# Thermodynamics of $d$-Dimensional Charged AdS (Anti-de Sitter) Black Holes: Hamiltonian Approach and Clapeyron Equation

## TL;DR

This paper explores the thermodynamics of $d$-dimensional charged AdS black holes using a Hamiltonian approach, analyzing phase transitions, coexistence regions, and deriving the Clapeyron equation to understand their thermodynamic behavior.

## Contribution

It introduces a symplectic geometric thermodynamic framework and extends the analysis of black hole thermodynamics to higher dimensions with a Hamiltonian perspective.

## Key findings

- Identified phase transition regions and coexistence curves.
- Derived the Clapeyron equation for these black holes.
- Analyzed $P$-$v$ diagrams across different dimensions and temperatures.

## Abstract

The study of thermodynamics in the view of the Hamiltonian approach is a newest tool to analyze the thermodynamic properties of the black holes. In this letter, we investigate the thermodynamics of $d$-dimensional ($d>3$) asymptotically AntideSitter black holes. A thermodynamic representation based on symplectic geometry is introduced in this letter. We extend the thermodynamics of $d-$dimensional charged AntideSitter black holes in the views of a Hamiltonian approach. Firstly, we study the thermodynamics in reduced phase space and correlate with the Schwarzschild solution. Then we enhance it in the extended phase space. In an extended phase space the thermodynamic equations of state are stated as constraints. We apply the canonical transformation to analyze the thermodynamics of said type of black holes. We plot $P$-$v$ diagrams for different dimensions $d$ taking the temperatures $T<T_c, T=T_c $ and $T>T_c$ and analyze the natures of the graphs and the dependencies on $d$. In theses diagrams, we point out the regions of coexistence. We also examine the phase transition by applying "Maxwell's equal area law" of the said black holes. Here we find the regions of coexistence of two phases which are also depicted graphically. Finally, we derive the "Clapeyron equation" and investigate the latent heat of isothermal phase transition.