On the Thermodynamics of Particles Obeying Monotone Statistics

Abstract

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.

Gas of Particles Obeying the Monotone Statistics

“…As explained [13], it was shown that there is some evidence that such a picture could provide the grand-partition function of the (block-)monotone case. The present paper is devoted to confirming this evidence.…”

“…Recently, the attempt to systematically investigate the thermodynamic properties of such exotic particles has been carried out in the series of papers [10][11][12][13][14]. The first study was that concerning the q-particles or quons, q ∈ [−1, 1].…”

“…These are called the strictly monotone (simply called "monotone" in most of the related literature and in the present paper) and the weakly monotone schemes. On the other hand, many (but not all significant) meaningful physical models are described by a one-particle Hamiltonian H with compact resolvent, acting on a, typically separable, Hilbert space H. Therefore, there is a completely natural order on the necessarily discrete spectrum σ(H) of the Hamiltonian H and, in addition, the set of normalised eigenvectors of H provides an orthonormal basis of H. Unfortunately, the energy-levels can have a degeneracy, and thus no total pre-order is inherited on the basis of H made of eigenvectors of H. To overcome this problem in view of possible physical applications, in [13] a new scheme was introduced, called block-monotone, which takes into account the natural partial order on the basis of H made of eigenvectors of H, established by considering the possible degeneracy of the energy-levels.…”

“…Since it is almost impossible to provide an explicit calculation of the block-monotone grand-partition function for most reasonable models, in the previously mentioned paper [13], we analysed a simple model, perhaps without degeneracy (and thus for which monotone and block-monotone schemes coincide). For such a model, we see that the correction factor 1/n!, connected to indistinguishability, automatically appears in the computation of the statistical weights corresponding to the various n-levels.…”

“…As in the previous model of [13], we explicitly estimated such a decimation in the case of a low density/high energy regime.…”