Most Probable Energy Distributions of Particles with Hierarchial Structures

TL;DR

This paper derives the most probable energy distributions for hierarchical multi-particle systems, extending classical statistical mechanics to complex nested structures with conservation laws, revealing effective Pauli principles and non-negative energy states.

Contribution

It introduces a combinatorial approach to determine energy distributions in hierarchical particle systems, generalizing Gibbs-Boltzmann distributions to nested structures.

Findings

Derived distributions for particles with hierarchical structures

Identified an effective Pauli principle in complex systems

Distributions describe particles without negative energy states

Abstract

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.