Stochastic relativistic viscous hydrodynamics from the Metropolis algorithm

We propose an algorithm for simulating stochastic relativistic fluid dynamics based on Metropolis updates. Each step of the algorithm begins with an update based on ideal hydrodynamics. This is followed by proposing random (spatial) momentum transfers between fluid cells, keeping the total energy fixed. These proposals are then accepted or rejected using the change in entropy as a statistical weight. The algorithm reproduces relativistic viscous hydrodynamics in the ``Density Frame", which is a formulation of viscous hydrodynamics we review and clarify here. This formulation is first order in time and requires no auxiliary dynamical fields such as $\Pi^{\mu\nu}$. The only parameters are the shear and bulk viscosities and the equation of state. By adopting the 3+1 split of general relativity, we extend the Metropolis algorithm to general space-time coordinates, such as Bjorken coordinates, which are commonly used to simulate heavy-ion collisions.