On the Stochastic Motion Induced by Magnetic Fields in Random Environments

Here, we study the Navier-Stokes equation for the motion of a passive particle based on the Fokker-Planck equation in an incompressible conducting fluid induced by a magnetic field subject to an exponentially correlated Gaussian force in three-time domains. For the hydro-magnetic case of velocity and the time-dependent magnetic field, the mean squared velocity for the joint probability density of velocity and the magnetic field has a super-diffusive form that scales as ∼t3 in t>>τ, while the mean squared displacement for the joint probability density of velocity and the magnetic field reduces to time ∼t4 in t<<τ. The motion of a passive particle for τ=0 and t>>τ behaves as a normal diffusion with the mean squared magnetic field being <h2(t)>∼t. In a short-time domain t<<τ, the moment in the magnetic field of the incompressible conducting fluid undergoes super-diffusion with μ2,0,2h∼t6, in agreement with our research outcome. Particularly, the combined entropy H(v,h,t) (H(h,v,t)) for an active particle with the perturbative force has a minimum value of ∼lnt2 (∼lnt2) in t>>τ (τ=0), while the largest displacement entropy value is proportional to lnt4 in t<<τ and τ=0.