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Journal of Experimental and Theoretical Physics

Statistical Properties of Many-Particle Systems with Hierarchical Structure

Mikhail Yu. Romanovsky

Published Feb 1, 2021• 0 citations
Publisher

Abstract

Many-particle systems contained inside other particles (problem 1), as well as systems of particles delimited inside other particles, which, in turn, are delimited inside the third particles (problem 2), are considered. Under natural assumptions about conserved quantities such as the total number of all particles, the total energy, and a number of others, the most probable energy distributions of particles, true (internal) and external observables, are studied. The results of problems 1 and 2 can be interpreted as the separation of bottom level particles into kinds with different distributions observed in different experiments. In single-particle observations in the full probability problem 1, an effective Pauli principle arises in a consistent manner.

DOI
10.1134/s1063776121020059

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Many-particle systems contained inside other particles (problem 1), as well as systems of particles delimited inside other particles, which, in turn, are delimited inside the third particles (problem 2), are considered. Under natural assumptions about conserved quantities such as the total number of all particles, the total energy, and a number of others, the most probable energy distributions of particles, true (internal) and external observables, are studied. The results of problems 1 and 2 can be interpreted as the separation of bottom level particles into kinds with different distributions observed in different experiments. In single-particle observations in the full probability problem 1, an effective Pauli principle arises in a consistent manner.
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.
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
Although many of the principles of quantum mechanics can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that are manifest only in systems containing two or more particles, and these will be the subject of the present chapter. We shall begin with some general statements and go on to consider the case of two interacting particles subject to no external forces, when we shall find that, in this case, the problem can be separated into one describing the behaviour of the centre of mass of the system and another describing the relative motion. We shall then consider the case of two noninteracting particles and show that these can often be treated independently as would be expected. In the case of indistinguishable particles, however, we shall find that a symmetry is imposed on the wave function that ensures that the behaviour of such particles is coupled even when they do not interact. We shall introduce the problem of interacting systems containing more than two identical particles, explaining what is meant by “fermions” and “bosons” and how their properties compare. These results will be illustrated by discussing the energy levels and optical spectra of the helium atom, starting with the approximation that the two electrons do not interact and extending the discussion to the realistic case using perturbation theory. The chapter closes with a consideration of the effects of particle indistinguishability on scattering theory.
The statistical problem of the distribution of sss quanta of equal energy ε0ε_0ε0​ and total energy EEE among NNN distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by p(κ)=(N+s−κ−2N−2)(N+s−1N−1)      ;      κ=0,1,2,⋯ ,sp(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, sp(κ)=(N−1N+s−1​)(N−2N+s−κ−2​)​;κ=0,1,2,⋯,s In this context, the special case (N=4N=4N=4, s=4s=4s=4) presented indicates that the alternative concept of most probable state is not valid for finite values of sss and NNN. In the present article we derive alternatively p(κ)p(κ)p(κ) by distributing sss quanta over NNN particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; sεo=E∼s ε_{o} = E \simsεo​=E∼ fixed, where the energy of the particles becomes continuous, p(κ)p(κ)p(κ) transforms to the Boltzmann law P(ε)=1⟨ε⟩e−ε⟨ε⟩      ;      0≤ε<+∞P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\inftyP(ε)=⟨ε⟩1​e−⟨ε⟩ε​;0≤ε<+∞ where ⟨ε⟩=E/N\langle ε\rangle = E/N⟨ε⟩=E/N. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.
The waiting time distribution w(τ)w(τ)w(τ), i.e. the probability for a delay τττ between two subsequent transition (`jumps') of particles, is a statistical tool in (quantum) transport. Using generalized Master equations for systems coupled to external particle reservoirs, one can establish relations between w(τ)w(τ)w(τ) and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that w(τ)w(τ)w(τ) usually contains additional information on system parameters and properties such as quantum coherence, the number of internal states, or the entropy of the current channels that participate in transport.
Many-particle systems play a fundamental role in physics where we encounter systems of molecules or systems of elementary particles in quantum field theory. Many-particle theories provide the basis for understanding the macroscopic behaviour of vast assemblies of interacting particles starting from their microscopic properties and leads to descriptions of entropy, statistical mechanics and fluid mechanics. Covering a wide range of topics and including more than 400 problems and solutions, this book is an ideal resource for students specialising in many-particle systems. Key features • Several hundred problems with solutions in each volume. • Problems range from basic introductory to advanced level. • Features worked examples as well as solutions to many problems. • Includes key concepts. • Covers essential complete physics course.