Quantitative Emergence Theory

Introduction

At different size scales, nature seems to be composed of qualitatively different physical systems of varying degrees of complexity. Ever since the work of Boltzmann, complex physical systems have been analyzed as statistical ensembles of particles. This means as a collection of random variables.

Such an approach has been tremendously successful resulting first in statistical mechanics, and then in thermodynamics, and ultimately paving the way for quantum mechanics. However, Botlzmann’s statistical treatment is not the only approach to modelling physical systems.

Herein lies an alternative approach to modelling physical systems which involves several mathematical theorems that allow one to systematically enumerate the microstates of an ensemble by modelling the physical system as a network of labelled nodes and edges and then enumerating all possible graphs on those nodes.

This results not only in the recapitulation of the macroscopic system properties of classical statistical mechanics but also a mechanistic explanation for the phenomenon of emergence.

It is concluded that emergent properties arise from the surplus information generated by a network of constituent parts being in a given state of “connectedness” as opposed to any other distinct state of connectedness.

Thus all emergent properties can be thought of as being rooted in a network of subatomic particles of the standard model represented as a network of nodes represented by qubits and edges representing entanglements and other mixed states between constituent qubits.

This surplus quantum information percolates upwards through different strata of size and complexity to manifest as emergent properties in the system in question. This can be shown to be true whether one is talking of bonds between atoms in individual molecules or alliances between nation states.