# Custom flow in overdamped Brownian dynamics

D. de las Heras, J. Renner, and M. Schmidt

Phys. Rev. E, **99**, 023306, (2019) DOI: 10.1103/PhysRevE.99.023306

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**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. To facilitate systematic analysis, here we present a general iterative scheme for the determination of the unique external force field that yields 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.

**Additional material/comments**:

There is a multitude of ways to generate flow in soft matter systems, ranging from microfluidic pumping to applying holographic laser tweezers or magnetic fields acting on suitably prepared particles. Precise control over the colloidal motion is crucial in important applications of colloidal science, such as cargo delivery with colloidal carriers, and lab-on-a-chip processing.
To drive the colloidal particles, external fields are typically applied in controlled ways, and the system under consideration is hence far from equilibrium. As the response of the colloids to such external perturbations can be very complex due to packing, correlations and the interactions between the particles, detailed and specifically tailor considerations are typically required to obtain the desired effect(s).

In this manuscript we develop a numerical method that allows to freely prescribe the desired motion of the system ("custom flow"). This is achieved by systematically constructing the precise form of the required external force field that produces the motion. Our method allows to deal with arbitrary non-equilibrium situations and constitutes an efficient and straightforward iterative method to solve what is a complex inverse statistical many-body problem.

We give a first-principles derivation of the method, based on a careful consideration of the force balance equation for overdamped Brownian dynamics, from which we obtain an iterative computational scheme. We formally prove the existence and uniqueness of the solution, based on the exact power functional framework, which is the (formally exact) generalisation of classical density functional theory to dynamical situations. In equilibrium, our method allows to realise the finite temperature (Mermin) functional map of density functional theory.

We show several examples of application of the method to generic flow problems in both steady state and time-dependent situations. In steady state we cover (i) longitudinal flow across potential barriers controlling the density modulation, (ii) the emergence of non-equilibrium forces despite the density profile being spatially constant, which contradicts dynamical density functional theory, (iii) a two-dimensional fully inhomogeneous flow situation, which demonstrates that flow can be controlled through arbitrary and complex density landscapes. Also, in time-dependent situations, our method allows a complete and unprecedented control over the whole evolution of a Brownian system. As an example, we show how to arbitrarily change the internal time-scale of a given dynamical process.

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