In P ( x , t ∣ ∣ x 0 ) , x = { x − − M , x − − M + 1 , … … , x M } is the set of particles' positions at time t and x 0 is the set of the particles' initial positions at the initial time t 0 (set to zero). Equation (1) is solved with the appropriate boundary conditions, which reflect the hard-sphere nature of the file: