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\newcommand{\lrbar}{\bar{l}_{\rho}^{\mathrm{2D}}}

\begin{document}
\title{Derivation of the mean radial displacement, $\lrbar$, in 2 dimensions}
\author{Markus Dittrich}
\date{\today}

%\pacs{}
\maketitle

\section{Introduction}
In the following we derive an expression for the mean radial displacement, $\lrbar$,
for diffusing 2D molecules in analogy to the well known expression for $\bar{l}_r$
in three dimensions. 


\section{Derivation}
Starting point is the solution of the diffusion equation in 2 dimensions
\begin{equation}
\frac{\partial c(\rho,t)}{\partial t} = D \nabla^2 c(\rho,t)
\end{equation}
which, for a point source of $M$ molecules released at the origin at time $t=0$ 
can be shown to be
\begin{equation}
c(\rho,t) = \frac{M}{4 \pi Dt} e^{- \frac{\rho^2}{4Dt}}
\end{equation}
(see also Eqs. 3.1, 3.2 in \cite{KERR2008}). Therefore, the probability
density for a molecule being in a radial shell of thickness $d\rho$ around the 
origin is given by
\begin{equation}
p^{\mathrm{2D}} (\rho,t) = \frac{(2 \pi \rho) d\rho}{4\pi Dt}  e^{- \frac{\rho^2}{4Dt}} 
\end{equation}          
or, in normalized coordinates $\tilde{\rho} = \frac{\rho}{\lambda}$ with the 
normalization constant $\lambda = \sqrt{4Dt}$
\begin{equation}
p^{\mathrm{2D}} (\tilde{\rho},t) = 2 e^{-\tilde{\rho}^2} (\tilde{\rho} d\tilde{\rho})
  \;\;\;\; .
\end{equation}
Hence, the mean radial displacement of a 2D molecule is given by
\begin{equation}
\lrbar = 2 \lambda \int_0^{\infty}  \tilde{\rho}^2 e^{- \tilde{\rho}^2} d\tilde{\rho}
\end{equation}
Since
\begin{eqnarray}
\int_0^{\infty} t^{2n} e^{-at^2} dt &=& \frac{\Gamma(n+\frac{1}{2})}{2a^{n+\frac{1}{2}}} 
  \nonumber \\
\Rightarrow \int_0^{\infty}  \tilde{\rho}^2 e^{- \tilde{\rho}^2} d\tilde{\rho} &=& 
  \frac{\Gamma(\frac{3}{2})}{2} = \frac{\sqrt{\pi}}{4} 
\end{eqnarray}
and from this

\begin{equation}
\boxed{
\lrbar = 2 \lambda \frac{\sqrt{\pi}}{4} = \sqrt{\pi D t}
}
\end{equation}

\begin{thebibliography}{99}
\bibitem{KERR2008} R. Kerr, T.M. Bartol, B. Kaminsky, M. Dittrich, J.C.J. Chang, S. Baden, 
T.J. Sejnowski, and J.R. Stiles, Fast Monte Carlo Simulation Methods for Biological 
Reaction-Diffusion Systems in Solution and on Surfaces (2008),
SIAM J. Sci. Comput., 30:3126-3149. 
\end{thebibliography}

\end{document}
