  We show that a variety of nonequilibrium dynamics of interacting many-body
systems are universally characterized by an elegant relation, which we call the
dynamic virial theorem. The out-of-equilibrium dynamics of quantum correlations
is entirely governed by Tan\textquoteright s contact. It gives rise to a series
of observable consequences and is closely related to experiments with ultracold
atoms. Especially, we show that the dynamic virial theorem provides an
experimentally accessible verification of maximum energy growth theorem [Qi et
al., Phys. Rev. Lett. 126, 240401 (2021)], which is encoded in the evolution of
the atomic cloud size during expansion. In addition, the dynamic virial theorem
leads to a simple thermodynamic relation of strongly interacting quantum gases
in the framework of two-fluid hydrodynamic theory, which holds in a wide range
of temperature. This thermodynamic relation is a kind of the out-of-equilibrium
analog of Tan's pressure relation at equilibrium. Our results provide
fundamental understanding of generic behaviors of interacting many-body systems
at nonequilibrium, and are readily examined in experiments with ultracold
atoms.
