Analytical Solutions for a Charged Particle with White, Thermal, and Active Noises in the Presence of a Uniform Magnetic Field

In this paper, we apply the double Fourier transform method to the two-dimensional Vlasov equations for a charged particle subjected to white noise, exponentially correlated Gaussian forces, trap forces and thermal and active noises in a magnetic field. By deriving the corresponding Fokker–Planck equation, analytical solutions for the joint probability density are obtained in different time domains. The mean squared displacement and velocity of a charged particle driven by white noise exhibits a super-diffusive behavior, scaling as ~t2 in the short-time regime, while it grows linearly with time (~t) in the long-time regime, in agreement with numerical simulations of the mean squared displacement. When thermal noise is included together with harmonic trap and viscous forces, the characteristic time scale increases as ~t2h+1 in the corresponding time domains, whereas the mean squared velocity scales as ~t2h+3. The moments of the joint probability density under thermal noise scale as ~t2h+5. Furthermore, when the persistent Hurst exponent h→1/2, the entropy of the joint probability density associated with thermal noise coincides with that obtained for active noise in both the short-time (t≪τ) and long-time (t≫τ) limits.