense homogeneous fermion gas despite of a non-scaled interaction, i.e., despite of non-vanishing statistical fluctuations. Finally, we indicate how the gained insight can be rigorously justified.