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Active temperature and velocity correlations produced by a swimmer suspension

The agitation produced in a fluid by a suspension of micro-swimmers in the low Reynolds number limit is studied. In this limit, swimmers are modeled as force dipoles all with equal strength. The agitation is characterized by the active temperature defined, as in kinetic theory, as the mean square velocity, and by the equal-time spatial correlations. Considering the phase in which the swimmers are homogeneously and isotropically distributed in the fluid, it is shown that the active temperature and velocity correlations depend on a single scalar correlation function of the dipole-dipole correlation function. By making a simple medium range oder model, in which the dipole-dipole correlation function is characterized by a single correlation length $k_0^{-1}$ it is possible to make quantitative predictions. It is found that the active temperature depends on the system size, scaling as $L^{4-d}$ at large correlation lengths $L\ll k_0^{-1}$, while in the opposite limit it saturates in three dimensions and diverges logarithmically with the system size in two dimensions. In three dimensions he velocity correlations decay as $1/r$ for small correlation lengths, while at large correlation lengths the transverse correlation function becomes negative at maximum separation $r\sim L/2$, effect that disappears as the system increases in size.