First, I will briefly review simulations of dynamical critical phenomena used to study the O(4) critical point in QCD. The numerical approach is based on a symplectic step which preserves the phase space, and a dissipative step based on the metropolis algorithm. This is a prototype for all dissipative stochastic systems based on an action and entropy principle.
Then I discuss how the approach can be generalized to relativistic fluid dynamics, using the advection-diffusion equation as a prototype. I show that the dissipative dynamics of the boosted fluctuating fluid can be simulated by making random transfers of charge between fluid cells, interspersed with ideal hydrodynamic timesteps. The random charge transfers are accepted or rejected in a Metropolis step using the entropy as a statistical weight. This procedure reproduces the expected strains of dissipative relativistic hydrodynamics in a specific (non-covariant) hydrodynamic frame known as the density-frame. The equations, while not covariant, are invariant under Lorentz transformations followed by a reparametrization of the hydrodynamic fields consistent with the derivative expansion. Numerical results, both with and without noise, are presented and compared to relativistic kinetics and to analytical expectations. An all order resummation of the density-frame gradient expansion reproduces the covariant dynamics in a specific model. Then I indicate how the approach can be generalized to relativistic stochastic hydrodynamics in a general coordinate system. A virtue of the approach is that there are no non-hydrodynamic modes, or additional non-hydrodynamic variables such as $\pi_{\mu\nu}$.
