The energy of any physical space domain of an atomic or molecular system houses a fractional number of particles. It has been properly described in the framework of a Grand-canonical distribution (GC) formulation for its ground state [1]. Such type of systems driven by Coulomb interactions present an interesting feature, its energy is a convex function of the number of particles. This fact simplifies the description. The density matrices (DM) of the electron distribution described in this way results in a convex combination of two states leading to a linear piecewise behavior which induces discontinuities for some fundamental physico-chemical descriptors such as chemical potential (electronegativity) or chemical hardness. Hence, they unable to treat properly the equalization (equilibrium) principles. To overcome this difficulty, two different interpretations has been proposed. On the one hand, the interaction of the system with an environment [2] or on the other, a parabolic interpolation including three states [3]. The first proposal must consider a stabilization process due to the environmental effects (solvation) and /or the exchange of electrons with its environment, while the last one, the introduction of the concept of "temperature" to allow equilibrium states instead of the ground states and thus to properly treat  the aforementioned discontinuities. We discuss and criticize such formulations from a rigorous mathematical point of view and proper physical basis for the treatment of this problem within the grand-canonical statistical distribution (GC) of few particles. This subject of study makes evident the fact of the adequate use of shared methods of statistical physics for systems of few particles. Pioneering examples can be found in Refs. [4,5].

[1] R. C. Bochicchio, D. Rial, J. Chem. Phys., 137, 226101 (2012) and references.

[2] R. C. Bochicchio, Theor. Chem. Acc., 134, 138 (2015) and references.

[3] R. A. Miranda-Quintana, P. W. Ayers, J. Chem. Phys. 144, 244112 (2016) and references.

[4] J. A. Medrano, R. C. Bochicchio, J. Mol. Struc. (THEOCHEM) 169, 33 (l988).

[5] L. Arrachea, N. Canosa, A. Plastino, M. Portesi, R. Rossignoli, in Condensed Matter Theories, Vol. 7, A.N. Protto and J. L. Aliaga, Plenum, N.Y., 1992.