Critical fluid dynamics in two and three dimensions

We describe a numerical method for simulating stochastic fluid dynamics near a critical point in the Ising universality class. This theory is known as model H, and is expected to govern the nonequilibrium dynamics of quantum chromodynamics (QCD) near a possible critical endpoint of the phase transition between a hadron liquid and the quark-gluon plasma. The numerical algorithm is based on a Metropolis scheme, and automatically ensures that the distribution function of the hydrodynamic variables in equilibrium is independent of the transport coefficients and only governed by the microscopic free energy. We verify dynamic scaling near the critical point of a two and three-dimensional fluid and extract the associated critical exponent $z$. We find $z\ensuremath{\simeq}3$ in three dimensions, and $z\ensuremath{\simeq}2$ for a two-dimensional fluid. In a finite system, we observe a crossover between the mean field value $z=4$ and the true critical exponent $z\ensuremath{\simeq}3$ ($z\ensuremath{\simeq}2$ in $d=2$). This crossover is governed by the values of the correlation length and the renormalized shear viscosity.