Langevin description of gauged scalar fields in a thermal bath

We study the dynamics of the oscillating gauged scalar field in a thermal bath. A Langevin type equation of motion of the scalar field, which contains both dissipation and fluctuation terms, is derived by using the real-time finite temperature effective action approach. The existence of the quantum fluctuation-dissipation relation between the non-local dissipation term and the Gaussian stochastic noise terms is verified. We find the noise variables are anti-correlated at equal-time. The dissipation rate for the each mode is also studied, which turns out to depend on the wavenumber.