The role of multiplicative noise in critical dynamics

Abstract

We study the role of multiplicative stochastic processes in the description of the dynamics of an order parameter near a critical point. We study equilibrium as well as out-of-equilibrium properties. By means of a functional formalism, we build the Dynamical Renormalization Group equations for a real scalar order parameter with Z2 symmetry, driven by a class of multiplicative stochastic processes with the same symmetry. We compute the flux diagram using a controlled ϵ-expansion, up to order ϵ2. We find that, for dimensions d=4−ϵ, the additive dynamic fixed point is unstable. The flux runs to a multiplicative fixed point driven by a diffusion function G(ϕ)=1+g∗ϕ2(x)/2, where ϕ is the order parameter and g∗=ϵ2/18 is the fixed point value of the multiplicative noise coupling constant. We show that, even though the position of the fixed point depends on the stochastic prescription, the critical exponents do not. Therefore, different dynamics driven by different stochastic prescriptions (such as Itô, Stratonovich, anti-Itô and so on) are in the same universality class.