Impurity dynamics in a zero-temperature gas

Abstract If energy is suddenly released in a localized region of space uniformly filled with identical stationary hard spheres, the outcome is a blast with an asymptotically spherical shock wave separating moving and stationary hard spheres. The radius R ( t ) of the region filled with the moving spheres grows as t 2 / ( d + 2 ) , where d is the spatial dimension. The simplest way to inject energy is to kick a few ‘impurity’ particles. Using hydrodynamics and kinetic theory, we argue that the typical displacement of an impurity scales as R imp ∼ λ ( R / λ ) ( 4 + 3 d 2 ) / ( 8 + 3 d 2 ) , where λ is the mean-free path in the initial state. The number of collisions experienced by each impurity grows as ( R / λ ) ( 8 + 2 d 2 ) / ( 8 + 3 d 2 ) , while its average speed decreases as t − d ( 8 − 2 d + 3 d 2 ) / [ ( 2 + d ) ( 8 + 3 d 2 ) ] . In 2 D , the predictions for impurity displacement, collision numbers, and speed are t 2 / 5 , t 2 / 5