Critical Behavior of a Heavy Particle in a Granular Fluid

Behavior analogous to a second order phase transition is observed for the homogeneous cooling state of a heavy impurity particle in a granular fluid. The order parameter $\phi$ is the ratio of impurity mean square velocity to that of the fluid, with a conjugate field $h$ proportional to the mass ratio. A parameter $\beta$, measuring the fluid cooling rate relative to the impurity--fluid collision rate, is the analogue of the inverse temperature. For $\beta <1$ the fluid is ``normal'' with $\phi =0$ at $h=0$, as in the case of a system with elastic collisions. For $\beta >1$ an ``ordered'' state with $\phi \neq 0$ occurs at $h=0$, representing an extreme breakdown of equipartition. Critical slowing and qualitative changes in the velocity distribution function for the impurity particle near the transition are noted