Product-Sum universality and Rushbrooke inequality in explosive percolation

TL;DR

This paper investigates explosive percolation on Erdős-Rényi networks, redefining thermodynamic quantities to establish its relation to continuous phase transitions and confirming universality class and Rushbrooke inequality adherence.

Contribution

It introduces thermodynamic analogs for explosive percolation and demonstrates their behavior aligns with continuous phase transition models, establishing universality class and critical relations.

Findings

PR and SR rules belong to the same universality class

Critical exponents satisfy Rushbrooke inequality

Behavior of entropy, specific heat, and susceptibility mimics thermal phase transitions

Abstract

We study explosive percolation (EP) on Erd\"{o}s-R\'{e}nyi network for product rule (PR) and sum rule (SR). Initially, it was claimed that EP describes discontinuous phase transition, now it is well-accepted as a probabilistic model for thermal continuous phase transition (CPT). However, no model for CPT is complete unless we know how to relate its observable quantities with those of thermal CPT. To this end, we define entropy, specific heat, re-define susceptibility and show that they behave exactly like their thermal counterparts. We obtain the critical exponents $\nu, \alpha, \beta$ and $\gamma$ numerically and find that both PR and SR belong to the same universality class and they obey the Rushbrooke inequality.