Transport properties of stochastic fluids

TL;DR

The paper investigates heat and momentum transport in stochastic fluids within Model H, focusing on both non-critical and critical regimes in two and three dimensions. It combines Metropolis-based simulations with Kubo relations to extract transport coefficients, revealing a logarithmic infrared divergence of the shear viscosity in 2D non-critical fluids and critical-scale divergences of both shear viscosity and thermal conductivity in 2D and 3D. The study compares Model H with a truncation model H0, finds robust critical scaling for thermal conductivity that generally exceeds the viscosity's, and discusses UV/IR regularization and discretization limitations intrinsic to lattice implementations. These results provide a baseline for understanding critical transport phenomena in fluctuating fluids and have potential implications for expanding systems such as the quark-gluon plasma, while highlighting the need for improved discretizations to fully capture renormalized transport properties.

Abstract

We study heat conduction and momentum transport in the context of stochastic fluid dynamics. We consider a fluid described by model H in the classification of Hohenberg and Halperin. We study both non-critical and critical fluids, and we investigate transport properties in two as well as three dimensions. Our results are based on numerical simulations of model H using a Metropolis algorithm, and we employ Kubo relations to extract transport coefficients. We observe the expected logarithmic divergence of the shear viscosity in a two-dimensional non-critical fluid. At a critical point, we find that the transport coefficients exhibit power-law scaling with the system size $L$. The strongest divergence is seen for the thermal conductivity $κ$ in two dimensions. We find $κ\sim L^{x_κ}$ with $x_κ=1.6\pm 0.1$. The divergence is weaker in three dimensions, $x_κ=1.25 \pm 0.3$, and the scaling exponent for the shear viscosity, $x_η$, is significantly smaller than $x_κ$ in both two and three dimensions.

Figures (7)

• Figure 1: Diagrammatic representation of one-loop corrections to the shear viscosity. Fig. (a) and (b) show the self-advection and order parameter contributions to the retarded self energy of the momentum density. Fig. (c) and (d) show the corresponding contributions to the Kubo integrand. The dashed line represents an insertion of the stress tensor.
• Figure 2: Diagrammatic representation of one-loop corrections to the conductivity. Fig. (a) and (b) show the advection contributions to the retarded self energy of the order parameter. Fig. (c) shows the corresponding contributions to the Kubo integrand. The dashed line represents an insertion of the current.
• Figure 3: Left panel: Correlation function $C_\Pi(t)$ of the stress tensor $T_{xy} = \pi_x \pi_y/\rho$ in a 2d non-critical fluid. The dynamics only contains the self-advection term, and the bare shear viscosity is $\eta_0=1$. The correlation function is shown in several different volumes $V=(La)^2$ with $L=16,24,36,40,48,64$. The gray curves show numerical fits $C_\Pi(t)\sim A \exp(- pt)/t$. Right panel: Scaling of the Kubo integral for viscosity with system size $L$. The orange line shows a fit of the form $\Delta\eta= \alpha \log(L)$ with $\alpha=0.0158$.
• Figure 4: Shear viscosity of a non-critical model H fluid in three dimensions, measured in a volume $V=(La)^3$ with $L=24$. We show the physical shear viscosity as a function of the bare value. The viscosity is extracted using the momentum density correlator (red squares), the decay of a shear wave (green diamonds), and the Kubo formula (blue circles).
• Figure 5: Thermal conductivity in a critical two-dimensional fluid governed by model H0. Left panel: Kubo integrand $C_\jmath(t)$ in different volumes $V=(La)^2$ for $L=16,24,\ldots,64$. Right panel: Kubo integral $\Delta\kappa$ as a function of system size, together with the best fit $\Delta\kappa\sim L^{x_\kappa}$ where $x_\kappa=1.7$.