Heat kernel for the linearized Poisson NernstPlanck equation
Abstract
The linearized of the PoissonNernstPlanck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under nonlocal boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its dual. For this, we take advantage of the representation of the resulvent operator and recover the heat kernel by applying the inverse Laplace transform.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.01585
 Bibcode:
 2021arXiv210401585W
 Keywords:

 Mathematical Physics;
 35K10;
 35K51;
 70F45
 EPrint:
 13 pages