We describe numerical simulations of the critical dynamics near the superfluid phase transition. The calculations are based on an implementation of a stochastic hydrodynamic theory known as model $F$ in the classification of Hohenberg and Halperin. This theory is expected to describe dynamic scaling near the lambda transition in liquid ${}^4\mathrm{He}$, Bose-Einstein condensation in ultracold atomic gases, and the superfluid transition in the unitary Fermi gas. Our simulation is based on a Metropolis algorithm previously applied to the critical endpoint of the liquid-gas phase transition in ordinary fluids. In the model $E$ truncation of model $F$ we obtain the expected dynamical exponent $z\simeq 3/2$. We observe the emergence of a propagating second sound mode at the phase transition. The second sound diffusivity $D_s$ is consistent with the scaling relation $D_s\sim {\xi }^{x_{\kappa }}$, where $\xi$ is the correlation length and $x_{\kappa }=1/2$.

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• Dynamic critical phenomena
• Fermi gases