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\begin{document}

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



\author[1]{Gershon Wolansky}%
\affil[1]{Technion Israel Institute of Technology Faculty of Physics}%


\vspace{-1em}



  
  \date{May 14, 2024}


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\begin{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.%
\end{abstract}%



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