Time Evolution of Entropy in a Growth model: Dependence on the Description

TL;DR

This paper investigates how entropy evolves over time in a growth model, highlighting the impact of description scale and heterogeneity on information loss and size space expansion.

Contribution

It introduces a detailed analysis of entropy dynamics in a Yule-type growth model considering both continuous and discrete element sizes, revealing the effects of heterogeneity and correlations.

Findings

Heterogeneity and correlations can cause information loss during coarse-graining.

Size space expansion depends on the description level, differing between continuous and discrete models.

Entropy evolution is influenced by the scale of description and element heterogeneity.

Abstract

Entropy plays a key role in statistical physics of complex systems, which in general exhibit diverse aspects of emergence on different scales. However, it still remains not fully resolved how entropy varies with the coarse-graining level and the description scale. In this paper, we consider a Yule-type growth model, where each element is characterized by its size being either continuous or discrete. Entropy is then defined directly from the probability distribution of the states of all elements as well as from the size distribution of the system. Probing in detail their relations and time evolutions, we find that heterogeneity in addition to correlations between elements could induce loss of information during the coarse-graining procedure. It is also revealed that the expansion of the size space domain depends on the description level, leading to a difference between the continuous description and the discrete one.