(a) Schematic diagram illustrating the motion of a single particle on 1D lattice frame. The motion starts at lattice site q0 of side length m where the particle moves either to the right or to the left with step size l=m. The probability of occurrence of visits to new lattice sites at time t (Pt) decreases as t increases. (b) Probability distribution (p) of finding a particle at different lattice sites after n steps (n=t/τ) calculated using 1D random diffusion theory (equation (2)). The s.d.‘s increase (green dashed lines), whereas the peak heights decrease (black dashed lines) with the square root of n (equation (2)) (c) Average probability of occurrence of visits to new lattice sites at time t (<Pt2D>) obtained from 100 simulated 2D random diffusion trajectories. The lattice size (m) was set to 160 nm. The step sizes of the trajectories were generated using equation (5) (r=160 nm). The red line shows the fitting to equation (4). The scaling exponent (β) obtained by the fitting is 0.133 for the random walk.