The role of multiplicative noise in critical dynamics

TL;DR

This paper investigates how multiplicative noise influences the critical dynamics of an order parameter, revealing a multiplicative fixed point and universality of critical exponents across stochastic prescriptions.

Contribution

It introduces a dynamical renormalization group analysis for multiplicative noise near critical points, identifying a new multiplicative fixed point and demonstrating universality of critical exponents.

Findings

Additive fixed point is unstable in 4-ε dimensions.

A multiplicative fixed point driven by a specific diffusion function is identified.

Critical exponents are independent of stochastic prescription.

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 built the Dynamical Renormalization Group equations for a real scalar order parameter with $Z_2$ symmetry, driven by a class o multiplicative stochastic processes with the same symmetry. We have computed the flux diagram, using a controlled $\epsilon$-expansion, up to order $\epsilon^2$. We have found that, for dimensions $d=4-\epsilon$, the additive dynamic fixed point is unstable. The flux runs to a {\em multiplicative fixed point} driven by a diffusion function $G(\phi)=1+g^*\phi^2({\bf x})/2$, where $\phi$ is the order parameter and $g^*=\epsilon^2/18$ 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 It\^o, Stratonovich, anti-It\^o and so on) are in the same universality class.