Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman–Kac Approach

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Abstract

We derive, through subordination techniques, a generalized Feynman–Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman–Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman–Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.
Original languageEnglish
Article number92
JournalJournal of Statistical Physics
Volume190
Issue number5
DOIs
Publication statusPublished - 27 Apr 2023

Bibliographical note

Funding Information:
The authors thank Eli Barkai for useful discussions. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol—http://www.bristol.ac.uk/acrc/.

Funding Information:
TK and LG acknowledge funding from, respectively, an Engineering and Physical Sciences Research Council (EPSRC) DTP student grant and the Biotechnology and Biological Sciences Research Council (BBSRC) Grant No. BB/T012196/1.

Publisher Copyright:
© 2023, The Author(s).

Structured keywords

  • Engineering Mathematics Research Group

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