  Based on the recently formulated chiral radiation transport theory for
left-handed neutrinos, we study the chiral transport of neutrinos near thermal
equilibrium in core-collapse supernovae. We first compute the near-equilibrium
solution of the chiral radiation transport equation under the relaxation time
approximation, where the relaxation time is directly derived from the effective
field theory of the weak interaction. By using such a solution, we
systematically derive analytic expressions for the nonequilibrium corrections
of the neutrino energy-momentum tensor and neutrino number current induced by
magnetic fields via the neutrino absorption on nucleons. In particular, we find
the nonequilibrium neutrino energy current proportional to the magnetic field.
We also discuss its phenomenological consequences such as the possible relation
to pulsar kicks.
