  Open quantum systems, when driven by a periodic field, can relax to effective
statistical ensembles that resemble their equilibrium counterparts. We consider
a class of problems in which a periodically- driven quantum system is allowed
to exchange both energy and particles with a thermal reservoir. We demonstrate
that, even for noninteracting systems, effective equilibration to the grand
canonical ensemble requires both fine tuning the system-bath coupling and
selecting a sufficiently simple driving protocol. We study a tractable subclass
of these problems in which the long-time steady state of the system can be
determined analytically, and demonstrate that the system effectively
thermalizes with fine tuning, but does not thermalize for general values of the
system-bath couplings. When the driven system does not thermalize, it supports
a tunable persistent current in the steady state without external bias. We
compute this current analytically for two examples of interest: 1) a driven
double quantum dot, where the current is interpreted as a DC electrical
current, and 2) driven Dirac fermions in graphene, where it is interpreted as a
valley current.
