Critical dynamics in systems controlled by fractional kinetic equations

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 φ m 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. nce 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 . in quantitative result is the calculation of the dynamical critical exponent z for α = 1 / 2 up to ε 2 . We have obtained for it the expression z ( 1 / 2 ) = 4 + 0.1555 ε 2 + O ( ε 3 ) .