Some Implications of a Scale-Invariant Model of Statistical Mechanics to Classical and Black Hole Thermodynamics.

A scale-invariant model of statistical mechanics is applied to described modified forms of four laws of classical thermodynamics. Following de Broglie formula $\lambda_{rk} =h/m_{k} v_{rk} $, frequency of matter waves is defined as $\nu_{rk} =k/m_{k} v_{rk} $ leading to stochastic definitions of (Planck, Boltzmann) universal constants ($h=m_{k} <\lambda_{rk} >c$, $k=m_{k} <\nu_{rk} >c)$, $\lambda_{rk} \nu_{rk} =c$, relating to spatiotemporal \textit{Casimir} vacuum fluctuations. Invariant Mach number $Ma_{\beta } =v/v_{r\beta } $ is introduced leading to hierarchy of ``supersonic'' flow separated by shock front, viewed as ``event-horizon'' EH$_{\beta }$, from subsonic flow that terminates at surface of stagnant condensate of ``atoms'' defined as ``black-hole'' BH$_{\beta }$ at scale $\beta $ thus resulting in hierarchy of embedded ``black holes'' at molecular- atomic-, electron-, photon-, tachyon-. . . scales, ad infinitum. Classical black hole will correspond to solid phase photon or \textit{solid-light}. It is argued that Bardeen-Carter-Hawking$^{\, }$(1973) first law of black hole mechanics $\delta M=(\kappa /8\pi )\delta A+\Omega_{H} \delta J+\Phi_{H} \delta Q$, instead of $dE=TdS-PdV$ suggested by Bekenstein (1973), is analogous to first law of thermodynamics expressed as $TdS=PdV+dE$ such that entropy of black hole, rather than to its horizon surface area, will be related to its total energy hence enthalpy $H=TS\mbox{\, }$leading to $S_{BH} =4kN$ in exact agreement with prediction of Major and Setter$^{,\, }$\textit{Class. Quant. Grav.} \textbf{18}(2-3), 5125 (2001).