Dynamic critical phenomena in disordered systems with finite geometry

We study the critical dynamics of hyper-cubic finite size system in the presence of quenched short-range correlated disorder. By using the random $T_c$ model A for the critical dynamics and the renormalization group method in the vicinity of the upper critical dimension $d=4$, we derive in first order of $ϵ$ the expression for the relaxation time. Its finite-size scaling behavior is discussed both analytically and numerically in details. This was made possible by analyzing carefully the finite–size effects on the Onsager kinetic coefficient. The obtained results are compared to those reported in the literature.