# Statistics effects in extremal black holes ensemble

## TL;DR

This paper investigates how different statistical distributions (Bose-Einstein, Fermi-Dirac, classical, infinite) influence the effective mass and time dilation in a grand canonical ensemble of extremal black holes, using mean field approximation.

## Contribution

It introduces a statistical framework for extremal black holes and analyzes the impact of various statistics on physical quantities using mean field approximation.

## Key findings

- Statistics significantly affect effective mass and time dilation.
- Different statistics produce distinguishable effects on black hole ensemble properties.
- Mean field approximation enables visualization of statistical impacts.

## Abstract

We consider the grand canonical ensemble of the static and extremal black holes, when the equivalence of the electric charge and mass of individual black hole is postulated. Assuming uniform distribution of black holes in space, we are finding the effective mass of test particle and mean time dilation at the admissible points of space, taking into account the gravitational action of surrounding black holes. Having specified the statistics that governs extremal black holes, we study its effect on those quantities. Here, the role of statistics is to assign a statistical weight to the configurations of certain fixed number of black holes. We borrow these weights from Bose-Einstein, Fermi-Dirac, classical and infinite statistics. Using mean field approximation, the aforementioned characteristics are calculated and visualized, what permits us to draw the conclusions on visible effect of each statistics.