  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.

Soft Condensed Matter;Statistical Mechanics