# On entropy, specific heat, susceptibility and Rushbrooke inequality in percolation

## TL;DR

This paper explores the behavior of entropy, specific heat, and susceptibility in percolation models on different lattices, demonstrating their similarity to thermal phase transitions and confirming the Rushbrooke inequality across universality classes.

## Contribution

It introduces definitions for entropy, specific heat, and susceptibility in percolation, showing their analogous behavior to thermal systems and verifying the Rushbrooke inequality.

## Key findings

- Entropy, specific heat, and susceptibility in percolation behave like their thermal counterparts.
- Rushbrooke inequality holds for different lattice types in percolation.
- Percolation models exhibit power-law behavior near the critical point.

## Abstract

We investigate percolation, a probabilistic model for continuous phase transition (CPT), on square and weighted planar stochastic lattices. In its thermal counterpart, entropy is minimally low where order parameter (OP) is maximally high and vice versa. Besides, specific heat, OP and susceptibility exhibit power-law when approaching the critical point and the corresponding critical exponents $\alpha, \beta, \gamma$ respectably obey the Rushbrooke inequality (RI) $\alpha+2\beta+\gamma\geq 2$. Their analogues in percolation, however, remain elusive. We define entropy, specific heat and redefine susceptibility for percolation and show that they behave exactly in the same way as their thermal counterpart. We also show that RI holds for both the lattices albeit they belong to different universality classes.