Critical dynamics in systems controlled by fractional kinetic equations

Abstract

The field theory renormalization group is used for analyzing the fractional Langevin equation with the order of the temporal derivative 0<α<1, fractional Laplacian of the order σ, and Gaussian noise correlator. The case of non-linearity φm with odd m≥3 is considered. It is proved that the model is multiplicatively renormalizable. Propagators were found in the momentum and coordinate representation, expressed in terms of Fox’s H functions.Existence of the dissipative scaling regime in the framework of the ε expansion for σ=2, α=1/l,l=1,2,… is proved. Requirement of the continuous dependence of the critical exponents on α imposes the condition m=3.The main quantitative result is the calculation of the dynamical critical exponent z for α=1/2 up to ε2. We have obtained for it the expression z(1/2)=4+0.1555ε2+O(ε3).