a quantum theory of a two and three-level laser with injected atomic coherence is developed by using a density operator method, to the best of our knowledge, for the first time. the initial atomic coherence plays an essential role. at steady state, the equation of motion for the density operator yields to exhibit laser without inversion and a phase locking but no threshold for the laser field. ...

This article is devoted to investigating the evolution of probability density function for power system excited by fractional stochastic noise. First, single-machine-infinite-bus (SMIB) model Gaussian noise (FGN) established. Second, we derive Fokker–Planck–Kolmogorov (FPK) equation proposed and solve FPK using finite difference method. Finally, numerical results verify that addition FGN would ...

We present a uniform L-stability estimate for mild solutions to the Fokker-Planck-Boltzmann equation. For stability estimate, we derive a Gronwall type estimate using dispersion estimates for mild solutions due to the hypoelliptic structure of the Vlasov-Fokker-Planck operator.