Figure 20.6 — Anomalous diffusion in a random potential Figure 20.6 — Anomalous diffusion in a random potential. (a) Theoretical [Eq. (20.14)] and simulated normalised autocorrelation function of the force field produced by the optical forces associated with a speckle pattern. The dashed line represents the theoretical normalised autocorrelation function of the speckle pattern intensity. (b) Subdiffusive mean squared displacements (MSD) for a Brownian particle moving in a static speckle as a function of the average force ⟨F⟩ and its deviation from Einstein’s free diffusion law (black line). (c) Superdiffusive MSD for a Brownian particle moving in a speckle pattern that varies on a time scale ξ ; τs [Eq. (20.15)] is the characteristic time scale of the motion of the particle due to the optical forces associated with the speckle pattern. The dots represent Einstein’s free diffusion law. (d) The effective diffusion of the motion at long time scales as a function of ξ /τs shows a transition from sub diffusion (Deff < D) to superdiffusion (Deft > D). The maximum value of the superdiffusion appears for ξ ≈ τs (τ ≈ 22.5 ms for ⟨F⟩ = 50 fN, τ ≈ 11.2 ms for ⟨F⟩ = 100 fN and τ ≈ 5.6 ms for ⟨F⟩ = 200 fN). Every mean point is averaged over 500 particle trajectories 100 s long, whose initial position is randomly chosen within the speckle field. The grey shaded areas represent one standard deviation around the average values. In all cases, the particle is a 250 nm radius polystyrene (np = 1.59) sphere immersed in water at 300 K.Adapted from Volpe et al., Sci. Rep., 4, 3936 (2014).