Statistical Transition of Bose Gas to Fermi Gas

Mathematical notes. 2018. Vol. 103. No. 6. P. 929–935.

Maslov V. P.

It is well known that the formula for the Fermi distribution is obtained from the formula
for the Bose distribution if the argument of the polylogarithm, the activity a, the energy, and the
number of particles change sign. The paper deals with the behavior of the Bose–Einstein distribution
as a → 0; in particular, the neighborhood of the point a = 0 is studied in great detail, and the
expansion of both the Bose distribution and the Fermi distribution in powers of the parameter a is
used. During the transition from the Bose distribution to the Fermi distribution, the principal term of
the distribution for the specific energy undergoes a jump as a → 0. In this paper,we find the value of
the parameter a, close to zero, but not equal to zero, for which the Bose distribution (in the statistical
sense) becomes zero. This allows us to find the point a, distinct from zero, at which a jump of the
specific energy occurs. Using the value of the number of particles on the caustic, we can obtain the
jump of the total energy of the Bose system to the Fermi system. Near the value a = 0, the author
uses Gentile statistics, whichmakes it possible to study the transition fromthe Bose statistics to the
the Fermi statistics in great detail. Here an important role is played by the self-consistent equation
obtained by the author earlier.

Keywords: Gentile statistics, Bose statistics, self-consistent equation, Fermi statistics, jump of specific energy