The linearized Poisson-Nernst-Planck system as heat flow on the interval under non-local boundary conditions

TL;DR

This paper analyzes the linearized Poisson-Nernst-Planck system, showing it can be modeled as a heat flow with non-local boundary conditions, and provides explicit spectral solutions with a stochastic interpretation.

Contribution

It introduces a novel reduction of the linearized PNP system to a damped heat equation with non-local boundary conditions, offering explicit eigenvalues and eigenstates.

Findings

Eigenvalues and eigenstates of the PNP system are explicitly characterized.

The heat kernel is reconstructed via inverse Laplace transform.

A stochastic interpretation as a jump-reflected Brownian particle is provided.

Abstract

The linearized of the Poisson-Nernst-Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non-local 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 adjoint. For this, we take advantage of the representation of the resulvent operator and recover the heat kernel by applying the inverse Laplace transform.