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\begin{document}
\title{On Irreversible Deposition on Disordered Substrates}
\author{E.~Ben-Naim and P.~L.~Krapivsky}
\address{Center for Polymer Studies and Department of Physics, 
Boston University, Boston, MA 02215}
\maketitle
\begin{abstract}
The kinetics of random
sequential adsorption of $k$-mers on a one-dimensional lattice,
occupied initially by point impurities with a random distribution, is
solved exactly. The solution of the process on a continuum substrate
is also derived from the discrete case.  
\end{abstract}
\begin{multicols}{2}
A number of processes in physics, chemistry and biology may be 
modelled by random sequential adsorption (RSA) on a lattice [1].  
In this model, particles arrive randomly and adsorb irreversibly
unless they are in the exclusion zone of previously adsorbed 
particles. In an arbitrary dimension, 
RSA processes reach a jamming configuration, where further 
adsorption events are impossible. The final coverage as well 
as the temporal approach to the jammed state are of interest. 
Exact analytical results have been obtained mainly in one 
dimension, where the problem is also known as the parking 
problem [1-3], and for the Bethe lattice [4,5]. 
In these studies, the substrate is usually assumed to be initially empty.  
In a very recent paper [6], the kinetics of RSA of $k$-mers on 
the one-dimensional lattice, occupied initially by point 
impurities with a Poissonian  distribution, has been investigated 
numerically (mostly for dimers). In this comment we aim to
to point out that for the dimer case the model has been solved
exactly for the dual process $A+A\to 0$ with 
immobile reactants [7,8]. Moreover, we  point out discrepancies between 
the exact solution and the simulation results. 
We also present an exact analytical solution for the $k$-mer case 
with arbitrary $k$ and exploit this solution to obtain the solution to
the same process on a line. 

In the RSA process, $k$-mers land uniformly on a lattice with 
a constant rate, to be taken as unity without loss of generality. 
An adsorption event is successful if all $k$ sites are empty.
Let $P_m(t)$ denote the probability that $m$ 
consecutive lattice sites are empty. The rate equations
for these probabilities are  
\begin{equation}
{dP_m(t)\over dt}=-(m-k+1)P_m(t)-2\sum_{j=1}^{k-1}P_{j+m}(t) 
\end{equation}
for $m \ge k$. The first term in the right-hand side describes 
adsorption events inside the original $m$-site sequence.
The next terms describe desorption events involving sites outside the
original sequence.  

Since the initial density of point impurities is $\rho_0$
and the distribution of impurities is random, $P_m(0)=(1-\rho_0)^m$.
Solving Eq.~(1) subject to these initial conditions yield 
\begin{equation}
P_m(t)=(1-\rho_0)^m \exp\Bigl[-(m-k+1)t
-2\sum_{j=1}^{k-1}{1-e^{-jt'}\over j}(1-\rho_0)^j\Bigr] .
\end{equation}
For an initially empty lattice, $\rho_0=0$, the solution of Eq.~(2) agrees with
the well-known exact solution [3].  

Similarly, one can write rate equations and solve them exactly for
the probabilities $P_m(t)$ with $m<k$. The most interesting quantity,
$P_1(t)$, satisfies the equation
\begin{equation}
{dP_1(t)\over dt}=-kP_k(t), 
\end{equation}
which involves only $P_k(t)$. Integrating Eq.~(3) we find the 
coverage $\rho(t)$, $\rho(t)=1-P_1(t)$,



\begin{equation}
\rho(t)=\rho_0+k(1-\rho_0)^k\int_0^t dt'\exp
\Bigl[-t'-2\sum_{j=1}^{k-1}{1-e^{-jt'}\over j}(1-\rho_0)^j\Bigr]
\end{equation}

\begin{figure}
%\centerline{\epsfxsize=8cm \epsfbox{fig1.ps}}
\centerline{\includegraphics[width=8cm]{fig1}}
\caption{ng coverage versus the initial impurity density. The exact
solution of equation (4) is plotted for the cases of $k=2$, $k=3$ and $k=4$.}
\end{figure}


The jamming coverage, $\rho_{\rm jam}\equiv\lim_{t\to\infty}\rho(t)$, for the case of dimer
deposition, $k=2$, is obtained from Eq.~(4),  
\begin{equation}
\rho_{\rm jam}=1-(1-\rho_0)\exp[-2(1-\rho_0)].
\end{equation}
Thus, we reproduced the result first derived in the 
context of two-particle annihilation reaction with immobile 
reactants [7,8]. From Eq.~(5) one sees that the jamming concentration has a minimum
$\rho_{\rm jam}^{\rm min}=1-e^{-1}/2=0.8160\ldots$ at $\rho_0=1/2$. 
This behaviour is reminiscent of the general $k$-mer
deposition problem where the jamming coverage is a nonmonotonic 
function of the impurity density (see Figure 1). Clearly, in the limit of a 
full initial state $\rho_0\ltwid 1$, $\rho_{\rm jam}\cong\rho_0$.
The simulational result of Milo\v sevi\'c and \v Svraki\'c exhibits
a minimum at $\rho_0\cong0.13$. Moreover, the minimal jamming density is
found to be $\rho_{\rm jam}^{\rm min}\cong0.8564$. Both values differ significantly
from the exact values.





The exact $k$-mer solution can be used to obtain the continuum limit.
In this limit, objects of unit length are deposited on a
one-dimensional line, initially occupied by point defects. The initial
density of the defects is set to $\lambda$. To attain this limit we
rescale the density $k\rho_0\to\lambda$ and the time $kt\to t$. Thus,
we take the limit $k\to\infty$ of Eq.~(4), with the rescaled density
and the rescaled time remaining finite, and obtain the following
continuum coverage function
\begin{equation}
\rho(t)=\int_0^t dt' 
\exp\Bigl[-\lambda -2\int_{\lambda}^{\lambda+t'} du{1-e^{-u}\over u}\Bigr] .
\end{equation}

In the limit $\lambda\to \infty$ this coverage approaches zero
exponentially, $\rho_{\rm jam}\cong
\lambda\exp(-\lambda)$. Conversely, when $\lambda\to 0$ the jamming
coverage approaches the well known R\'enyi number $\rho_{\rm
jam}=0.7475\ldots$.  Unlike the lattice case, the coverage
monotonically decreases to zero, as the density of impurities
increases.



Using the exact solution for the density, we study the approach to the
jamming limit. In the lattice case we find from Eq.~(4) an exponential
approach $\rho_{\rm jam}-\rho(t)\sim \exp(-t)$. This decay was confirmed by the
simulation performed by Milo\v sevi\'c and \v Svraki\'c.
Interestingly, in the continuum case the approach is slower, and from
Eq.~(6) one finds algebraic decay  $\rho_{\rm jam}-\rho(t)\sim t^{-1}$.

\bigskip
We thank Vladimir Privman and Sidney Redner for useful discussions.
We also greatfully acknowledge grant \#DAAH04-93-G-0021, NSF grant
\#DMR-9219845, and the Donors of The Petroleum Research Fund, administered 
by the American Chemical Society, for partial support of this
research.


\begin{thebibliography}{99}

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\bibitem{}  A.~R\'enyi, 
{\sl Publ. Math. Inst. Hung. Acad. Sci.} {\bf 3},  109 (1958).
\bibitem{}  J.~J.~Gonzalez, P.~C.~Hemmer, and J.~S.~Hoye, 
{\sl Chem. Phys.} {\bf 3}, 228 (1974).
\bibitem{}  J.~W.~Evans, {\sl J. Math. Phys.} {\bf 25}, 2527 (1984).
\bibitem{}  Y.~Fan and J.~K.~Percus, {\sl Phys. Rev. A} {\bf 44}, 5099 (1991).
\bibitem{}  D.~Milo\v sevi\'c and N.~M.~\v Svraki\'c, 
{\sl J. Phys. A} {\bf 26}, L1061 (1993).
\bibitem{}  V.~M.~Kenkre and H.~M.~van~Horn, {\sl Phys. Rev. A} {\bf 23}, 
3200 (1981).
\bibitem{}  S.~M.~Majumdar and V.~Privman, {\sl J. Phys. A} {\bf 26}, 
L743 (1993).
\end{thebibliography} 
\end{multicols}{2}
\end{document}

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