# Fractional-order phase transition of charged AdS black holes

## TL;DR

This paper introduces fractional derivatives to classify phase transitions more precisely, revealing that charged AdS black holes and Van der Waals fluids undergo a fractional-order phase transition of order 4/3 at the critical point.

## Contribution

It proposes a fractional derivative-based classification of phase transitions, identifying a 4/3-order transition in charged AdS black holes and Van der Waals fluids.

## Key findings

- Charged AdS black holes exhibit a 4/3-order phase transition.
- Van der Waals fluids also show a 4/3-order transition at criticality.
- The method refines the understanding of phase transition orders beyond traditional classifications.

## Abstract

Employing the fractional derivatives, we construct a more elaborate classification of phase transitions compared to the original Ehrenfest classification. In this way, a thermodynamic system can even undergo a fractional-order phase transition. We use this method to restudy the charged AdS black hole and Van der Waals fluids and find that at the critical point they both have a $4/3$-order phase transition, but not the previously recognized second-order one.