# Dynamical networking using Gaussian fields

## TL;DR

This paper introduces a new method using Gaussian fields to model dynamic interactions in complex systems, showing how particles bind and unbind over time.

## Contribution

A novel field theoretical approach for modeling dynamic networking using Gaussian fields and Martin–Siggia–Rose generating functionals.

## Key findings

- The formalism accounts for spatial and temporal constraints in particle networking via statistical weights.
- Application to cross-linking polymers shows system collapse when networking is introduced.
- Repulsive time-dependent potential above a minimum strength prevents system collapse.

## Abstract

A novel field theoretical approach towards modelling dynamic networking in complex systems is presented. An equilibrium networking formalism which utilises Gaussian fields is adapted to model the dynamics of particles that can bind and unbind from one another. Here, networking refers to the introduction of instantaneous co-localisation constraints and does not necessitate the formation of a well-defined transient or persistent network. By combining this formalism with Martin–Siggia–Rose generating functionals, a weighted generating functional for the networked system is obtained. The networking formalism introduces spatial and temporal constraints into the Langevin dynamics, via statistical weights, thereby accounting for all possible configurations in which particles can be networked to one another. A simple example of Brownian particles which can bind and unbind from one another demonstrates the tool and that this leads to results for physical quantities in a collective description. Applying the networking formalism to model the dynamics of cross-linking polymers in a mixture, we can calculate the average number of networking instances. As expected, the dynamic structure factors for each type of polymer show that the system collapses once networking is introduced, but that the addition of a repulsive time-dependent potential above a minimum strength prevents this. The examples presented in this paper indicate that this novel approach towards modelling dynamic networking could be applied to a range of synthetic and biological systems to obtain theoretical predictions for experimentally verifiable quantities.