On the Uniform Convergence of Ergodic Averages for $$C^*$$-Dynamical Systems

The aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor n! (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of “exclusion principle” analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected.

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“…For some properties of monotone systems, including their ergodic properties, see, e.g., [18] and the literature cited therein.…”

Section: Preliminariesmentioning

confidence: 99%

On the Thermodynamics of Particles Obeying Monotone Statistics

Ciolli

Fidaleo

Marullo

2023

Entropy

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“…It is still unknown whether such a scheme can have potential applications to physics. Since the ergodic behaviour of any (classical or quantum) stochastic process is relevant for the investigation of its main properties (e.g., [21]), we refer the reader to [22] for a systematic investigation of the ergodic properties of such non conventional quantum stochastic processes.…”

Section: Introductionmentioning

confidence: 99%

Gas of Particles Obeying the Monotone Statistics

Fidaleo

2023

Entropy

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The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional environment. This investigation can have consequences in two fundamental respects. The first one concerns the applicability of the (block-)monotone statistics to concrete physical models, yet completely unknown. Since the formula for the degeneracy of the energy-levels of the one-particle Hamiltonian of a free particle is very involved, the second aspect might be related to the, highly nontrivial, investigation of the expected thermodynamics of the free gas of particles obeying the block-monotone statistics in arbitrary spatial dimensions. A final section contains a comparison between the various (block, strict, and weak) monotone schemes with the Boltzmann statistics, which describes the gas of classical particles. It is seen that the block-monotone statistics, which takes into account the degeneracy of the energy-levels, seems the unique one having realistic physical applications.

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