  High-mobility 2D electron systems in a perpendicular magnetic field exhibit
zero resistance states (ZRS) when driven with microwave radiation. We study the
nonequilibrium phase transition into this ZRS using phenomenological equations
of motion to describe the current and density fluctuations. We focus on two
models for the transition into a time-independent steady state. Model-I assumes
rotational invariance, density conservation, and symmetry under shifting the
density globally by a constant. This model is argued to describe physics on
small length scales where the density does not vary appreciably from its mean.
The ordered state that arises in this case breaks rotational invariance and
consists of a uniform current and transverse Hall field. We discuss some
properties of this state, such as stability to fluctuations and the appearance
of a Goldstone mode associated with the continuous symmetry breaking. Using
dynamical renormalization group techniques, we find that with short-range
interactions this model can admit a continuous transition described by
mean-field theory, whereas with long-range interactions the transition is
driven first-order. Model-II, which assumes only rotational invariance and
density conservation and is argued to be appropriate on longer length scales,
is shown to predict a first-order transition with either short- or long-range
interactions. We discuss implications for experiments, including scaling
relations and a possible way to detect the Goldstone mode in the case of a
continuous transition into the ZRS, as well as possible signatures of a
first-order transition in larger samples. We also point out the connection of
our work to the well-studied phenomenon of `flocking'.
