**Brownian motion**

### Background

Objects in miniaturized systems could be moving in an environment with varying configurations, temperatures and fluid properties, among other factor. Thermal fluctuations are important to the motion of such objects and must be included in any fundamental modeling scheme.The previously developed Brownian dynamics (BD) and Stokesian dynamics (SD) approaches for the Brownian motion of particles are based on the Langevin equation for particle motion. A random force is included in the particle equation of motion. Although these techniques are effective in many cases, using these techniques to objects of irregular shapes and to cases where the fluid exhibits varying properties is not straightforward. This is mainly because the properties of the random force in the particle equations depend on the hydrodynamic interactions, which in turn depends on the particle positions, shapes and the fluid properties.

### Our approach

In light of the above issues we considered fully resolved simulation of fluid-particle motion to be an excellent tool to fundamentally investigate the motion of micron scale particles in varying fluid environments. Hence, our primary objective was to devise a convenient way to incorporate the effect of thermal fluctuations in the fully resolved schemes.

We have proposed an approach where the thermal fluctuations are included in the fluid equations (instead of in the particle equations as in BD and SD approaches) via random stress terms. Solving these fluctuating hydrodynamic (FHD) equations coupled with the rigid motion constraint in the particle domain results in the Brownian motion of the particles. The random stress in the fluid equations is easy to calculate unlike the random terms in the SD approach. It has to be ensured, however, that the random stresses for the discrete system satisfy the so called â€˜fluctuation dissipation theoremâ€™. We call this approach the **Fluctuating Immersed Material (FIMAT) dynamics** approach.

We have tested our approach for a variety of cases including single spheres, single ellipsoids and many spheres by considering quasi-steady simulations in the long time limit. Translational and rotational diffusion of the particles were considered. Unsteady simulations were also performed to test the short and long time behavior of the velocity correlations. The method correctly reproduces the algebraic tail of the velocity correlation at long times. In all the test cases, the agreement with theoretical values was found to be very good. The references and some sample results are given below.

### Uniqueness, impact and outlook

The uniqueness of this approach is that it can be applied to general configurations and particle shapes, potentially in a variety of fluid environments, without any additional complexity. It can model short and long time thermal motion of the particles. The translational as well as the rotational diffusion of the particle are simulated simultaneously.

It is to be noted that this approach captures the algebraic tail in the velocity autocorrelation function consistent with the molecular time autocorrelation functions unlike the Langevin approach (e.g. in BD) which gives an exponential tail in the velocity autocorrelation function.

This approach can be easily incorporated into existing fluid flow solvers based on the Navier-Stokes equations. Extension of the method to droplets and elastic bodies immersed in fluids should be considered in the future.

### Applications

We are interested in applying this tool to understand the dynamics of intracellular processes, e.g. motion of motor proteins and to develop multiscale techniques. The FHD approach is also being used to study slip near hydrophobic surfaces.

Biomaterials, such as cellular materials contain biomolecules suspended in the aqueous component of the cytoplasm. Examples of biomolecules include motor proteins and polymerizing/de-polymerizing filaments which use chemical energy and produce motion or generate force (mechanical energy). These are active processes. It is essential to link the microscale dynamics of the active processes to the mechanics of the cell material. We are interested in using our FIMAT dynamics approach to study such problems.