Multiplicative noise and the diffusion of conserved densities

Jingyi Chao,Thomas Schäfer

doi:10.1007/jhep01(2021)071

Abstract

Stochastic fluid dynamics governs the long time tails of hydrodynamic correlation functions, and the critical slowing down of relaxation phenomena in the vicinity of a critical point in the phase diagram. In this work we study the role of multiplicative noise in stochastic fluid dynamics. Multiplicative noise arises from the dependence of transport coefficients, such as the diffusion constants for charge and momentum, on fluctuating hydrodynamic variables. We study long time tails and relaxation in the diffusion of a conserved density (model B), and a conserved density coupled to the transverse momentum density (model H). Careful attention is paid to fluctuation-dissipation relations. We observe that multiplicative noise contributes at the same order as non-linear interactions in model B, but is a higher order correction to the relaxation of a scalar density and the tail of the stress tensor correlation function in model H.

Highlights

We study long time tails and relaxation in the diffusion of a conserved density, and a conserved density coupled to the transverse momentum density
In this work we studied the role of multiplicative noise in the theory of a conserved density coupled to the transverse momentum density of a fluid
It fits into the standard long time, large wavelength, expansion of hydrodynamic correlation functions

Summary

Diffusion

We study the diffusion of a conserved density ψ(x, t). The diffusion equation is given by δF [ψ]. Where L is a noise kernel that we will specify below Correlation functions of this theory are computed from solutions of the diffusion equation, averaged over the noise distribution in eq (2.4). Showed how to write this noise average in terms of a stochastic field theory [25,26,27]. Time reversal invariance can be used to derive fluctuation-dissipation relations For this purpose we define the response function as the derivative of ψ(t) h with respect to the external field in eq (2.3). In momentum space eq (2.14) is equivalent to This is the standard FD relation in the case κ(ψ) = κ0, but for a field dependent diffusion constant the left hand side of eq (2.15) includes the vertex function of the composite operator [κ(ψ)ψ]

Response and correlation functions

Shear modes and model H

Advection of the scalar density

Renormalization of the shear viscosity

Conclusions and outlook

A Fluctuation-dissipation relation