
On Using a Lie Group to Represent Increments of Axial Motion in Suspension Flows
Eugene Eckstein
Department of Biomedical Engineering, University of Memphis, TN 38152 October 8, 2008, 4:00pm, Manning Hall 201 Refreshments served at 3:30pm, Manning Hall 222
The equation relating increments of displacement over time for the OrnsteinUhlenbeck
process (physicistâ€™s Brownian motion) and persistent random walk was found to fit
increments of axial position as observed in a moving reference frame when the sign
of the time increment was reversed. The equation then has an exponential growth term
and all studies must be done with a targeted distance increment or time of observation.
The main topic is a means of describing this type of data by assuming that the constants
of integration for motion as observed in four dimensions (position, axial momentum,
balance of kinetic energy and internal energy, and free energy per mass). These constants
are to be selected from the Lie group for rotations, SO(3), which can be considered
as quaternions showing the change of position on the unit 3sphere. Gilmore (http://www.physics.drexel.edu/~bob/LieGroups.html) treats related ideas in chapter 16 of his book, many chapters of which are available
at the web site.
Primarily, he notes that symmetries in the Lie group provide a means of bringing physical
principles found by Galileo and Einstein to the organization of such data by means
of a differential equation. L.F. Richardson, an editor for G.I. Taylorâ€™s 1921 paper
on persistent random walk (Proc. London Math. Soc. Ser. 2, 20, 196), noted that these
walks described the separation of two particles.
The unit threesphere, which surrounds the origin for the moving reference frame,
would seem to provide a sound means to understand this remark in respect to the objects
in suspension flows. The vector from the origin to any path on the manifold pierces
the three sphere and so, the intersection point provides an implicit record of events.
These ideas appear to have direct application to far more than suspension flows.
Student Collaborators: J. Lavine, M. Leggas, B. Ma, V. Bhal Faculty Collaborators: J. Goldstein, J. Marchetta, M. Kiani, D. Goetz
