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 ∼t2h+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 ∼t4h+2, consistent with our analytical results. Notably, as h→1/2, the entropy of the joint probability density with thermal noise ζth(t) coincides with that obtained for active noise ζac(t) in both t≫τ and τ=0 limits.