A Brownian particle's trajectory is parameterized by its self-diffusion coefficient D through the Einstein-Smoluchowsky equation
where d is the number of dimensions of trajectory data. The angle brackets indicate a thermodynamic average over many starting times t for a single particle or over many particles for an ensemble. While eqn. (15) can be used to measure D directly, fitting the histogram of particle displacements to the expected Gaussian distribution
also affords consistency checks. The offset reflects secular drift in the sample of particles perhaps due to flow in the supporting fluid, while is a normalization constant. Information regarding the particle dynamics appears in the width of the distribution . Typical diffusion data for spheres with radius nm appear in Fig. 6(a) together with a least squares fit to eqn. (16).
The long-time self-diffusion coefficient D can be extracted from the time dependence of the distribution function's width through
The additive offset arises in part from rapid short-time diffusion and in part from measurement errors which contribute . Non-linear evolution of can reflect such effects as caging in dense suspensions , non-Newtonian behavior in the suspending fluid, or two-dimensional corrections for geometrically confined suspensions. For the spheres in the example data, the slope of the fit to eqn. (17) shown in Fig. 6(b) indicates a self-diffusion coefficient of m /s.
The quality of the fit to eqn. (16) is a sensitive test of the proper functioning of the image processing software. For instance, the histogram of displacement probabilities shows strong modulation with a wavelength of one pixel if the size w of the convolution kernels used for sub-pixel position refinement is too small or if the image has an uncorrected bright background. A strong peak at zero displacement usually indicates that the software is mistaking motionless image defects (such as dust on the optics) for actual particles. Outliers and shoulders on the histogram usually signify unreliable particle identifications and could be warnings of poor image quality or an inappropriate choice of system parameters.
In addition to providing values of D, diffusion measurements on tracer particles can be used to measure suspension properties at very small length scales such as the local viscosity of the suspending medium. The self diffusion coefficient for isolated Brownian spheres is given by the Stokes-Einstein equation,
where is the viscosity of the suspending fluid and is the sphere's radius. For the spheres in the example data, m /s at the experimental temperature C, in good agreement with the measured value.
A variety of hydrodynamic effects tend to reduce the the observed diffusion coefficient below the Stokes-Einstein value. Hydrodynamic coupling between the sphere and a flat wall (such as the microscope cover slip) a distance h away is both predicted  and measured  to reduce the lateral diffusivity to approximately
for . Similarly, coupling between a highly charged particle and its surrounding counterions can reduce the particle's diffusivity by as much as 10 percent when the particle's dimensions are comparable to the Debye-Hückel screening length . The double-layer effect was minimized in the example data in Fig. 6 by ensuring that the screening length was much shorter than the sphere radius.
Extracting local-scale information is greatly simplified if extraneous coupling to the system's walls and neighboring particles can be minimized by restricting observations to dilute suspensions far from walls. Measurements at high dilution can become prohibitively time consuming, however, since particles readily diffuse out of the observation volume and away from experimentally desirable configurations. Optical trapping provides a means to reproduce useful arrangements of particles and thereby to take maximum advantage of the accuracy and resolution offered by digital video microscopy.