Mutual information as a measure of mixing efficiency in viscous fluids*

Fluid mixing at the microscale is of paramount importance in biological organisms and in artificial systems. A particular challenge to microscale mixing is posed by the time-reversibility of flows at low Reynolds numbers. Mixing therefore requires an interplay between advection and diffusion. Here we introduce mutual information between particle positions before and after mixing as a measure of mixing efficiency. We demonstrate its application in a Couette flow in an annulus and show that the mixing efficiency depends in a non-trivial way on the time sequence of rotation. We show that under this measure, the mixing efficiency is symmetric upon time reversal of the actuation sequence. Among all sequences with the same rotation angle, the ones with optimal mixing consist of a fast rotation in the middle of the time interval, or in some cases two symmetrically arranged. We also determine mutual information from Brownian dynamics simulations using data compression algorithms and demonstrate that advanced neural network based compression algorithms can be applied to estimate mutual information to a high accuracy. Our results show that mutual information provides a universal and assumption-free measure of mixing efficiency in microscale flows. Furthermore, we expect that our formalism will also be applicable to more complex mixing situations in active systems. The current work is to study mixing caused by different types of microswimmers.