Correlations and fluctuations of stress and velocity in suspensions of swimming microorganisms

Abstract

Active systems, which are driven out of equilibrium, can produce long range correlations and large fluctuations that are not restricted by the fluctuation-dissipation theorem. We consider here the fluctuations and correlations in suspensions of swimming microorganisms that interact hydrodynamically. Modeling the organisms as force dipoles in Stokes flow and considering run-and-tumble and rotational diffusion models of their orientational dynamics allow derivation of closed form results for the stress fluctuations in the long-wave limit. Both of these models lead to Lorentzian distributions, in agreement with some experimental data. These fluctuations are not restricted by the fluctuation-dissipation theorem, as is explicitly verified by comparing the fluctuations with the viscosity of the suspension. In addition to the stress fluctuations in the suspension, we examine correlations between the organisms. Because of the hydrodynamic interactions, the velocities of two organisms are correlated even if the positions and orientations are uncorrelated. We develop a theory of the velocity correlations in this limit and compare with the results of computer simulations. We also formally include orientational correlations in the theory; and comparing with simulations, we are able to show that these are important even in the dilute limit and are responsible in large part for the velocity correlations. While the orientation correlations cannot as yet be predicted from this theory, by inserting the results from simulations into the theory it is possible to properly determine the form of the swimmer velocity correlations. These correlations of orientations are also the key to understanding the spatial correlations of the fluid velocity. Through simulations we show that the orientational correlations decay as r−2 with distance—inserting this dependence into the theory leads to a logarithmic dependence of the velocity fluctuations on the size of the system.