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Experimental methods based on single particle tracking (SPT) are beingincreasingly employed in the physical and biological sciences, where nanoscaleobjects are visualized with high temporal and spatial resolution. SPT can probeinteractions between a particle and its environment but the price to be paid isthe absence of ensemble averaging and a consequent lack of statistics. Here weaddress the benchmark question of how to accurately extract the diffusionconstant of one single Brownian trajectory. We analyze a class of estimatorsbased on weighted functionals of the square displacement. For a certain choiceof the weight function these functionals provide the true ensemble averageddiffusion coefficient, with a precision that increases with the trajectoryresolution.

Optimal fits of diffusion constants from single time data points of Brownian trajectories