Description of high energy collision cascades within the multifractal theory framework

The random nature of both the energy lost during a collision event and the length traveled by a particle between two successive collisions leads to introduce probability density functions (pdf) to define their evolution in a displacement cascade. Using the Hard Sphere approximation to describe the collision cross section as a toy model to handle the physics of energy dissipation in a displacement cascade, analytical expressions for these pdf can be calculated within the BCA. As these functions exhibit long tails, a description of a collision cascade by a deterministic fractal is not able to capture physical key parameters associated with the energy dissipation. The multifractal formalism is then the only natural framework to define the fractal feature of a displacement cascade. Applying such a formalism, the bifractal behavior of displacement cascade has been clearly pointed out. This feature is clearly due to the long tails of pdf describing collision events.