# Distribution in the Geometrically Growing System and Its Evolution

## TL;DR

This paper presents a theory for geometrically growing systems that explains power-law distributions and their evolution, highlighting unique convexity features and the system's tendency to flatten over time, without relying on complex economic models.

## Contribution

The paper introduces a new statistical theory for geometrically growing systems that accounts for power-law phenomena and their evolution, including a convexity feature absent in traditional models.

## Key findings

- Explains power-law distributions in demographic, economic, and pandemic data.
- Identifies convexity in the low-size distribution part.
- Shows the distribution tends to flatten as the system evolves.

## Abstract

Recently, we developed a theory of a geometrically growing system. Here we show that the theory can explain some phenomena of power-law distribution including classical demographic and economic and novel pandemic instances, without introduction of delicate economic models but only on the statistical way. A convexity in the low-size part of the distribution is one peculiarity of the theory, which is absent in the power-law distribution. We found that the distribution of the geometrically growing system could have a trend to flatten in the evolution of the system so that the relative ratio of size within the system increases. The system can act as a reverse machine to covert a diffusion in parametric space to a concentration in the size distribution.