# Custom flow in overdamped Brownian Dynamics

## TL;DR

This paper introduces a general iterative simulation scheme to determine the unique external force field needed to produce a specified flow in overdamped Brownian systems, facilitating systematic analysis of complex colloidal flows.

## Contribution

It presents a novel computational method based on the force balance equation that can tailor flow fields and densities, extending the power functional framework for non-equilibrium systems.

## Key findings

- The method converges to a unique external force field for prescribed flows.
- It can generate conservative forces for target density profiles in equilibrium.
- The approach provides detailed physical insights into complex flow behaviors.

## Abstract

When an external field drives a colloidal system out of equilibrium, the ensuing colloidal response can be very complex and obtaining a detailed physical understanding often requires case-by-case considerations. In order to facilitate systematic analysis, here we present a general iterative scheme for the determination of the unique external force field that yields a prescribed inhomogeneous stationary or time-dependent flow in an overdamped Brownian many-body system. The computer simulation method is based on the exact one-body force balance equation and allows to specifically tailor both gradient and rotational velocity contributions, as well as to freely control the one-body density distribution. Hence compressibility of the flow field can be fully adjusted. The practical convergence to a unique external force field demonstrates the existence of a functional map from both velocity and density to external force field, as predicted by the power functional variational framework. In equilibrium, the method allows to find the conservative force field that generates a prescribed target density profile, and hence implements the Mermin-Evans classical density functional map from density distribution to external potential. The conceptual tools developed here enable one to gain detailed physical insight into complex flow behaviour, as we demonstrate in prototypical situations.