Modeling diffusion in restricted systems using the heat kernel expansion.
The averaged return-to-origin probability of finding a diffusing particle within a volume or in the neighborhood of the surface of a bounded region can be separated into a volume and a surface integral of the corresponding probability densities. However with the usual treatments (e.g., the commonly encountered diffusion propagator approach) there is no clear method to separate the integration of the diffusion propagators in each domain. Here we propose a general procedure based on applying the heat kernel expansion in restricted diffusion problems for the Green's function of the diffusion equation on an arbitrary region with an arbitrary boundary condition. We apply this method to the treatment of surface reaction rate in a sphere subject to the reflecting boundary condition. We determine that the rate of diffusion of a particle from the interior to the surface of the sphere changes by the square root of time plus some extra correction terms. Further, we are able to relate the diffusion propagator to the invariant properties of the region. Also in this approach we investigate how the heat kernel expansion can be applied to the problem of determining the return-to-origin probability, where we obtain a more precise result for the expansion of this probability in the case of a sphere. The advantage of this method lies in its generality and applicability to any geometrical boundary configuration and any kind of boundary condition.
Ghadirian
Bahman
B
Nanoscale Organisation and Dynamics, College of Health and Science, University of Western Sydney, New South Wales 1797, Australia.
Stait-Gardner
Tim
T
Castillo
Reynaldo
R
Price
William S
WS
D004058:Diffusion; D006358:Hot Temperature; D007700:Kinetics; D008956:Models, Chemical; D011336:Probability; D013269:Stochastic Processes