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Closed quantum systems far from thermal equilibrium can show universaldynamics near attractor solutions, known as non-thermal fixed points,generically in the form of scaling behavior in space and time. A systematicclassification and comprehensive understanding of such scaling solutions aretasks of future developments in non-equilibrium quantum many-body theory. Inthis tutorial review, we outline several analytical approaches to non-thermalfixed points and summarize corresponding numerical and experimental results.The analytic methods include a non-perturbative kinetic theory derived withinthe two-particle irreducible effective-action formalism, as well as alow-energy effective field theory framework. As one of the driving forces ofthis research field are numerical simulations, we summarize the main results ofexemplary cases of universal dynamics in ultracold Bose gases. This encompassesquantum vortex ensembles in turbulent superfluids as well as recently observedreal-time instanton solutions in one-dimensional spinor condensates.

Universal dynamics and non-thermal fixed points in quantum fluids far from equilibrium