Many-particle Problems

Calvin R. Putnam

Michael Y. Romanovsky

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.

MASSIMO RUDAN

Mikhail Yu. Romanovsky

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.

Andrej B. Arbuzov, T. V. Kopylova, A.B. Zhunisbayev

The problem of the description of two interacting particles is considered. It is shown that it can be reduced to the description of one particle in an external static potential even in a relativistic case. The method is based on the Maupertuis least action principle. The proposed method is verified for known problems in two-particle problems in classical mechanics, as well as in non-relativistic and relativistic quantum mechanics.