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.
Keywords: quantum probability; quantum statistical mechanics; thermodynamics.