# Entropy Alternatives for Equilibrium and Out-of-Equilibrium Systems

## TL;DR

This paper introduces new entropy-related measures to analyze complex systems, showing their advantages over traditional entropy in capturing dynamic behaviors.

## Contribution

The novel non-repeatability and sorted mutability functions are introduced for analyzing dynamical behaviors in complex systems.

## Key findings

- Non-repeatability and mutability are sensitive to temporal data ordering, unlike traditional entropy.
- Sorted mutability provides deeper insights into the critical behavior of magnetic and seismic systems.
- The proposed measures are validated using established systems and compared with Shannon entropy.

## Abstract

We introduce a novel entropy-related function, non-repeatability, designed to capture dynamical behaviors in complex systems. Its normalized form, mutability, has been previously applied in statistical physics as a dynamical entropy measure associated with any observable stored in a sequential file. We now extend this concept by calculating the sorted mutability for the same data file previously ordered by increasing or decreasing value. To present the scope and advantages of these quantities, we analyze two distinct systems: (a) Monte Carlo simulations of magnetic moments on a square lattice, and (b) seismic time series from the United States Geological Survey catalog. Both systems are well established in the literature, serving as robust benchmarks. Shannon entropy is employed as a reference point to assess the similarities and differences with the proposed measures. A key distinction lies in the sensitivity of non-repeatability and mutability to the temporal ordering of data, which contrasts with traditional entropy definitions. Moreover, sorted mutability reveals additional insights into the critical behavior of the systems under study.