Gas of Particles Obeying the Monotone Statistics.

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.

On the Thermodynamics of Particles Obeying Monotone Statistics.

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 , 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.

On the Thermodynamics of the -Particles.

Since the grand partition function Zq for the so-called -particles (i.e., quons), q∈(-1,1), cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for q=0, and its -deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to q∈[-1,1].

Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations.

In order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers Z , that is that generated by left and right hand-side partial shifts, the monoid of all strictly increasing maps whose range has finite complement, and finally the collection of all strictly increasing maps of Z . We show that such three monoids are strictly ordered, and the second-named one is the semidirect product between the first and the action of Z generated by the one-step shift. Even if the definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the last monoid on the indices, we see that spreadability can be directly stated in terms of invariance with respect to the action of the first monoid.

Uniform Convergence of Cesaro Averages for Uniquely Ergodic -Dynamical Systems.

Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n ∑ k = 0 n - 1 λ - n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to "measurable non-continuous" eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener-Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.