Bose-Einstein Statistics and Fermi-Dirac Statistics: A Logical Error

The critical analysis of Bose-Einstein statistics and Fermi-Dirac statistics---consequence of Bose's method---is proposed. The main result of the analysis is as follows. (1) In accordance with the definition, Bose-Einstein (B-E) and Fermi-Dirac (F-D) distribution functions $f_{(B-E)}^s $, $f_{(F-D)}^s $ are the average values of the random quantity: $f^s\equiv {\varepsilon ^s} \mathord{\left/ {\vphantom {{\varepsilon ^s} {\varepsilon _1^s }}} \right. \kern-\nulldelimiterspace} {\varepsilon _1^s }$, $\varepsilon ^s\equiv \sum\limits_r {\varepsilon _r^s p_r^s } $, $p_r^s =p_0^s \,\exp \,\left[ {-\,(\alpha +\beta \varepsilon _1^s )r} \right]^$, $r=0,\;1,\;\ldots {\kern 1pt}\quad (B-E)$, $r=0,\;1\quad (F-D)$ where $f^s$ is the average number of the noninteracting monoenergetic identical quantum particles in the $s$-layer cell; $\varepsilon _1^s $ is energy of one particle of kind $s$; $p_r^s $ is the probability that energy takes on the value $\varepsilon _r^s =\varepsilon _1^s r\equiv {(\alpha +\beta \varepsilon _1^s )r} \mathord{\left/ {\vphantom {{(\alpha +\beta \varepsilon _1^s )r} \beta }} \right. \kern-\nulldelimiterspace} \beta $; $1 \mathord{\left/ {\vphantom {1 {\beta \equiv T}}} \right. \kern-\nulldelimiterspace} {\beta \equiv T}$ is temperature; $\alpha \equiv -\beta \mu $ is degeneration parameter; $\mu $ is chemical potential. (2) In accordance with the logic law of identity, $p_r^s \equiv p_r^s $, $\varepsilon _r^s =\varepsilon _1^s r\equiv {(\alpha +\beta \varepsilon _1^s )r} \mathord{\left/ {\vphantom {{(\alpha +\beta \varepsilon _1^s )r} \beta }} \right. \kern-\nulldelimiterspace} \beta $. Hence, $\alpha \equiv 0$. Thus, $\mu \equiv 0$ and, consequently, Bose-Einstein statistics and Fermi-Dirac statistics represent logical error.