# Entropy Maximization, Time Emergence, and Phase Transition

## TL;DR

This paper explores entropy maximization in finite systems, linking it to biological and physical concepts like time emergence and phase transitions.

## Contribution

The paper introduces a novel interpretation of Lagrange multipliers as time-oriented measures and applies entropy maximization to biological models without thermodynamic limits.

## Key findings

- Entropy maximization is applied to biological competition models using the Canonical Ensemble.
- Lagrange multipliers are reinterpreted with time units to describe the Arrow of Time.
- Phase transitions in growth dynamics are modeled without requiring a thermodynamic limit.

## Abstract

We survey developments in the use of entropy maximization for applying the Gibbs Canonical Ensemble to finite situations. Biological insights are invoked along with physical considerations. In the game-theoretic approach to entropy maximization, the interpretation of the two player roles as predator and prey provides a well-justified and symmetric analysis. The main focus is placed on the Lagrange multiplier approach. Using natural physical units with Planck’s constant set to unity, it is recognized that energy has the dimensions of inverse time. Thus, the conjugate Lagrange multiplier, traditionally related to absolute temperature, is now taken with time units and oriented to follow the Arrow of Time. In quantum optics, where energy levels are bounded above and below, artificial singularities involving negative temperatures are eliminated. In a biological model where species compete in an environment with a fixed carrying capacity, use of the Canonical Ensemble solves an instance of Eigen’s phenomenological rate equations. The Lagrange multiplier emerges as a statistical measure of the ecological age. Adding a weak inequality on an order parameter for the entropy maximization, the phase transition from initial unconstrained growth to constrained growth at the carrying capacity is described, without recourse to a thermodynamic limit for the finite system.