Dislocations in a vortex lattice and complexity of chlamydomonas ciliary beating

Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)




Kenneth W. Foster


Chlamydomonas, Vortex lattice, Ciliary beating, Phototaxis, Superconductors

Subject Categories

Biological and Chemical Physics | Physical Sciences and Mathematics | Physics


For the first topic the moving dislocations interrupt an orchestrating transport of vortices, leading to the different velocities of vortices at the different parts of a vortex lattice. Since the correlation of displacement grows algebraically in two dimensions rather than logarithmically in three dimensions, we emphasize the movement of edge dislocations on a single copper oxide plane. Effect of moving dislocations is particularly examined in connection to the velocity-force characteristics of vortices. Under the neutrality condition, the density of Burgers vectors of dislocations emerges in the equations of motion of vortices as a source term. Time evolution of the density of Burgers vectors is governed by a Fokker-Planck equation in which the drift and diffusion coefficients describe the interaction of dislocations and the thermal fluctuation, respectively. To find the Green's function of Fokker-Planck equation a perturbation series in the orders of drift coefficient which generally possesses the spatiotemporal dependence is constructed, analogous to the Born series of the time-dependent Schr¨odinger equation. In contrast, the drift coefficient shows up only with the even orders and the sign in a series alternates. Dislocations slow the velocity of vortices below their linear flux flow velocity, like the pinning. Free dislocations are more efficient to slow the velocity of vortices than interacting dislocations.

For the second topic the adaptation of Chlamydomonas ciliary beating to light stimulation during its phototaxis is studied by adopting a notion of memory believed to account for the slower responses. The influence of the past ciliary beating on the present one is expressed in terms of memory time estimated by a saturating point of Lipschitz number. Mutant cells seem to possess a memory time longer than wild type cells. Under a dark environment the ciliary beating shows strong time variability suitable for a temporal self-similarity study. The scaling exponent estimated by a detrended fluctuation analysis falls into two regimes, possibly signifying a behavioral transition, at least in the statistical sense, of ciliary beating. Using the time-delay coordinates of ciliary beating, the phase space of unknown differential equations underlying the phototaxis shows an 8-like shape.


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