  This work explores a fundamental dynamical structure for a wide range of
many-body quantum systems under periodic driving. Generically, in the
thermodynamic limit, such systems are known to heat up to infinite temperature
states after infinite-time evolution, irrespective of dynamical details. In the
present study, instead of considering infinitely long-time scale, we aim to
provide a framework to understand the long but finite time behavior, namely the
transient dynamics. In the analysis, we focus on the Floquet-Magnus (FM)
expansion that gives a formal expression of the effective Hamiltonian on the
system. Although in general the full series expansion is not convergent in the
thermodynamics limit, we give a clear relationship between the FM expansion and
the transient dynamics. More precisely, we rigorously show that a truncated
version of the FM expansion accurately describes the exact dynamics for a
finite-time scale. Our result reveals a reliable time scale of the validity of
the FM expansion, which can be comparable to the experimental time scale.
Furthermore, we discuss several dynamical phenomena, such as the effect of
small integrability breaking, efficient numerical simulation of periodically
driven systems, dynamical localization and thermalization. Especially on
thermalization, we discuss generic scenario of the prethermalization phenomenon
in periodically driven systems.
