  We consider a collection of active agents and present an effective
single-particle theory to describe the dynamical behaviour of the system in the
steady state. Unlike their equilibrium counterparts, active particles display
phase separation at much lower packing densities, marked by continuous
formation and dissolution of clusters. To model such behavior, we replace the
effect of many-particle interactions by a time-dependent local density. Further
we propose a unique auto-correlation of the local density with a characteristic
correlation time. The correlation time monotonically increases with packing
density and unveils a transition at a critical density. In experimental setups,
this correlation time is easily quantifiable by monitoring changes in density
around a tagged particle. By utilizing this local density and its correlation,
we calculate the effective diffusivity of the particles. Finally, we compare
the results of the effective single particle theory with numerical simulations
and find good agreement between them across a range of packing densities.
