  It was recently suggested that the sign of particle drift in inhomogeneous
temperature or turbulence depends on the particle inertia: weakly inertial
particles localize near minima of temperature or turbulence intensity (effects
known as thermophoresis and turbophoresis), while strongly inertial particles
fly away from minima in an unbounded space. The problem of a particle near
minima of turbulence intensity is related to that of two particles in a random
flow, so that the localization-delocalization transition in the former
corresponds to the path-coalescence transition in the latter. The transition is
signaled by the sign change of the Lyapunov exponent that characterizes the
mean rate of particle approach to the minimum (which could be wall or another
particle). Here we solve analytically this problem for inelastic collisions and
derive the phase diagram for the transition in the inertia-inelasticity plane.
An important feature of the phase diagram is the region of inelastic collapse:
if the restitution coefficient of particle velocity is smaller than some
critical value, then the particle is localized for any inertia. We present
direct numerical simulations which support the theory and in addition reveal
the dependence of the transition of the flow correlation time, characterized by
the Stokes number.
