Random sets of stadiums in square and collective behavior of bacteria

Collective motion of swimmers can be detected by hydrodynamic interactions through the eective (macroscopic) viscosity. It follows form the general hydrodynamics that the eective viscosity of non-dilute random suspensions depends on the shape of particles and of their spacial probabilistic distribution. Therefore, a comparative analysis of disordered and collectively interacting particles of the bacteria shape can be done in terms of the probabilistic geometric parameters which determine the eective viscosity. In this paper, we develop a quantitative criterion to detect the collective behavior of bacteria. This criterion is based on the basic statistic moments (e-sums or generalized Eisenstein-Rayleigh sums) which characterize the high-order correlation functions. The locations and the shape of bacteria are modeled by stadiums randomly embedded in medium without overlapping. This shape models can be considered as improvement of the previous segment model.We calculate the e-sums of the simulated disordered sets and of the observed experimental locations of bacteria subtilis. The obtained results show a dierence between these two sets that demonstrates the collective motion of bacteria.