How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems

TL;DR

This paper explores how multiplicity influences entropy and extends the maximum entropy principle to complex, non-ergodic systems by deriving generalized entropies from microscopic properties.

Contribution

It demonstrates that the maximum entropy principle remains valid for complex systems when their relative entropy is properly factorized, introducing a new derivation of generalized entropies from microscopic dynamics.

Findings

Generalized entropies are compatible with the maximum entropy principle in complex systems.

Path-dependent processes with memory require specific generalized entropies.

First exact derivation of a generalized entropy from microscopic properties.

Abstract

The maximum entropy principle (MEP) is a method for obtaining the most likely distribution functions of observables from statistical systems, by maximizing entropy under constraints. The MEP has found hundreds of applications in ergodic and Markovian systems in statistical mechanics, information theory, and statistics. For several decades there exists an ongoing controversy whether the notion of the maximum entropy principle can be extended in a meaningful way to non-extensive, non-ergodic, and complex statistical systems and processes. In this paper we start by reviewing how Boltzmann-Gibbs-Shannon entropy is related to multiplicities of independent random processes. We then show how the relaxation of independence naturally leads to the most general entropies that are compatible with the first three Shannon-Khinchin axioms, the (c,d)-entropies. We demonstrate that the MEP is a perfectly consistent concept for non-ergodic and complex statistical systems if their relative entropy can be factored into a generalized multiplicity and a constraint term. The problem of finding such a factorization reduces to finding an appropriate representation of relative entropy in a linear basis. In a particular example we show that path-dependent random processes with memory naturally require specific generalized entropies. The example is the first exact derivation of a generalized entropy from the microscopic properties of a path-dependent random process.