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 [16],
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 [17] and measured [18] 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 [19].
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.
