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The first part of the paper deals with the behavior of the Bose--Einsteindistribution as the activity $a\to 0$. In particular, the neighborhood of thepoint $a=0$ is studied in great detail, and the expansion of both the Bosedistribution and the Fermi distribution in powers of the parameter $a$ is used.This approach allows to find the value of the parameter $a_0$, for which theBose distribution (in the statistical sense) becomes zero. In the second partof the paper, the process of separation of a nucleon from the atom's nucleus isstudied from the mathematical point of view. At the moment when the nucleontears away from the fermionic nucleus, the nucleus becomes a boson. Weinvestigate the further transformations of bosonic and fermionic separationstates in a small neighborhood of the pressure $P$ equal to zero. We useinfinitely small quantities to modify the parastatistical distribution. Ourconception is based on interpolation formulas yielding expansions in powers ofthe density. This method differs from those in other models based on theinteraction potential between two or three particles. We obtain new importantrelations connecting the temperature with the chemical potential during theseparation of a nucleon from the atom's nucleus. The obtained relations allowus to construct, on an antipode of sorts of the Hougen--Watson P-Z chart, thevery high temperature isotherms corresponding to nuclear matter. It is provedmathematically that the passage of particles satisfying the Fermi--Diracdistribution to the Bose--Einstein distribution in the neighborhood of pressure$P$ equal to zero occurs in a region known as the "halo".

Transition from the Bose gas to the Fermi gas through a Nuclear Halo