Statistical Transition of Bose Gas to Fermi Gas

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 $a\to 0$; 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 $a\to 0$. 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, which makes it possible to study the transition from the 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.