%----- LaTeX file ------ %----- Figures included using LaTeX picture mode ----- BODY %----- LaTeX file ------ \documentstyle[11pt]{article} \topmargin -0.3in \oddsidemargin 0in \textheight 8.7in \textwidth 6.63in \parskip 4pt plus 1pt minus .5pt \textfloatsep 33pt plus 3pt minus 3pt \pretolerance=5000 \setcounter {topnumber}{3} \renewcommand {\topfraction}{0.8} \setcounter {totalnumber}{4} \renewcommand {\textfraction}{0.2} \renewcommand {\floatpagefraction}{0.7} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \def\appendix{\par % Have \appendix say \setcounter{section}{0} % `Appendix A', not just `A' \setcounter{subsection}{0} \def\thesection{Appendix \Alph{section}} \renewcommand{\theequation}{\Alph{section}.\arabic{equation}} } \begin{document} \title{Exact Results for the Asymmetric Simple Exclusion Process with a Blockage\thanks{ Supported in part by NSF Grant DMR92-13424}} \author{S. A. Janowsky\thanks{Supported in part by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 90-07206}\hspace{.25em}\thanks{Address after August 1993: Department of Mathematics, University of Texas, Austin, TX 78712}\hspace{.25em} and J. L. Lebowitz\\[.3cm] Departments of Physics and Mathematics\\ Rutgers University\\ New Brunswick, New Jersey 08903} \date{June 1993} \maketitle \renewcommand{\baselinestretch}{1.5}\large\normalsize \vspace{-.5cm} \begin{abstract} We present new results for the current as a function of transmission rate in the one dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one to $r<1$. Exact finite volume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a gap in allowed density corresponding to a nonequilibrium ``phase transition'' in the infinite system. A series expansion in $r$, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Pad\'e approximants based on this series, which make specific assumptions about the nature of the singularity at $r=1$, match numerical data for the ``infinite'' system to a part in $10^4$. \end{abstract} \renewcommand{\baselinestretch}{1.7}\large\normalsize \section{Introduction} The one dimensional totally asymmetric simple exclusion process (TASEP) is a continuous-time sto\-chastic process in which particles on a one dimensional lattice jump independently and randomly at unit rate to vacant neighboring sites on their immediate right~\cite{reviews}. It corresponds to a Kawasaki exchange dynamics~\cite{Kawa} at infinite temperature and infinite electric field~\cite{KLS}. The stationary state of this system for $N$ particles on a ring of $K$ sites, $K\ge N$, gives equal weight to all $K\choose N$ permissible configurations. This measure goes over, in the limit $K\rightarrow\infty$, $N/K \rightarrow \rho$, to the product measure with occupation probability $\rho$. The TASEP is thus the simplest driven diffusive lattice-gas model whose dynamics does not satisfy detailed balance~\cite{KLS}. It is also, for the infinite lattice, an example of a microscopic system from which one can derive Euler-like hydrodynamical equations~\cite{dMP}, {\em e.g.}\ the Burgers equation. In an earlier work~\cite{JL} we introduced a variant of the TASEP where the jump rate across one bond of the system was reduced from 1 to $r$, $0 \sqrt{1-4J_{\rm max}(r)}/2$ is permitted. The problem thus is to find bounds on $J_{\rm max}(r)$. A very simple bound on $J_{\rm max}(r)$ can be obtained by noting that if we remove the right (or left) half of the system we are left with a system of $L$ sites with input (removal) rate $a$ and removal (input) rate $r$. Calling $\hat{J}(L; \alpha,\beta)$ the current in a system of $L$ sites with input rate $\alpha$, removal rate $\beta$, and all ``internal'' jump rates now being unity, we clearly have % \begin{equation} J_L(r,a) \le \hat{J}(L; a,r) = \hat{J}(L; r,a), \label{halfsys} \end{equation} % where the last equality, as well as exact formulae for $\hat{J}(L; \alpha,\beta)$, have been computed in \cite{DDM,DEHP}. In particular for $a\ge r$ and $r\le 1/2$ we have $\lim_{L\rightarrow\infty}\hat{J}(L; \alpha,r) = r(1-r)$ so that $J_{\rm max}(r) \le r(1-r)$ for $r\le 1/2$. For $a$ and $r$ both greater than $1/2$ the right hand side of (\ref{halfsys}) approaches $1/4$ so it yields no new information---just a proof that $J_{\rm max}(r)$ cannot exceed $1/4$. To obtain better bounds we note that for any configuration, since the maximum rate at which a particle attempts to jump on to any site is bounded by one, the current in a system of size $L$ with boundary conditions $\alpha=\beta=1$ cannot increase as $L$ increases. Note that $\alpha=\beta=1$ corresponds to keeping the site $-L$ always occupied and the site $L+1$ always vacant in a system of size $L'>L$. Thus $J_L(r)$ is monotonically decreasing in $L$, and for every finite value of $L$, $J_L(r)$ is an upper bound for $J_\infty(r)$. (Similar arguments show that $J_L(r,a)$ is nondecreasing in $r$ and $a$.) These bounds prove the existence of a gap in the set of stationary measures of the infinite system; the results for finite $L$ (illustrated in fig.~\ref{finite}) show that wherever there exists an $L$ such that % \begin{figure} \begin{center} % GNUPLOT: LaTeX picture \setlength{\unitlength}{0.2409pt} \ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi \begin{picture}(1790,999)(130,120) \sbox{\plotpoint}{\rule[-.175pt]{.35pt}{.35pt}}% \put(264,158){\rule[-.175pt]{390.499pt}{.35pt}} \put(264,158){\rule[-.175pt]{.35pt}{216.328pt}} \put(264,158){\rule[-.175pt]{4.818pt}{.35pt}} \put(250,158){\makebox(0,0)[r]{0.00}} \put(1865,158){\rule[-.175pt]{4.818pt}{.35pt}} \put(264,286){\rule[-.175pt]{4.818pt}{.35pt}} \put(250,286){\makebox(0,0)[r]{0.05}} \put(1865,286){\rule[-.175pt]{4.818pt}{.35pt}} \put(264,415){\rule[-.175pt]{4.818pt}{.35pt}} \put(250,415){\makebox(0,0)[r]{0.10}} \put(1865,415){\rule[-.175pt]{4.818pt}{.35pt}} \put(264,543){\rule[-.175pt]{4.818pt}{.35pt}} 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\put(1050,400){\makebox(0,0)[t]{$J_3(r,1/2)$}} \put(1050,405){\line(0,1){300}} \put(1046,716){\vector(-1,3){0}} % \put(1500,450){\makebox(0,0)[t]{$J_5(r,1)$}} \put(1500,455){\line(0,1){390}} \put(1500,848){\vector(0,1){0}} % \end{picture} \end{center} \caption{$J_L(r,1)$ and $J_L(r,1/2)$ for several values of $L$. Where $J_L(r,1) < 1/4$ there must be a gap in the allowed stationary measures of the infinite system.} \label{finite}\end{figure} % $J_L(r) < 1/4$ there is a forbidden range of currents. (One result from our simulations is that $J_{15000}(0.8) = 0.24979(5)$, indicating that the gap exists at least up to $r=0.8$.) We note here also that letting $a\rightarrow\infty$ in $J_L(r,a)$ corresponds to reducing the size of the system from $2L$ to $2(L-1)$ so that $\lim_{a\rightarrow\infty} J_L(r,a) = J_{L-1}(r).$ Using monotonicity in $a$ this implies again that $J_L(r)$ is monotone nonincreasing in $L$. For the case $\alpha=\beta=1/2$ the sequence $J_L(r,1/2)$ appears to be increasing in $L$, and we believe but cannot prove that $J_\infty(r)\ge J_L(r,1/2)$. Proving such an inequality is nontrivial since $J_\infty(r)$ is the {\em maximum} current the infinite system can have; there exist stationary measures where the current is less than $J_L(r,1/2)$. Thus our results for finite $L$ only provide rigorous one-sided bounds. Note also that $J_L(1,1/2) = 1/4$ for all $L$, c.f.~\cite{DDM}. \subsection*{Series expansion} While there is no apparent pattern to the ``raw'' expressions for the current (\ref{J1-J1a})--(\ref{J3}), (\ref{J4-eq}), one does emerge if we examine a Taylor expansion around $r=0$. We obtain: \begin{eqnarray} \label{J1Taylor} J_1(r) &=& r - \frac32 r^2 + \frac94 r^3 - \frac{27}8 r^4 + \frac{81}{16} r^5 - \frac{243}{32} r^6 + O(r^7), \\[5pt] J_2(r) &=& r - \frac32 r^2 + \frac{19}{16} r^3 - \frac{257}{256} r^4 + \frac{24105}{4096} r^5 - \frac{829297}{65536} r^6 + O(r^7),\\[5pt] J_3(r) &=& r - \frac32 r^2 + \frac{19}{16} r^3 - \frac{21535}{27648} r^4 + \frac{919407829}{214990848} r^5 - \frac{7398899579671}{417942208512} r^6 + O(r^7), \\[5pt] J_4(r) &=& r - \frac32 r^2 + \frac{19}{16} r^3 - \frac{21535}{27648} r^4 + \frac{77729356627}{146767085568} r^5 \nonumber\\[-6pt] \label{J4Taylor}\\[-6pt] &&\quad + \frac{1067903077191004635349}{126214320739011526656} r^6 + O(r^7).\nonumber \end{eqnarray} There is a clear pattern: as we increase the size of the system the low-order coefficients stop changing after a certain point. Assuming the continuation of this behavior, and including the results for the size 10 system, we have \begin{eqnarray} J_\infty(r)\! &=& r - \frac32 r^2 + \frac{19}{16} r^3 - \frac{21535}{27648} r^4 + \frac{77729356627}{146767085568} r^5 -.3278724755(1) r^6 + O(r^7) \nonumber\\[-8pt] \label{taylor}\\[-4pt] &=& r - \frac32 r^2 + \frac{19}{2^4} r^3 - \frac{5\cdot 59\cdot73}{2^{10}\cdot 3^3} r^4 + \frac{13\cdot 33613\cdot 177883}{2^{26}\cdot 3^7} r^5 -.3278724755(1) r^6 + O(r^7);\nonumber \end{eqnarray} in the second part of (\ref{taylor}) we show the prime factorization of the coefficients; the denominators appear deceptively simple while examination of the numerators proves less instructive. Including the dependence of the boundary terms ({\em i.e.}\ taking $a\ne 1$) does not significantly alter this behavior: \begin{eqnarray} J_1(r,a) &=& r - \frac3{2a} r^2 + \frac9{4a^2} r^3 -\frac{27}{8 a^3} r^4 + O(r^5),\\ J_2(r,a) &=& r - \frac32 r^2 + \frac{9 a^4 + 18 a^3 + 7 a^2 -8 a - 7}{ 4 a^2 (a+1)^2} r^3\nonumber\\ &&{}- \frac{ 54 a^7 + 216 a^6 + 300 a^5 + 64 a^4 - 294 a^3 - 370 a^2 - 188 a -39}{ 16 a^3 (a + 1)^4} r^4 + O(r^5),\\ J_3(r,a) &=& r - \frac32 r^2 + \frac{19}{16} r^3 + \Big[514 a^{14} + 6939 a^{13} + 41551 a^{12} + 144387 a^{11} + 316671 a^{10}\nonumber\\ &&{} + 432661 a^9 + 285181 a^8 - 176743 a^7 - 702157 a^6 - 944908 a^5 - 799104 a^4 - 457504 a^3\nonumber\\ &&{} - 172480 a^2 - 38528 a - 3840\Big]r^4 \Big/ \left[256 a^3 (1 + 2a)(a+2)^3 (a+1)^7\right] + O(r^5).\label{J3aT} \end{eqnarray} The dependence on the boundary appears one term earlier than in (\ref{J1Taylor})--(\ref{J4Taylor}), but otherwise the structure is the same. In fact, the assumption regarding the behavior of the Taylor coefficients for progressively larger systems expressed in (\ref{J1Taylor})--(\ref{J3aT}) can be proven to be correct: \begin{theorem}\label{asymtheorem} Fix $L_2$. Then for $L_1 \le L_2$, $J_{L_2}(r) = J_{L_1}(r) + O(r^{L_1 + 2})$. \end{theorem} \begin{theorem}\label{asymtheorem2} Fix $L_2$. Then for $L_1 \le L_2$ and $a>0$, $J_{L_2}(r,a) = J_{L_1}(r) + O(r^{L_1 + 1})$. \end{theorem} Thus we see a small system not only bounds but also provides a good approximation (at least for small $r$) for any (finite) system that is larger. This also strongly suggests that the approach of $J_L(r)$ to $J_\infty(r)$ is exponential for $r<1$. Of course the rate of the exponential approach vanishes at $r=1$, where the convergence becomes algebraic (see (\ref{DDM-result})). We reserve the proof for \ref{proof}. \section{Pad\'e Approximants} The Taylor series given in (\ref{taylor}) gives an accurate measure of the current for small $r$, but for large $r$ it is less successful. In fact, we can see that the series must break down by $r=1$: although one commonly thinks of $r$ as being a transmission rate $\le 1$, it is perfectly acceptable to take $r>1$ in the generator (\ref{generator}). It is also fairly easy to see, by comparison with a system with a boundary at the origin, that the infinite volume current $J_\infty(r)$ will have the same value for all $r\ge 1$, namely $J_\infty(1) = 1/4$, so that there must be a nonanalyticity for some $r\le 1$ in $J_\infty(r)$. One can also examine the coefficients of (\ref{taylor}) and see that they apparently decrease (in magnitude) rather slowly, indicating that the radius of convergence of the series is most probably 1. Some numerical analysis indicates that there is no discontinuity in any of the derivatives of $J_\infty(r)$ at $r=1$; this evidence leads us to hypothesize that there is an essential singularity at $r=1$ in $J_\infty(r)$. An alternative which cannot be ruled out by our results is that $J_\infty(r) = 1/4$ for $r>r_1$ with $r_1<1$. Numerical results for the structure of $J_L(r)$ would seem however to argue in favor of a changeover at $r=1$. We thus look for a function of the appropriate form for the current. The ``simplest'' function with an essential singularity at $r=1$ that also gives $J_\infty(0) = 0$ and $J_\infty(1) = 1/4$ is a function of the form \begin{equation} J_\infty(r) = 1/4 - \exp[f(r)]/4, \end{equation} where $f(r)$ has a simple pole at $r=1$. We therefore examined functions of the form \begin{equation} f(r) = \frac14 - \frac14\exp\left[\frac{p(r)}{(1-r)q(r)}\right]; \end{equation} specifically, we used our Taylor series to fit the following Pad\'e approximants: \begin{eqnarray} J^{33}(r) &=& \frac14 - \frac14\exp\left[\frac{-4r(1 + a_1 r + a_2 r^2)}{ (1-r)(1 + b_1 r + b_2 r^2)}\right], \\[5pt] J^{43}(r) &=& \frac14 - \frac14\exp\left[\frac{-4r(1 + a_1' r + a_2' r^2 + a_3' r^3)}{(1-r)(1 + b_1' r + b_2' r^2)}\right], \\[5pt] J^{34}(r) &=& \frac14 - \frac14\exp\left[\frac{-4r(1 + a_1'' r + a_2'' r^2)}{ (1-r)(1 + b_1'' r + b_2'' r^2 + b_3'' r^3)}\right]. \end{eqnarray} The actual values of the coefficients of the Pad\'e approximants are not particularly revealing, as one might expect. The Pad\'e functions do, however, appear to be converging pointwise: for $r\in [0,1]$ the maximum difference between any of $J^{33}$, $J^{43}$ and $J^{34}$ is less than $2\times 10^{-5}$. Of course we are not interested in how well the different approximants approximate each other, but how well they approximate $J_\infty(r)$. We thus must compare the approximants with numerical simulations. We see in fig.~\ref{J-r-fig} that the approximation is within the error bounds of the simulations; we happened to have plotted $J^{43}$ but any of the approximants would have fit the data as well. In comparison, a Pad\'e approximant that behaves quadratically or quartically at $r=1$ (as opposed to exponentially) looks qualitatively similar but does not fall % \begin{figure} \begin{center} % GNUPLOT: LaTeX picture \setlength{\unitlength}{0.2409pt} \ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi \sbox{\plotpoint}{\rule[-.175pt]{.35pt}{.35pt}}% \begin{picture}(1500,860)(160,100) \sbox{\plotpoint}{\rule[-.175pt]{.35pt}{.35pt}}% \put(264,158){\rule[-.175pt]{336.296pt}{.35pt}} \put(264,158){\rule[-.175pt]{.35pt}{183.807pt}} \put(264,158){\rule[-.175pt]{4.818pt}{.35pt}} \put(242,158){\makebox(0,0)[r]{0.00}} \put(1640,158){\rule[-.175pt]{4.818pt}{.35pt}} \put(264,305){\rule[-.175pt]{4.818pt}{.35pt}} \put(242,305){\makebox(0,0)[r]{0.05}} \put(1640,305){\rule[-.175pt]{4.818pt}{.35pt}} \put(264,451){\rule[-.175pt]{4.818pt}{.35pt}} \put(242,451){\makebox(0,0)[r]{0.10}} \put(1640,451){\rule[-.175pt]{4.818pt}{.35pt}} \put(150,525){\makebox(0,0)[r]{$J(r)$}} 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Estimated errors for the numerical calculations are less than the thickness of the lines, ranging from $5\times 10^{-5}$ to $2\times 10^{-4}$.} \label{J-r-fig} \end{figure} % within the error bounds of the simulations. The simulation results were obtained by direct simulation of the TASEP dynamics; progressively larger systems were used until the current reached an asymptotic value. For small values of $r$ this occurred quite quickly, but the needed system size grows quite rapidly as one approaches $r=1$: for $r=0.7$ we needed to investigate systems with 6400 sites and for $r=0.8$ we needed to investigate systems with 15000 sites. Our results can also be viewed as the determination of the phase diagram for the system. As mentioned earlier the current $J_\infty(r)$ is the {\em maximum} stationary current permitted by the infinite system---with some boundary conditions the current may be less. Accepting \cite{Bramson} that the stationary measure is asymptotically a product measure then $J_\infty = \rho_\infty(1-\rho_\infty)$, and the bound on the current is equivalent to a bound on the allowable range of densities. 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\put(918,728){\rule[-.175pt]{.35pt}{.964pt}} \put(919,732){\rule[-.175pt]{.35pt}{1.686pt}} \put(920,739){\rule[-.175pt]{.35pt}{.843pt}} \put(921,742){\rule[-.175pt]{.35pt}{.843pt}} \put(922,746){\rule[-.175pt]{.35pt}{1.927pt}} \put(923,754){\rule[-.175pt]{.35pt}{3.373pt}} \put(924,768){\rule[-.175pt]{.35pt}{3.613pt}} % \multiput(315,180)(30,0){41}{\makebox(0,0){.}} \multiput(375,210)(30,0){37}{\makebox(0,0){.}} \multiput(435,240)(30,0){33}{\makebox(0,0){.}} \multiput(495,270)(30,0){30}{\makebox(0,0){.}} \multiput(525,300)(30,0){28}{\makebox(0,0){.}} \multiput(585,330)(30,0){24}{\makebox(0,0){.}} \multiput(615,360)(30,0){22}{\makebox(0,0){.}} \multiput(645,390)(30,0){6}{\makebox(0,0){.}} \multiput(1065,390)(30,0){6}{\makebox(0,0){.}} \multiput(675,420)(30,0){5}{\makebox(0,0){.}} \multiput(1065,420)(30,0){4}{\makebox(0,0){.}} \multiput(735,450)(30,0){14}{\makebox(0,0){.}} \multiput(765,480)(30,0){12}{\makebox(0,0){.}} \multiput(795,510)(30,0){10}{\makebox(0,0){.}} \multiput(825,540)(30,0){8}{\makebox(0,0){.}} \multiput(855,570)(30,0){6}{\makebox(0,0){.}} \multiput(885,600)(30,0){4}{\makebox(0,0){.}} \multiput(915,630)(30,0){2}{\makebox(0,0){.}} \multiput(915,660)(30,0){2}{\makebox(0,0){.}} % % \put(504,302){{\makebox(0,0){$\bullet$}}} \put(511,306){{\makebox(0,0){$\bullet$}}} \put(517,311){{\makebox(0,0){$\bullet$}}} \put(524,315){{\makebox(0,0){$\bullet$}}} \put(531,320){{\makebox(0,0){$\bullet$}}} \put(537,324){{\makebox(0,0){$\bullet$}}} \put(544,329){{\makebox(0,0){$\bullet$}}} \put(550,333){{\makebox(0,0){$\bullet$}}} \put(557,338){{\makebox(0,0){$\bullet$}}} \put(563,342){{\makebox(0,0){$\bullet$}}} \put(570,347){{\makebox(0,0){$\bullet$}}} \put(576,351){{\makebox(0,0){$\bullet$}}} \put(583,356){{\makebox(0,0){$\bullet$}}} \put(589,360){{\makebox(0,0){$\bullet$}}} \put(595,365){{\makebox(0,0){$\bullet$}}} \put(602,369){{\makebox(0,0){$\bullet$}}} \put(608,374){{\makebox(0,0){$\bullet$}}} \put(614,378){{\makebox(0,0){$\bullet$}}} \put(621,383){{\makebox(0,0){$\bullet$}}} \put(627,387){{\makebox(0,0){$\bullet$}}} \put(633,392){{\makebox(0,0){$\bullet$}}} \put(639,396){{\makebox(0,0){$\bullet$}}} \put(645,401){{\makebox(0,0){$\bullet$}}} \put(652,405){{\makebox(0,0){$\bullet$}}} \put(658,410){{\makebox(0,0){$\bullet$}}} \put(664,414){{\makebox(0,0){$\bullet$}}} \put(670,419){{\makebox(0,0){$\bullet$}}} \put(676,423){{\makebox(0,0){$\bullet$}}} \put(683,428){{\makebox(0,0){$\bullet$}}} \put(689,432){{\makebox(0,0){$\bullet$}}} \put(695,437){{\makebox(0,0){$\bullet$}}} \put(701,441){{\makebox(0,0){$\bullet$}}} \put(708,446){{\makebox(0,0){$\bullet$}}} \put(714,450){{\makebox(0,0){$\bullet$}}} \put(721,455){{\makebox(0,0){$\bullet$}}} \put(727,459){{\makebox(0,0){$\bullet$}}} \put(733,464){{\makebox(0,0){$\bullet$}}} \put(740,468){{\makebox(0,0){$\bullet$}}} \put(747,473){{\makebox(0,0){$\bullet$}}} \put(753,477){{\makebox(0,0){$\bullet$}}} \put(760,482){{\makebox(0,0){$\bullet$}}} \put(767,486){{\makebox(0,0){$\bullet$}}} \put(774,491){{\makebox(0,0){$\bullet$}}} \put(782,495){{\makebox(0,0){$\bullet$}}} \put(789,500){{\makebox(0,0){$\bullet$}}} \put(797,504){{\makebox(0,0){$\bullet$}}} \put(805,509){{\makebox(0,0){$\bullet$}}} \put(814,513){{\makebox(0,0){$\bullet$}}} \put(823,518){{\makebox(0,0){$\bullet$}}} \put(832,522){{\makebox(0,0){$\bullet$}}} \put(843,526){{\makebox(0,0){$\bullet$}}} \put(855,531){{\makebox(0,0){$\bullet$}}} \put(868,535){{\makebox(0,0){$\bullet$}}} \put(877,538){{\makebox(0,0){$\bullet$}}} \put(887,540){{\makebox(0,0){$\bullet$}}} % \put(895,540.5){{\makebox(0,0){$\bullet$}}} \put(908,541){{\makebox(0,0){$\bullet$}}} \put(920,541){{\makebox(0,0){$\bullet$}}} % \put(1354,295){{\makebox(0,0){$\bullet$}}} \put(496,295){{\makebox(0,0){$\bullet$}}} \put(1362,289){{\makebox(0,0){$\bullet$}}} \put(488,289){{\makebox(0,0){$\bullet$}}} \put(1370,283){{\makebox(0,0){$\bullet$}}} \put(480,283){{\makebox(0,0){$\bullet$}}} \put(1378,278){{\makebox(0,0){$\bullet$}}} \put(472,278){{\makebox(0,0){$\bullet$}}} \put(1386,273){{\makebox(0,0){$\bullet$}}} \put(464,273){{\makebox(0,0){$\bullet$}}} \put(1393,269){{\makebox(0,0){$\bullet$}}} \put(457,269){{\makebox(0,0){$\bullet$}}} \put(1400,265){{\makebox(0,0){$\bullet$}}} \put(450,265){{\makebox(0,0){$\bullet$}}} \put(1407,261){{\makebox(0,0){$\bullet$}}} \put(443,261){{\makebox(0,0){$\bullet$}}} \put(1414,257){{\makebox(0,0){$\bullet$}}} \put(436,257){{\makebox(0,0){$\bullet$}}} \put(1421,253){{\makebox(0,0){$\bullet$}}} \put(429,253){{\makebox(0,0){$\bullet$}}} \put(1428,249){{\makebox(0,0){$\bullet$}}} \put(422,249){{\makebox(0,0){$\bullet$}}} \put(1435,245){{\makebox(0,0){$\bullet$}}} \put(415,245){{\makebox(0,0){$\bullet$}}} \put(1442,241){{\makebox(0,0){$\bullet$}}} \put(408,241){{\makebox(0,0){$\bullet$}}} \put(1449,237){{\makebox(0,0){$\bullet$}}} \put(401,237){{\makebox(0,0){$\bullet$}}} \put(1456,233){{\makebox(0,0){$\bullet$}}} \put(394,233){{\makebox(0,0){$\bullet$}}} \put(1462,229){{\makebox(0,0){$\bullet$}}} \put(388,229){{\makebox(0,0){$\bullet$}}} \put(1469,225){{\makebox(0,0){$\bullet$}}} \put(381,225){{\makebox(0,0){$\bullet$}}} \put(1475,221){{\makebox(0,0){$\bullet$}}} \put(375,221){{\makebox(0,0){$\bullet$}}} \put(1482,217){{\makebox(0,0){$\bullet$}}} \put(368,217){{\makebox(0,0){$\bullet$}}} \put(1489,213){{\makebox(0,0){$\bullet$}}} \put(361,213){{\makebox(0,0){$\bullet$}}} \put(1495,209){{\makebox(0,0){$\bullet$}}} \put(355,209){{\makebox(0,0){$\bullet$}}} \put(1502,205){{\makebox(0,0){$\bullet$}}} \put(348,205){{\makebox(0,0){$\bullet$}}} \put(1509,201){{\makebox(0,0){$\bullet$}}} \put(341,201){{\makebox(0,0){$\bullet$}}} \put(1515,197){{\makebox(0,0){$\bullet$}}} \put(335,197){{\makebox(0,0){$\bullet$}}} \put(1522,193){{\makebox(0,0){$\bullet$}}} \put(328,193){{\makebox(0,0){$\bullet$}}} \put(1529,189){{\makebox(0,0){$\bullet$}}} \put(321,189){{\makebox(0,0){$\bullet$}}} \put(1535,185){{\makebox(0,0){$\bullet$}}} \put(315,185){{\makebox(0,0){$\bullet$}}} \put(1542,181){{\makebox(0,0){$\bullet$}}} \put(308,181){{\makebox(0,0){$\bullet$}}} \put(1549,177){{\makebox(0,0){$\bullet$}}} \put(301,177){{\makebox(0,0){$\bullet$}}} \put(1556,174){{\makebox(0,0){$\bullet$}}} \put(294,174){{\makebox(0,0){$\bullet$}}} \put(1563,170){{\makebox(0,0){$\bullet$}}} \put(287,170){{\makebox(0,0){$\bullet$}}} \put(1570,166){{\makebox(0,0){$\bullet$}}} \put(280,166){{\makebox(0,0){$\bullet$}}} \put(1578,162){{\makebox(0,0){$\bullet$}}} \put(272,162){{\makebox(0,0){$\bullet$}}} \put(1585,158){{\makebox(0,0){$\bullet$}}} \put(265,158){{\makebox(0,0){$\bullet$}}} % \put(1346,302){{\makebox(0,0){$\bullet$}}} \put(1339,306){{\makebox(0,0){$\bullet$}}} \put(1333,311){{\makebox(0,0){$\bullet$}}} \put(1326,315){{\makebox(0,0){$\bullet$}}} \put(1319,320){{\makebox(0,0){$\bullet$}}} \put(1313,324){{\makebox(0,0){$\bullet$}}} \put(1306,329){{\makebox(0,0){$\bullet$}}} \put(1300,333){{\makebox(0,0){$\bullet$}}} \put(1293,338){{\makebox(0,0){$\bullet$}}} \put(1287,342){{\makebox(0,0){$\bullet$}}} \put(1280,347){{\makebox(0,0){$\bullet$}}} \put(1274,351){{\makebox(0,0){$\bullet$}}} \put(1267,356){{\makebox(0,0){$\bullet$}}} \put(1261,360){{\makebox(0,0){$\bullet$}}} \put(1255,365){{\makebox(0,0){$\bullet$}}} \put(1248,369){{\makebox(0,0){$\bullet$}}} \put(1242,374){{\makebox(0,0){$\bullet$}}} \put(1236,378){{\makebox(0,0){$\bullet$}}} \put(1229,383){{\makebox(0,0){$\bullet$}}} \put(1223,387){{\makebox(0,0){$\bullet$}}} \put(1217,392){{\makebox(0,0){$\bullet$}}} \put(1211,396){{\makebox(0,0){$\bullet$}}} \put(1205,401){{\makebox(0,0){$\bullet$}}} \put(1198,405){{\makebox(0,0){$\bullet$}}} \put(1192,410){{\makebox(0,0){$\bullet$}}} \put(1186,414){{\makebox(0,0){$\bullet$}}} \put(1180,419){{\makebox(0,0){$\bullet$}}} \put(1174,423){{\makebox(0,0){$\bullet$}}} \put(1167,428){{\makebox(0,0){$\bullet$}}} \put(1161,432){{\makebox(0,0){$\bullet$}}} \put(1155,437){{\makebox(0,0){$\bullet$}}} \put(1149,441){{\makebox(0,0){$\bullet$}}} \put(1142,446){{\makebox(0,0){$\bullet$}}} \put(1136,450){{\makebox(0,0){$\bullet$}}} \put(1129,455){{\makebox(0,0){$\bullet$}}} \put(1123,459){{\makebox(0,0){$\bullet$}}} \put(1117,464){{\makebox(0,0){$\bullet$}}} \put(1110,468){{\makebox(0,0){$\bullet$}}} \put(1103,473){{\makebox(0,0){$\bullet$}}} \put(1097,477){{\makebox(0,0){$\bullet$}}} \put(1090,482){{\makebox(0,0){$\bullet$}}} \put(1083,486){{\makebox(0,0){$\bullet$}}} \put(1076,491){{\makebox(0,0){$\bullet$}}} \put(1068,495){{\makebox(0,0){$\bullet$}}} \put(1061,500){{\makebox(0,0){$\bullet$}}} \put(1053,504){{\makebox(0,0){$\bullet$}}} \put(1045,509){{\makebox(0,0){$\bullet$}}} \put(1036,513){{\makebox(0,0){$\bullet$}}} \put(1027,518){{\makebox(0,0){$\bullet$}}} \put(1018,522){{\makebox(0,0){$\bullet$}}} \put(1007,526){{\makebox(0,0){$\bullet$}}} \put(995,531){{\makebox(0,0){$\bullet$}}} \put(982,535){{\makebox(0,0){$\bullet$}}} \put(973,538){{\makebox(0,0){$\bullet$}}} \put(963,540){{\makebox(0,0){$\bullet$}}} % \put(955,540.5){{\makebox(0,0){$\bullet$}}} \put(942,541){{\makebox(0,0){$\bullet$}}} \put(930,541){{\makebox(0,0){$\bullet$}}} % \end{picture} \end{center} \caption{Phase diagram of allowed asymptotic density {\em vs.}\ transmission. The heavy line indicates the region excluded by currently available exact results; the thin line and the shaded region are determined by the Pad\'e results.} \label{phase_r-fig} \end{figure} % in fig.~\ref{phase_r-fig}. If the overall density of the finite system in a periodic box, $\rho$, was in the disallowed region, the system would have to segregate with high density $\rho_-$ to the left of the blockage and low density $\rho_+$ to the right of it, with $\rho_- + \rho_+ = 1$ and $\rho = c\rho_- + (1-c)\rho_+$ for some fraction $c$, $00$ and all $0\le r\le 1$. \renewcommand{\baselinestretch}{1}\large\normalsize \appendix \renewcommand{\thesection}{Appendix \Alph{section}} \section{The function $J_4(r)$}\label{J4} \setcounter{equation}{0} $J_4(r) =$ \begin{equation}\label{J4-eq} {\scriptsize\frac{ \begin{array}{c} 2r\cdot\big( 46594570041553208865114157983928672r^{34} +\\ 3272256807527488050658050196010194008r^{33} + 111157036726242688664509666570575952692r^{32} +\\ 2432978692972890221457116915305608070176r^{31} + 38563042893543905939312199308038000267376r^{30} +\\ 471629643887135562049943024025386831761624r^{29} + 4630830690747494363914241555175156474737952r^{28} +\\ 37501116089445029526599189649856279114367080r^{27} + 255349882255809833270215616484973179447971604r^{26} +\\ 1483034668343761760933311314169905866593307200r^{25} + 7427024657245470907468866632968858025038621224r^{24} +\\ 32341852979577601689520720487367994974417590384r^{23} + 123258959443962771434544300470844688653283339220r^{22} +\\ 413187952609538688738716350764927640900234526696r^{21} + 1222925851525408590841791369809880777445578325556r^{20} +\\ 3204670630666608783584495910901402180619797197008r^{19} + 7449424997833123151978707539741892038741323482132r^{18} +\\ 15378015964501993765574252081802020182175729230264r^{17} + 28201950376870887559972172287807374530071235163796r^{16} +\\ 45931932106228251497346831830246712730250907133712r^{15} + 66367346077080103767105589115636872559989214550441r^{14} +\\ 84922961375392487277902239726735636329302249943128r^{13} + 95985607018161363007960499193162341535912061049744r^{12} +\\ 95495899491963130911426581603975014896548923261952r^{11} + 83250539026363674821927230131182759997615987109888r^{10} +\\ 63222693774680812665162035869098746973122020442112r^9 + 41512233585213949665760166256158119415119899262976r^8 +\\ 23337512260240138424809534978398421177693582131200r^7 + 11088988610809857322221895274024151043294063755264r^6 +\\ 4375523126050494396079936095098710196366170128384r^5 + 1398264149827785557648870482515232428810049683456r^4 +\\ 348505991819536772424594468421800136094590697472r^3 + 63703892226452562097992685657738901012902576128r^2 +\\ 7610495543799775598845584755474871742038540288r + 446443113770610009364511999835819193606864896\big) \end{array}} { \begin{array}{c} \big( 289620280319036422192318811402048808r^{35} + 20347660058076078580887803589027266100r^{34} +\\ 691509180090958636822780801273385528578r^{33} + 15143171306946285250897143672457580856112r^{32} +\\ 240156757626348422508288967223318773219730r^{31} + 2939013682631225236236144632768990651743996r^{30} +\\ 28878453872089870194234547560112273497410248r^{29} + 234055024231518053113128876309994685465865998r^{28} +\\ 1595219951014207692709829729475814422350403348r^{27} + 9274911880978847804107170124884025274187759308r^{26} +\\ 46507371429345382752005370961862741048375464600r^{25} + 202819905961978964692531821554688508829966150639r^{24} +\\ 774305913910573093515211395294773846677490470023r^{23} + 2600902978686057892600556891176773816130767065580r^{22} +\\ 7716564464804046735459659032377460737774938262294r^{21} + 20279584392798156542425998004346657157480554084969r^{20} +\\ 47304820202361946730446972991019456869802234988211r^{19} + 98064294514560130382369885216651184747849062158254r^{18} +\\ 180769182668553783438231100925433451981380541595764r^{17} + 296287842239143809420984903211913226411657347513603r^{16} +\\ 431494541664969696255270976652324836280276021938501r^{15} + 557613386760147331351017000992052900309321526028898r^{14} +\\ 638162101110332154357279278430725658354907392715680r^{13} + 645076256680960920649709526072469900202092833324704r^{12} +\\ 573956167631320085801939332326120871725603509536768r^{11} + 447552460656613237448046131746142189157548579971072r^{10} +\\ 304176946559301647309430508519196663546007393796096r^9 + 178937028617243446184563625330908654841920346914816r^8 +\\ 90290026293392171865547954006547530297965379846144r^7 + 38609332952461801047936627883941955240573198663680r^6 +\\ 13757984465655294150768749140417907410320746348544r^5 + 3986654552595495753325700874754842100218571259904r^4 +\\ 904824015385555468140765882234506884944960159744r^3 + 151187962701066997262421713581553533038335361024r^2 +\\ 16560320428911381225784705510457201064897675264r + 892886227541220018729023999671638387213729792\big) \end{array}}} \end{equation} \renewcommand{\baselinestretch}{1.7}\large\normalsize \section{Proof of Theorem~\protect\ref{asymtheorem}}\label{proof} \setcounter{equation}{0} To prove theorem~\ref{asymtheorem}, we will need to consider systems with unequal numbers of sites to the left and to the right of the blockage, say $L_-$ and $L_+$, respectively; $J_{L_-,L_+}(r)$ will represent the current in such a system with $\alpha=\beta=1$. We will prove the following lemma: \begin{lemma} For $L_-' \le L_-$ and $L_+' \le L_+$, \[ J_{L_-,L_+}(r) = J_{L_-',L_+}(r) + O(r^{L_-' + 2})\quad\hbox{and}\quad J_{L_-,L_+}(r) = J_{L_-,L_+'}(r) + O(r^{L_+' + 2}). \] \end{lemma} Theorem~\ref{asymtheorem} is a direct consequence: for $L_2\ge L_1$, $J_{L_2}(r)\equiv J_{L_2,L_2}(r) = J_{L_2,L_1}(r) + O(r^{L_1 + 2}) =J_{L_1,L_1}(r) + O(r^{L_1 + 2}) \equiv J_{L_1}(r) + O(r^{L_1 + 2})$. Now we prove the lemma, considering only the case $L_+' \le L_+$ since the case $L_-' \le L_-$ is the same by symmetry. We will need to consider probabilities of certain collections of configurations. We write \begin{equation}\label{marginal-notation} \Pr ( \eta_{-L_-+1}, \eta_{-L_-+2}, \ldots, \eta_0 ; \eta_1, \eta_2, \ldots, \eta_N )_{L_-,L_+'} \end{equation} for $0\le N\le L_+'$. This represents the marginal probability of the configuration at sites $-L_-+1, -L_-+2, \ldots N$; the rest of the sites can take arbitrary values. The semicolon indicates the position of the blockage in the system. Let $e_R(\eta)$ be the number of the $\eta_1, \eta_2, \ldots, \eta_N$ that take the value 1, {\em i.e.}\ $e_R(\eta)=\sum_{i=1}^N \eta_i$. For $r$ small particles to the right of the blockage are rare and can be treated as excitations; $e_R(\eta)$ can be thought of as the (right) excitation number in the system. Our proof proceeds by induction: Suppose we know {\em all} probabilities of the form\\ $\Pr ( \eta_{-L_-+1}, \eta_{-L_-+2}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_N )_{L_-,L_+'}$ to a certain accuracy, namely to order $e_R(\{\eta_1, \ldots, \eta_N\}) + k$ in $r$. Then (for $N > 1$) we can (show that we can) compute all probabilities of the form $\Pr ( \eta_{-L_-+1}, \eta_{-L_-+2}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_{N-1} )_{L_-,L_+'}$ up to order $e_R(\{\eta_1, \ldots, \eta_{N-1}\}) + k + 1$ in $r$. This is sufficient to prove our lemma, and in fact more general results: for any size system (with $L_+' \ge N$) we can compute $\Pr ( \eta_{-L_-+1}, \eta_{-L_-+2}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_N )_{L_-,L_+'}$ up to order $e_R(\eta)$ simply by making use of the fact that excitations are created at rate $r$, providing the initial step in the induction. This estimate (and as a result all following estimates) is independent of $L_+'$ and thus also valid for all $L_+ \ge L_+'$. Since the current is \begin{equation} J(r) = r\Pr (1;0) = r[1 - \Pr (0;0) - \Pr (1;1) - \Pr (0;1)] \end{equation} if we can iterate the induction step $j$ times we know the current to order $2+j$. So let us return to the induction step. Consider the transitions that occur between the different values of $\eta_{-L_-+1}, \ldots \eta_{N-1}$: we will give estimates on the rates of these transitions in the stationary state. (The rate of a transition $\tau\rightarrow\tau'$, ${\rm rate\,} [\tau\rightarrow\tau']$, where $\tau$ and $\tau'$ are sets of configurations, is \begin{equation} {\rm rate\,} [\tau\rightarrow\tau'] = \lim_{\Delta t\rightarrow0} (\Delta t)^{-1} \Pr(\tau'\, {\rm at~time\,}t+\Delta t | \tau\, {\rm at~time\,}t). \end{equation} Thus all (elementary) transitions between individual configurations occur at rate 0, 1 or $r$.) Transitions that increase $e_R(\{\eta_1, \ldots, \eta_{N-1}\})$ involve a jump across the blockage, and thus simply have rate $r$. Transitions that keep $e_R(\{\eta_1, \ldots, \eta_{N-1}\})$ fixed do not involve sites outside of $\{ -L_-+1, -L_-+2, \ldots, N-1 \}$ and so have rates that are simple integers. Transitions that decrease $e_R(\{\eta_1, \ldots, \eta_{N-1}\})$ require that we know something about site $N$, since the only way to reduce $e_R(\{\eta_1, \ldots, \eta_{N-1}\})$ is for a particle to move from site $N-1$ to site $N$. So \begin{eqnarray}\lefteqn{ {\rm rate\,}\left[ e_R(\{\eta_1, \ldots, \eta_{N-2},1\}) \rightarrow e_R(\{\eta_1, \ldots, \eta_{N-2},0\}) \right]}\nonumber\\ &=& \frac{\Pr ( \eta_{-L_-+1}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_{N-2},1,0)_{L_-,L_+'}} {\Pr ( \eta_{-L_-+1}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_{N-2},1)_{L_-,L_+'}}\label{rate1}\\ &=& 1 - \frac{\Pr ( \eta_{-L_-+1}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_{N-2},1,1)_{L_-,L_+'}} {\Pr ( \eta_{-L_-+1}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_{N-2},1,1)_{L_-,L_+'} + \Pr ( \eta_{-L_-+1}, \ldots, \eta_0 ; \eta_1, \ldots, \eta_{N-2},1,0)_{L_-,L_+'}}\nonumber \end{eqnarray} The numerator of (\ref{rate1}) is $O(e_R(\{\eta_1,\ldots, \eta_{N-1}\}) + 1)$ and we know it to $O(e_R(\{\eta_1,\ldots, \eta_{N-1}\}) + k + 1)$ by the induction hypothesis. The denominator of (\ref{rate1}) is $O(e_R(\{\eta_1,\ldots, \eta_{N-1}\}))$ and we know it to $O(e_R(\{\eta_1,\ldots, \eta_{N-1}\}) + k)$ by the induction hypothesis. Thus the rate in (\ref{rate1}) is $1-O(r)$ and we know it to $O(e_R(\{\eta_1,\ldots, \eta_{N-1}\}) + k + 1)$. % \begin{figure} \begin{center} \setlength{\unitlength}{1.2cm} \begin{picture}(13.7,2.4)(.1,1.35) \multiput(1,2)(2,0){5}{\circle*{.35}} \put(10.9,2){\makebox(0,0){$\cdots$}} \put(12.8,2){\circle*{.35}} \put(1,1.55){\makebox(0,0){0}} \put(3,1.55){\makebox(0,0){1}} \put(5,1.55){\makebox(0,0){2}} \put(7,1.55){\makebox(0,0){3}} \put(9,1.55){\makebox(0,0){4}} \put(12.9,1.55){\makebox(0,0){$N\!\!-\!1$}} \multiput(2,2.05)(2,0){4}{\oval(1.7,.35)[t]} \multiput(2,1.95)(2,0){4}{\oval(1.7,.35)[b]} \put(10,2.05){\oval(1.7,.35)[tl]} \put(10,1.95){\oval(1.7,.35)[bl]} \put(11.8,2.05){\oval(1.7,.35)[tr]} \put(11.8,1.95){\oval(1.7,.35)[br]} \multiput(10.,2.225)(0,-.45){2}{\line(1,0){.65}} \multiput(11.8,2.225)(0,-.45){2}{\line(-1,0){.65}} \multiput(1,2.55)(2,0){5}{\circle{.8}} \put(12.8,2.55){\circle{.8}} \multiput(1.09,2.95)(2,0){5}{\vector(1,0){0}} \put(13.09,2.95){\vector(1,0){0}} \multiput(2.05,2.225)(2,0){5}{\vector(1,0){0}} \multiput(1.95,1.775)(2,0){5}{\vector(-1,0){0}} \put(11.85,2.225){\vector(1,0){0}} \put(11.75,1.775){\vector(-1,0){0}} \multiput(1.,3.24)(2,0){5}{\makebox(0,0){\scriptsize exact: $O(1)$}} \put(12.8,3.24){\makebox(0,0){\scriptsize exact: $O(1)$}} \multiput(2,2.4)(2,0){5}{\makebox(0,0){\scriptsize exact: $r$}} \multiput(2,1.55)(2,0){5}{\makebox(0,0){\scriptsize $1-O(r)$}} \multiput(2,1.3)(2,0){5}{\makebox(0,0){\scriptsize to order $k+1$}} \put(11.8,2.4){\makebox(0,0){\scriptsize exact: $r$}} \put(11.8,1.55){\makebox(0,0){\scriptsize $1-O(r)$}} \put(11.8,1.3){\makebox(0,0){\scriptsize to order $k+1$}} \end{picture} \end{center} \caption{Markov Chain-like representation for the transition rates in the partial system. The indices on the ``states'' are the $e_R(\eta)$ which are altered by the ``horizontal'' transitions. The ``circular'' transitions are those between states with the same $e_R(\eta)$. }\label{rates} \end{figure} % Fig.~\ref{rates} thus represents our transitions, where we group together states with a common $e_R$. It is clear that it is consistent to solve for the probabilities in stationary state of the subsystem represented by fig.~\ref{rates} to $O(e_R(\{\eta_1,\ldots, \eta_{N-1}\}) + k + 1)$. This completes the induction step and thus the proof of the lemma. The proof of theorem~\ref{asymtheorem2} proceeds in identical fashion. The only difference is that in (\ref{marginal-notation}) one is limited to considering $0\le N < L_+'$ instead of $0\le N\le L_+'$, ensuring that the transition rates for changing the excitation number do not depend on the boundary conditions. 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