A non-conserving coagulation model with extremal dynamics

TL;DR

This paper introduces a coagulation model with extremal dynamics that interpolates between various physical models, analyzing how primary and secondary variables grow under different parameters and estimating related exponents.

Contribution

It presents a novel coagulation process that unifies multiple physical models and provides accurate estimates of growth exponents across parameter ranges.

Findings

Exponent α_ω varies monotonically with ω.

Exponent β_ω has a maximum at ω=0.

The model interpolates between known physical models.

Abstract

A coagulation process is studied in a set of random masses, in which two randomly chosen masses and the smallest mass of the set multiplied by some fixed parameter $\omega\in [-1,1]$ are iteratively added. Besides masses (or primary variables), secondary variables are also considered that are correlated with primary variables and coagulate according to the above rule with $\omega=0$. This process interpolates between known statistical physical models: The case $\omega=-1$ corresponds to the strong disorder renormalisation group transformation of certain disordered quantum spin chains whereas $\omega=1$ describes coarsening in the one-dimensional Glauber-Ising model. The case $\omega=0$ is related to the renormalisation group transformation of a recently introduced graph with a fat-tail edge-length distribution. In the intermediate range $-1<\omega<1$, the exponents $\alpha_{\omega}$ and $\beta_{\omega}$ that characterise the growth of the primary and secondary variable, respectively, are accurately estimated by analysing the differential equations describing the process in the continuum formulation. According to the results, the exponent $\alpha_{\omega}$ varies monotonically with $\omega$ while $\beta_{\omega}$ has a maximum at $\omega=0$.