Statistical mechanics of exploding phase spaces: Ontic open systems

TL;DR

This paper explores the behavior of phase spaces that grow super-exponentially in complex systems, proposing a new entropy measure that remains extensive and could aid in defining probability measures.

Contribution

It introduces an axiomatic entropy for exploding phase spaces and demonstrates its extensivity across various ensembles, addressing limitations of standard ensemble theory.

Findings

Entropy remains extensive in super-exponentially growing phase spaces.

Standard ensemble theory can break down in complex systems with emergent states.

Proposes a new entropy measure useful for defining probability in complex systems.

Abstract

The volume of phase space may grow super-exponentially ("explosively") with the number of degrees of freedom for certain types of complex systems such as those encountered in biology and neuroscience, where components interact and create new emergent states. Standard ensemble theory can break down as we demonstrate in a simple model reminiscent of complex systems where new collective states emerge. We present an axiomatically defined entropy and argue that it is extensive in the micro-canonical, equal probability, and canonical (max-entropy) ensemble for super-exponentially growing phase spaces. This entropy may be useful in determining probability measures in analogy with how statistical mechanics establishes statistical ensembles by maximising entropy.