Product-Sum universality and Rushbrooke inequality in explosive percolation

Abstract

We study explosive percolation (EP) on Erdös-Rényi 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 $ν, α, β$ and $γ$ numerically and find that both PR and SR belong to the same universality class and they obey the Rushbrooke inequality.