Fokker-Planck equation for the probability density of a spin-boson system with impurities
We consider the dynamics of a spin-boson system with impurities coupled to a thermal bath. We use the slave-boson method to account for the site vacancies and no-double occupancy constraint of the system. We employ the positive P-representation and coherent state formalism to map the quantum master equation to a Fokker-Planck equation (FPE). In the FPE, the drift vector has six complex expressions for each site, and the diffusion matrix on a site has four expressions for each source of noise. For future work, these results provide the starting equations for analysis of the nonequilibrium dynamics of spin-bosons systems with impurities.