# Langevin dynamics for ramified structures

## TL;DR

This paper introduces a generalized Langevin formalism to model transport in ramified structures like combs, deriving analytical expressions for mean square displacement and validating them with simulations across various types of secondary branch motions.

## Contribution

It develops a unified Langevin-based framework for transport in ramified structures, encompassing different secondary branch dynamics and extending to higher dimensions.

## Key findings

- Analytical MSD expressions match Monte Carlo simulations.
- MSD for Gaussian noise is independent of noise color.
- Framework applies to finite and infinite secondary branches.

## Abstract

We propose a generalized Langevin formalism to describe transport in combs and similar ramified structures. Our approach consists of a Langevin equation without drift for the motion along the backbone. The motion along the secondary branches may be described either by a Langevin equation or by other types of random processes. The mean square displacement (MSD) along the backbone characterizes the transport through the ramified structure. We derive a general analytical expression for this observable in terms of the probability distribution function of the motion along the secondary branches. We apply our result to various types of motion along the secondary branches of finite or infinite length, such as subdiffusion, superdiffusion, and Langevin dynamics with colored Gaussian noise and with non-Gaussian white noise. Monte Carlo simulations show excellent agreement with the analytical results. The MSD for the case of Gaussian noise is shown to be independent of the noise color. We conclude by generalizing our analytical expression for the MSD to the case where each secondary branch is $n$ dimensional.