Numerical simulations of non-relativistic stochastic fluids via the Metropolis algorithm

TL;DR

This work presents a Metropolis-based numerical algorithm for simulating non-relativistic stochastic fluids in two dimensions, integrating fluctuations and dissipation by replacing dissipative fluxes with stochastic noise and an accept-reject step. The method splits time evolution into ideal advection and stochastic-dissipative updates, using the Kurganov-Tadmor scheme for the ideal part and a Metropolis-update of stochastic fluxes to maintain correct equilibrium behavior while mitigating multiplicative-noise issues. The authors validate the approach through macroscopic equilibration tests and microscopic measurements of fluxes, and they extract renormalized transport coefficients from shear-shear correlators, illustrating the method’s ability to capture fluctuation-induced modifications to transport properties. The framework is adaptable to other hydrodynamic theories and dimensions, with potential extensions to relativistic hydrodynamics and critical dynamics, providing a robust tool for investigating stochastic fluids in a computationally stable manner.

Abstract

Stochastic hydrodynamics provides a dynamical framework for the evolution of fluctuations in heavy-ion collisions, but poses significant challenges in numerical simulations. We present an algorithm for the simulation of non-relativistic stochastic fluids in two spatial dimensions in a box. We use the robust Metropolis algorithm, handling fluctuations and dissipation at once by systematically replacing dissipative terms in the hydrodynamic equations by random forces. The algorithm can easily be modified for numerical simulations of other hydrodynamic theories. We present test cases as well as numerical calculations of the renormalization of shear viscosity.

Figures (4)

• Figure 1: Mesh refinement via zero-padding Fourier interpolation in real (top) and Fourier space (bottom). The finer lattice has the same volume, such that its Fourier spectrum has additional modes only in the padding around the original spectrum. The refinement is effectively a trigonometric interpolation, copying the spectrum and setting modes in the padding to zero. After the hydrodynamic evolution step, there can be small occupations of the fast modes in the padding. These are removed in the coarsening step, such that the coarse configuration is given by the slow envelope of the fine configuration.
• Figure 2: Left: Fluxes transporting the conserved quantities in and out of cell $\alpha$ with faces $q_\alpha\in\{l,r,t,b\}$. Right: Possible orientations of L-update.
• Figure 3: Averaged Metropolis fluxes recovering viscous hydrodynamics to first order in the shear channel.
• Figure 4: Left: Numerical shear-shear correlators $F(\Delta t, k)$ for $L=80$, $\Delta x=1$, $L_\eta=6$ and $L_\kappa=L_\zeta=0$ in grand-canonical ensemble with $\beta=0.4$, $\mu=10.0$. Middle: Parabolic fit to decay constants $\Gamma_k$ extracted from fits in left panel. Right: Renormalization of shear viscosity. $L_{\eta, \text{ren}}$ was extracted from shear-shear correlators, while $L_\eta$ denotes the microscopic bare value. Since results in the given region agreed for different mesh refinement factors, no mesh refinement was used for these calculations.