The Second Law of Thermodynamics: Mathematical Error

The critical analysis of the generally accepted foundations of thermodynamics is proposed. Within the framework of the work [1], the following statement is proved: Gibbs's quantum canonical distribution $f_n =f_0 \exp \,(-{E_n } \mathord{\left/ {\vphantom {{E_n } {T)}}} \right. \kern-\nulldelimiterspace} {T)}$ (where$E_n $, $n=0,\;1,\;\ldots $, $f_n $, $T$ are the energy of the subsystem, probability, and temperature, respectively) defines the correct relation of the thermal energy $Q$ of the subsystem to the entropy $S$ of the subsystem and the temperature $T$. This relation has the form: $S=Q \mathord{\left/ {\vphantom {Q T}} \right. \kern-\nulldelimiterspace} T$ and $\mathop {\lim }\limits_{T\to \,0\,} S=0$ (where $Q\equiv \sum\limits_{n=0}^\infty {E_n } f_n $, $S\equiv \sum\limits_{n=0}^\infty {S_n f_n } $, $S_n \equiv {E_n } \mathord{\left/ {\vphantom {{E_n } {T=-\ln \,({f_n } \mathord{\left/ {\vphantom {{f_n } {f_0 )}}} \right. \kern-\nulldelimiterspace} {f_0 )}}}} \right. \kern-\nulldelimiterspace} {T=-\ln \,({f_n } \mathord{\left/ {\vphantom {{f_n } {f_0 )}}} \right. \kern-\nulldelimiterspace} {f_0 )}})$. Consequence: the second law (i.e. $dS={dQ} \mathord{\left/ {\vphantom {{dQ} T}} \right. \kern-\nulldelimiterspace} T)$ of thermodynamics represents mathematical error. Ref.: [1] T.Z. Kalanov, ``On the main errors underlying statistical physics.'' Bulletin of the APS, Vol. 47, No. 2 (2005), p. 164.