Biased Random Process of Randomly Moving Particles with Fixed Mean and Curl

In this study, we investigate the behavior of randomly moving particles with a defined average energy, focusing on their biased motion around a specific point. These particles' velocity magnitudes follow a Maxwell distribution with a specific scale parameter. Our analysis derives the stochastic rotation rules for these particles and introduces the Ito equation conditions for biased stochastic rotation at a given curl value. We also consider the special relativistic-like effects in this context. The findings are validated through a representative example.