Reconciling fractional entropy and black hole entropy compositions

Abstract: This study investigates the implications of adopting fractional entropy in the area law framework and demonstrates its natural alignment with an isothermal description of black hole composition. We discuss the Zeroth law compatibility of the fractional entropy and define an empirical temperature for the horizon. We highlight the distinction between the empirical and conventional Hawking temperatures associated with the black holes. Unlike the Hawking temperature, this empirical temperature appears universal, and its proximity to the Planck temperature suggests a possible quantum gravity origin. We also establish the connection between these temperatures. Furthermore, extending the conventional fractional parameter $q$, constrained between 0 and 1, we establish that any positive real number can bound $q$ under the concavity condition, provided the log of micro-state dimensionality exceeds $q-1$. Specifically, for black holes, $q = 2$, necessitating micro-state dimensionality greater than $e$, thereby excluding the construction of black hole horizon states with two level bits or qubits. We also identify the connection between the validity of the second law and information fluctuation complexity. The second law requires that the variance of information content remain smaller than the area of the black hole horizon. This constraint naturally gives rise to a Boltzmann-Gibbs-like entropy for the black hole, which, in contrast to the canonical formulation, is associated with its mass rather than its area. Equilibrium distribution analysis uncovers multiple configurations, in which the one satisfying the prerequisites of probability distribution exhibits an exponent stretched form, revealing apparent deviation from the Boltzmann distribution.