From non-Brownian Functionals to a Fractional Schrödinger Equation

We derive backward and forward fractional Schr\"odinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fra...

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Bibliographic Details
Published in:arXiv.org 2009-09
Main Authors: Turgeman, Lior, Carmi, Shai, Barkai, Eli
Format: Article
Language:English
Subjects:
Schrodinger equation
Online Access:Get full text
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Summary:We derive backward and forward fractional Schr\"odinger type of equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [PRL {\bf 96}, 230601 (2006)] provide the correct fractional framework for the problem at hand. In the limit of normal diffusion we recover the Feynman-Kac treatment of Brownian functionals. For applications, we calculate the distribution of occupation times in half space and show how statistics of anomalous functionals is related to weak ergodicity breaking.
ISSN:2331-8422
DOI:10.48550/arxiv.0909.0144