Recent development on fragmentation, aggregation and percolation

TL;DR

This paper reviews the influence of Robert Ziff on fragmentation, aggregation, and percolation, highlighting recent work on entropy, critical exponents, and explosive transitions using Monte Carlo simulations.

Contribution

It introduces new entropy-based methods to analyze percolation, connecting critical phenomena with order-disorder transitions and explosive behavior.

Findings

Entropy helps define specific heat and critical exponents.

Percolation obeys Rushbrooke inequality.

Entropy and order parameter reveal transition nature.

Abstract

In this article, I have outlined how an accomplished researcher like Robert Ziff has influenced a new generation of researchers across the globe like gravity as an action-at-a-distance. In the 80s Ziff made significant contributions to the kinetics of fragmentation followed by the kinetics of aggregation. Here, I will discuss fractal and multifractal that emerges in fragmentation and aggregation processes where the dynamics is governed by non-trivial conservation laws. I have then discussed my recent works and results on percolation where I made extensive use of Newman-Ziff fast Monte Carlo algorithm. To this end, I have defined entropy which paved the way to define specific heat and show that the critical exponents of percolation obey Rushbrooke inequality. Besides, we discuss how entropy and order parameter together can help us to check whether the percolation is accompanied by order-disorder transition or not. The idea of entropy also help to explain why encouraging smaller cluster to grow faster than larger clusters makes the transition explosive.