On the Motion of a Charged Colloid with a Harmonic Trap

In this study, we derive the Fokker–Planck equation for a colloidal particle subject to a harmonic trap and viscous forces under the influence of a magnetic field. We then extend the analysis to a charged colloid driven by both thermal and active noises in the same magnetic environment. Finally, the case of a charged colloid experiencing a harmonic trap together with thermal and active noises is investigated. Analytical solutions for the joint probability density are obtained in the limits of t « τ,  t » τ, and τ = 0. For a colloid under a harmonic trap and magnetic field, the mean squared displacement exhibits a superdiffusive scaling proportional to t³ in the short-time regime (t « τ), while the mean squared velocity scales as ~t when τ = 0. For a charged colloid with thermal noise, the mean-squared displacement follows a superdiffusive form ~t^2h+1  for  t « τ, and the mean squared velocity again scales linearly with time for τ = 0. When the active noise is included together with a harmonic trap, the characteristic time scale grows as ~t⁴ in the short-time regime, while the mean squared velocity becomes normally diffusive at τ = 0. In the long-time limit (t » τ) and for τ = 0, the moments of the joint probability density under combined thermal and active noises scale as ~t^4h+2, consistent with our analytical results. Notably, as h→1/2, the entropy of the joint probability density with thermal noise ζ_th(t) coincides with that obtained for active noise ζ_ac(t) in both t » τ and τ = 0 limits.