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Fractional Schroedinger Equations and Biological Computation

Banff International Research Station for Mathematical Innovation and Discovery

In order to justify certain model equations proposed in the biophysics literature for charge transport on polymers like DNA and protein, we consider a general class of discrete nonlinear Schroedinger equations on lattices, and prove that in the continuum limit, the limiting dynamics are given by a nonlinear Schroedinger equation (NLS) with a fractional Laplacian. In particular, a range of fractional powers arise from long-range lattice interactions in this limit, whereas the usual NLS with the non-fractional Laplacian arises from short-range interactions. We also obtain equations of motion for the expected position and momentum, the fractional counterpart of the well-known Newtonian equations of motion for the standard Schroedinger equation, and use a numerical method to suggest that the nonlocal Laplacian introduces decoherence, but that effect can be mitigated by the nonlinearity. Joint work with Gigliola Staffilani, Enno Lenzmann, and Yanzhi Zhang. Time permitting, I will talk about recent work defining biophysical machines that out-perform Turing machines, in joint work with Onyema Osuagwu and Daniel Inafuku.

Mathematics; Partial differential equations; Fourier analysis

video/mp4

Author affiliation: University of Illinois

2020-01-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73040

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/