# An entropy for groups of intermediate growth

## TL;DR

This paper explores a new entropy functional, the δ-entropy, for systems with phase spaces modeled by groups of intermediate growth, exemplified by the first Grigorchuk group, linking group growth to statistical mechanics.

## Contribution

It introduces the δ-entropy as a novel entropy functional suitable for systems with phase spaces of intermediate growth, using the first Grigorchuk group as a key example.

## Key findings

- Demonstrates the δ-entropy's applicability to groups of intermediate growth.
- Connects group growth properties with statistical mechanics frameworks.
- Suggests implications for the evolution of systems in complex phase spaces.

## Abstract

One of the few accepted dynamical foundations of non-additive "non-extensive") statistical mechanics is that the choice of the appropriate entropy functional describing a system with many degrees of freedom should reflect the rate of growth of its configuration or phase space volume. We present an example of a group, as a metric space, that may be used as the phase space of a system whose ergodic behavior is statistically described by the recently proposed $\delta$-entropy. This entropy is a one-parameter variation of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from the power-law entropies that have been recently studied. We use the first Grigorchuk group for our purposes. We comment on the connections of the above construction with the conjectured evolution of the underlying system in phase space.