On single-file and less dense processes

The diffusion process of N hard rods in a 1D interval of length is studied using scaling arguments and an asymptotic analysis of the exact N -particle probability density function (PDF). In the class of such systems, the universal scaling law of the tagged particle's mean absolute displacement reads, ⟨| r |⟩∼⟨| r |⟩ free / n μ , where ⟨| r |⟩ free is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent μ is given by, μ=1/(1+ a ), where a is associated with the particles' density law of the system, ρ∼ρ 0 L - a , 0⩽ a ⩽1. The scaling law for ⟨| r |⟩ leads to, ⟨| r |⟩∼ρ 0 ( a −1)/2 (⟨| r |⟩ free ) (1+ a )/2 , an equation that predicts a smooth interpolation between single-file diffusion and free-particle diffusion depending on the particles' density law, and holds for any underlying dynamics. In particular, for normal diffusion, with a Gaussian PDF in space for any value of a (deduced by a complementary analysis), and, , for anomalous diffusion in which the system's particles all have the same power-law waiting time PDF for individual events, ψ∼t -1-β , 0<β<1. Our analysis shows that the scaling ⟨ r 2 ⟩∼ t 1/2 in a “standard” single file is a direct result of the fixed particles' density condition imposed on the system, a =0.