Diverging velocity fluctuations and enhanced tracer diffusivities in interacting swimmer suspensions

Evidence from both experiments and simulations suggests that suspensions of rear-actuated micro-swimmers exhibit interesting phenomena, such as long ranged orientational order, enhanced tracer particle diffusivities as well as collective motion, among others. We present an analytical characterization of the fluid velocity correlations and tracer diffusivities in an interacting swimmer suspension. Pair-interactions lead to a logarithmically divergent velocity variance, and a non-decaying covariance, in a suspension of straight swimmers at O(nL³)²; n being the swimmer number density and L being its characteristic length (nL³ << 1). Incorporating orientation decorrelation in the form of tumbles leads to the said divergence being cut-off at distances of order the long but finite run length - Uτ, where τ represents the swimmer's mean run duration and U the swimming speed. The covariance shows a weak logarithmic decay at distances r ∼ O(Uτ), and an eventual O(1/r) decay for r >> Uτ. The O(nL³)² tracer diffusivity exhibits a linear divergence with Uτ. Interestingly, the mean square displacement exhibits a broad crossover from ballistic to diffusive regimes, with the transition extending from times of O(L/U) to O(τ), for Uτ/L >> 1, giving the impression of an anomalous exponent.