Abstract
Developed primarily for motion on discrete lattices representative of molecular crystals described in Chap. 11, the defect technique was extended to the continuum in this chapter and analyzed in novel ways for simple 1-dimensional systems, the motion without defects being described by the diffusion equation. The first focus was on taking the continuum limit of discrete lattice results. The second was to obtain and catalog 1-dimensional results for perfect and imperfect absorption. The third was to do the same for higher dimensional systems with high symmetry, explicit results being obtained in each of these cases in terms of known special functions such as the error function and Bessel functions. The basic equation was next generalized to one in which, before the introduction of the defect, the moving particles have a tendency to be attracted to a center, necessitating, therefore, the replacement of the diffusion equation by the Smoluchowski equation. Novel results were displayed in this scenario including an effect involving non-monotonicity relative to the strength of the attraction. Finally, a brief discussion was given of how to build a theory of the coalescence of signaling receptor clusters in immune cells by reversing the defect technique.
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Notes
- 1.
Several of these results were derived by Kathrin Spendier in collaboration with me as part of her Ph.D. dissertation work (Spendier 2012). Others were compiled by her from sources in the literature and cited appropriately as noted. She was an exceptional student equally at ease with theoretical research and experimental work.
- 2.
For the rare reader who decides to compare expressions to those in the original publication, here is a note so confusion can be avoided. In this book, ξ ′ refers to what one gets by evaluating ξ at 𝜖 = 1∕τ H. This is opposite to the usage in the original publication and has been done here for convenience.
- 3.
The diffusion length derives its name from the fact that, ignoring factors of order of unity, it is the distance the exciton would cover in the host in its radiative lifetime, while moving with the diffusion constant D.
- 4.
- 5.
As a keynote lecture on “Theoretical framework for the description of signal receptor cluster aggregation in cells” at the Fourteenth International Membrane Research Forum, Kyoto, Japan, on March 16, 2013.
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Kenkre, V.M.(. (2021). The Defect Technique in the Continuum. In: Memory Functions, Projection Operators, and the Defect Technique. Lecture Notes in Physics, vol 982. Springer, Cham. https://doi.org/10.1007/978-3-030-68667-3_12
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