% The following is a LaTeX source file for the paper % (appeared in Phys Rev A 45 618--625 (1992)) % `Finite Size Effects and Shock Fluctuations in the Asymmetric Simple % Exclusion Process' by Janowsky/Lebowitz % % Can use ams.sty to get correct `\lessapprox' character; see below. % BODY \documentstyle[11pt]{article} \textwidth 6.25in \textheight 8.6in \topmargin -.25in \oddsidemargin 0.2in \parskip 2.8pt plus 1pt minus .5pt \pretolerance=2000 \renewcommand{\baselinestretch}{1.2} \newcommand{\lessapprox}{<\approx} % should use a special AMS char for this. % character 2F in msam \author{Steven A. Janowsky\thanks{Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 90-07206}\hspace{.25em} and Joel L. Lebowitz\\ Departments of Mathematics and Physics\\ Rutgers University\\New Brunswick, NJ 08903} \title{Finite Size Effects and Shock Fluctuations in the Asymmetric Simple Exclusion Process\thanks{Supported in part by National Science Foundation grant DMR 89-18903}} \date{January, 1992} \begin{document}\maketitle \begin{abstract} We consider a system of particles on a lattice of $L$ sites, set on a circle, evolving according to the asymmetric simple exclusion process, {\em i.e.}\ particles jump independently to empty neighboring sites on the right (left) with rate $p$ (rate $1-p$), $1/2
0$. If we use the approximation (\ref{rhoapprox}), we must
have $|\rho_{\rm avg} - 1/2| < (1-r)/(2(1+r))$. If this is not the case we no
longer have the basic structure of two near-uniform regions meeting at a shock
front, with local perturbations near the blockage; instead we have only one
approximately uniform phase. Apparently, if the current in the uniform phase
is sufficiently small, there is no need for phase segregation to reduce the
bulk current to that at the blockage~\cite{Krug}. Similar behavior is
observed when we examine an infinite system with one site blocked (see
Section~\ref{infinite}).
\subsection*{Tracking the Shock}
While the average shock location is easy to determine, finding its microscopic
position or even defining it precisely at a given time is nontrivial. Instead
of determining the shock position directly, we track it through the use of a
{\bf second class particle}~\cite{BCFG}. The second class particle is an
extra particle added to the system, which is treated as a hole in exchanges
with particles and as a particle in exchanges with holes. In other words, if
we select site $i$ and it is occupied by an ordinary particle, it moves to
site $i+1$ either if that site is empty or if it is occupied by the second
class particle; in the latter case the second class particle simultaneously
jumps (to the left) to site $i$, {\em i.e.}\ there is an exchange between the
first and second class particles. If site $i$ is occupied by a second class
particle, it moves to site $i+1$ only if site $i+1$ is unoccupied; if site
$i+1$ is occupied it stays put. Therefore, if the second class particle is in
a high density region (of ordinary particles), it will be forced to the left,
while if it is in a low density region it will be able to move to the right.
A shock consists of an abrupt change from a low density region to a high
density region. To the left of a shock (a low density region) a second class
particle moves predominantly to the right, while on the right side (a high
density region) it moves predominantly to the left. The second class particle
therefore executes a biased random walk with drift towards the shock position.
Simple analysis shows and computer simulations confirm that the inherent
fluctuations of this random walk are small compared with the movement of the
shock---namely that if we fix the densities on either side of the shock
(${\rho_{\rm low}}, {\rho_{\rm high}}$) and consider different values of the
system size $L$, the motion of the second class particle about the shock
position will have fixed variance while the variance of the shock position
itself will grow with system size. Thus analyzing the motion of the second
class particle will allow us to indirectly examine the shock motion.
An alternative method of finding the shock is suggested by the surface growth
model. Recalling equation~(\ref{surfaceheight}), we see that surface height
increases in low (particle) density regions and decreases in high density
regions, so that the shock location should correspond intuitively to the
location of the maximum height. Unfortunately this position need not be
(microscopically) unique, so we prefer the use of the second class particle.
It suggests, however, that in addition to the shock position we study other
features from the growth model, such as the difference between the maximum and
minimum surface heights, which is equivalent to studying the number of
particles on either side of the interface in the ASEP.
As a third possibility we could ignore the microscopic shock position entirely
and simply study the width of the time-averaged shock density profile, such as
that in figure~\ref{prof35}. However, this method obscures the distinction
between the instantaneous microscopic width of the shock and the width due to
fluctuations in the shock's position (particularly relevant in the extension
to two-dimensional models), and the width is difficult to determine accurately
because of lattice effects. In any case, we expect that as the system size
goes to infinity that rescaling the shock profile by the standard deviation of
the shock fluctuation (determined by any of the three methods) should yield a
well-defined limiting shape.
We sample the position of the second class particle after allowing the system
to reach a steady state. For each system we obtain a distribution (histogram)
of shock (second class particle) positions and compute the variance of the
obtained distribution; a typical run consisted of approximately 25,000 samples
where 15 sweeps ($15\times L$ monte carlo steps) were made between samples.
We then compare the variances obtained from systems with different values of
the parameters. We also perform the same set of computations for the number
of particles found between site~1 and the second class particle position.
In figure~\ref{var55} we plot the standard deviation of the interface position
{\em vs.}\ system size for average density 0.55 and blocking rate $r=0.35$.
\begin{figure}[hbt]
\begin{center}
% GNUPLOT: LaTeX picture
\setlength{\unitlength}{0.33pt}
\begin{picture}(1323,660)(167,40)
\newcommand{\mD}{\makebox(0,0){$\diamond$}}
\newcommand{\uP}{\usebox{\plotpoint}}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\put(264,113){\line(1,0){10}}
\put(1436,113){\line(-1,0){10}}
\put(264,145){\line(1,0){10}}
\put(1436,145){\line(-1,0){10}}
\put(264,173){\line(1,0){10}}
\put(1436,173){\line(-1,0){10}}
\put(264,197){\line(1,0){10}}
\put(1436,197){\line(-1,0){10}}
\put(264,219){\line(1,0){20}}
\put(1436,219){\line(-1,0){20}}
\put(242,219){\makebox(0,0)[r]{10}}
\put(264,364){\line(1,0){10}}
\put(1436,364){\line(-1,0){10}}
\put(264,448){\line(1,0){10}}
\put(1436,448){\line(-1,0){10}}
\put(264,508){\line(1,0){10}}
\put(1436,508){\line(-1,0){10}}
\put(264,554){\line(1,0){10}}
\put(1436,554){\line(-1,0){10}}
\put(242,554){\makebox(0,0)[r]{50}}
\put(264,592){\line(1,0){10}}
\put(1436,592){\line(-1,0){10}}
\put(264,624){\line(1,0){10}}
\put(1436,624){\line(-1,0){10}}
\put(264,652){\line(1,0){10}}
\put(1436,652){\line(-1,0){10}}
\put(303,113){\line(0,1){10}}
\put(303,652){\line(0,-1){10}}
\put(401,113){\line(0,1){10}}
\put(401,652){\line(0,-1){10}}
\put(470,113){\line(0,1){10}}
\put(470,652){\line(0,-1){10}}
\put(524,113){\line(0,1){10}}
\put(524,652){\line(0,-1){10}}
\put(568,113){\line(0,1){10}}
\put(568,652){\line(0,-1){10}}
\put(605,113){\line(0,1){10}}
\put(605,652){\line(0,-1){10}}
\put(637,113){\line(0,1){10}}
\put(637,652){\line(0,-1){10}}
\put(666,113){\line(0,1){10}}
\put(666,652){\line(0,-1){10}}
\put(691,113){\line(0,1){20}}
\put(691,652){\line(0,-1){20}}
\put(691,78){\makebox(0,0){1000}}
\put(858,113){\line(0,1){10}}
\put(858,652){\line(0,-1){10}}
\put(956,113){\line(0,1){10}}
\put(956,652){\line(0,-1){10}}
\put(1025,113){\line(0,1){10}}
\put(1025,652){\line(0,-1){10}}
\put(1079,113){\line(0,1){10}}
\put(1079,652){\line(0,-1){10}}
\put(1123,113){\line(0,1){10}}
\put(1123,652){\line(0,-1){10}}
\put(1160,113){\line(0,1){10}}
\put(1160,652){\line(0,-1){10}}
\put(1192,113){\line(0,1){10}}
\put(1192,652){\line(0,-1){10}}
\put(1221,113){\line(0,1){10}}
\put(1221,652){\line(0,-1){10}}
\put(1246,113){\line(0,1){20}}
\put(1246,652){\line(0,-1){20}}
\put(1246,78){\makebox(0,0){10000}}
\put(1413,113){\line(0,1){10}}
\put(1413,652){\line(0,-1){10}}
\put(264,113){\line(1,0){1172}}
\put(1436,113){\line(0,1){539}}
\put(1436,652){\line(-1,0){1172}}
\put(264,652){\line(0,-1){539}}
\put(850,33){\makebox(0,0){System Size}}
\put(303,176){\mD}
\put(303,178){\mD}
\put(303,180){\mD}
\put(568,264){\mD}
\put(568,266){\mD}
\put(568,268){\mD}
\put(691,336){\mD}
\put(691,338){\mD}
\put(691,340){\mD}
\put(833,365){\mD}
\put(833,368){\mD}
\put(833,370){\mD}
\put(1025,443){\mD}
\put(1025,446){\mD}
\put(1025,450){\mD}
\put(1246,552){\mD}
\put(1246,555){\mD}
\put(1246,558){\mD}
\put(1359,602){\mD}
\put(1359,605){\mD}
\put(1359,608){\mD}
\sbox{\plotpoint}{\rule[-0.125pt]{1.10pt}{.25pt}}
\put(271,129){\uP}
\put(273,130){\uP}
\put(276,131){\uP}
\put(278,132){\uP}
\put(280,133){\uP}
\put(282,134){\uP}
\put(284,135){\uP}
\put(286,136){\uP}
\put(288,137){\uP}
\put(290,138){\uP}
\put(293,139){\uP}
\put(295,140){\uP}
\put(298,141){\uP}
\put(300,142){\uP}
\put(303,143){\uP}
\put(305,144){\uP}
\put(308,145){\uP}
\put(310,146){\uP}
\put(312,147){\uP}
\put(315,148){\uP}
\put(317,149){\uP}
\put(320,150){\uP}
\put(322,151){\uP}
\put(324,152){\uP}
\put(326,153){\uP}
\put(328,154){\uP}
\put(330,155){\uP}
\put(332,156){\uP}
\put(334,157){\uP}
\put(337,158){\uP}
\put(339,159){\uP}
\put(342,160){\uP}
\put(344,161){\uP}
\put(347,162){\uP}
\put(349,163){\uP}
\put(352,164){\uP}
\put(354,165){\uP}
\put(356,166){\uP}
\put(359,167){\uP}
\put(361,168){\uP}
\put(364,169){\uP}
\put(366,170){\uP}
\put(369,171){\uP}
\put(371,172){\uP}
\put(374,173){\uP}
\put(375,174){\uP}
\put(377,175){\uP}
\put(379,176){\uP}
\put(381,177){\uP}
\put(383,178){\uP}
\put(386,179){\uP}
\put(388,180){\uP}
\put(391,181){\uP}
\put(393,182){\uP}
\put(396,183){\uP}
\put(398,184){\uP}
\put(400,185){\uP}
\put(403,186){\uP}
\put(405,187){\uP}
\put(408,188){\uP}
\put(410,189){\uP}
\put(413,190){\uP}
\put(415,191){\uP}
\put(418,192){\uP}
\put(419,193){\uP}
\put(421,194){\uP}
\put(423,195){\uP}
\put(425,196){\uP}
\put(427,197){\uP}
\put(429,198){\uP}
\put(432,199){\uP}
\put(434,200){\uP}
\put(437,201){\uP}
\put(439,202){\uP}
\put(442,203){\uP}
\put(444,204){\uP}
\put(447,205){\uP}
\put(449,206){\uP}
\put(451,207){\uP}
\put(454,208){\uP}
\put(456,209){\uP}
\put(459,210){\uP}
\put(461,211){\uP}
\put(463,212){\uP}
\put(465,213){\uP}
\put(467,214){\uP}
\put(469,215){\uP}
\put(471,216){\uP}
\put(473,217){\uP}
\put(476,218){\uP}
\put(478,219){\uP}
\put(481,220){\uP}
\put(483,221){\uP}
\put(486,222){\uP}
\put(488,223){\uP}
\put(491,224){\uP}
\put(493,225){\uP}
\put(495,226){\uP}
\put(498,227){\uP}
\put(500,228){\uP}
\put(503,229){\uP}
\put(505,230){\uP}
\put(507,231){\uP}
\put(509,232){\uP}
\put(511,233){\uP}
\put(513,234){\uP}
\put(515,235){\uP}
\put(517,236){\uP}
\put(519,237){\uP}
\put(522,238){\uP}
\put(525,239){\uP}
\put(528,240){\uP}
\put(530,241){\uP}
\put(532,242){\uP}
\put(534,243){\uP}
\put(537,244){\uP}
\put(539,245){\uP}
\put(541,246){\uP}
\put(544,247){\uP}
\put(547,248){\uP}
\put(550,249){\uP}
\put(551,250){\uP}
\put(553,251){\uP}
\put(555,252){\uP}
\put(557,253){\uP}
\put(559,254){\uP}
\put(561,255){\uP}
\put(563,256){\uP}
\put(566,257){\uP}
\put(569,258){\uP}
\put(572,259){\uP}
\put(574,260){\uP}
\put(576,261){\uP}
\put(578,262){\uP}
\put(581,263){\uP}
\put(583,264){\uP}
\put(585,265){\uP}
\put(588,266){\uP}
\put(591,267){\uP}
\put(594,268){\uP}
\put(595,269){\uP}
\put(597,270){\uP}
\put(599,271){\uP}
\put(601,272){\uP}
\put(603,273){\uP}
\put(605,274){\uP}
\put(607,275){\uP}
\put(610,276){\uP}
\put(613,277){\uP}
\put(616,278){\uP}
\put(618,279){\uP}
\put(620,280){\uP}
\put(622,281){\uP}
\put(625,282){\uP}
\put(627,283){\uP}
\put(629,284){\uP}
\put(632,285){\uP}
\put(635,286){\uP}
\put(638,287){\uP}
\put(639,288){\uP}
\put(641,289){\uP}
\put(643,290){\uP}
\put(645,291){\uP}
\put(647,292){\uP}
\put(649,293){\uP}
\put(651,294){\uP}
\put(654,295){\uP}
\put(657,296){\uP}
\put(660,297){\uP}
\put(662,298){\uP}
\put(664,299){\uP}
\put(666,300){\uP}
\put(669,301){\uP}
\put(671,302){\uP}
\put(673,303){\uP}
\put(676,304){\uP}
\put(679,305){\uP}
\put(682,306){\uP}
\put(684,307){\uP}
\put(686,308){\uP}
\put(688,309){\uP}
\put(690,310){\uP}
\put(692,311){\uP}
\put(694,312){\uP}
\put(696,313){\uP}
\put(698,314){\uP}
\put(700,315){\uP}
\put(702,316){\uP}
\put(705,317){\uP}
\put(708,318){\uP}
\put(711,319){\uP}
\put(713,320){\uP}
\put(715,321){\uP}
\put(717,322){\uP}
\put(720,323){\uP}
\put(722,324){\uP}
\put(724,325){\uP}
\put(727,326){\uP}
\put(730,327){\uP}
\put(733,328){\uP}
\put(734,329){\uP}
\put(736,330){\uP}
\put(738,331){\uP}
\put(740,332){\uP}
\put(742,333){\uP}
\put(744,334){\uP}
\put(746,335){\uP}
\put(749,336){\uP}
\put(752,337){\uP}
\put(755,338){\uP}
\put(757,339){\uP}
\put(759,340){\uP}
\put(761,341){\uP}
\put(764,342){\uP}
\put(766,343){\uP}
\put(768,344){\uP}
\put(771,345){\uP}
\put(774,346){\uP}
\put(777,347){\uP}
\put(778,348){\uP}
\put(780,349){\uP}
\put(782,350){\uP}
\put(784,351){\uP}
\put(786,352){\uP}
\put(788,353){\uP}
\put(790,354){\uP}
\put(793,355){\uP}
\put(796,356){\uP}
\put(799,357){\uP}
\put(801,358){\uP}
\put(803,359){\uP}
\put(805,360){\uP}
\put(808,361){\uP}
\put(810,362){\uP}
\put(812,363){\uP}
\put(815,364){\uP}
\put(818,365){\uP}
\put(821,366){\uP}
\put(822,367){\uP}
\put(824,368){\uP}
\put(826,369){\uP}
\put(828,370){\uP}
\put(830,371){\uP}
\put(832,372){\uP}
\put(834,373){\uP}
\put(837,374){\uP}
\put(840,375){\uP}
\put(843,376){\uP}
\put(845,377){\uP}
\put(847,378){\uP}
\put(849,379){\uP}
\put(852,380){\uP}
\put(854,381){\uP}
\put(856,382){\uP}
\put(859,383){\uP}
\put(862,384){\uP}
\put(865,385){\uP}
\put(866,386){\uP}
\put(868,387){\uP}
\put(870,388){\uP}
\put(872,389){\uP}
\put(874,390){\uP}
\put(876,391){\uP}
\put(878,392){\uP}
\put(881,393){\uP}
\put(884,394){\uP}
\put(887,395){\uP}
\put(889,396){\uP}
\put(891,397){\uP}
\put(893,398){\uP}
\put(896,399){\uP}
\put(898,400){\uP}
\put(900,401){\uP}
\put(903,402){\uP}
\put(906,403){\uP}
\put(909,404){\uP}
\put(910,405){\uP}
\put(912,406){\uP}
\put(914,407){\uP}
\put(916,408){\uP}
\put(918,409){\uP}
\put(920,410){\uP}
\put(922,411){\uP}
\put(925,412){\uP}
\put(928,413){\uP}
\put(931,414){\uP}
\put(933,415){\uP}
\put(935,416){\uP}
\put(937,417){\uP}
\put(940,418){\uP}
\put(942,419){\uP}
\put(944,420){\uP}
\put(947,421){\uP}
\put(950,422){\uP}
\put(953,423){\uP}
\put(955,424){\uP}
\put(957,425){\uP}
\put(959,426){\uP}
\put(961,427){\uP}
\put(963,428){\uP}
\put(965,429){\uP}
\put(967,430){\uP}
\put(969,431){\uP}
\put(972,432){\uP}
\put(975,433){\uP}
\put(977,434){\uP}
\put(979,435){\uP}
\put(981,436){\uP}
\put(984,437){\uP}
\put(986,438){\uP}
\put(988,439){\uP}
\put(991,440){\uP}
\put(994,441){\uP}
\put(997,442){\uP}
\put(999,443){\uP}
\put(1001,444){\uP}
\put(1003,445){\uP}
\put(1005,446){\uP}
\put(1007,447){\uP}
\put(1009,448){\uP}
\put(1011,449){\uP}
\put(1013,450){\uP}
\put(1015,451){\uP}
\put(1017,452){\uP}
\put(1020,453){\uP}
\put(1023,454){\uP}
\put(1026,455){\uP}
\put(1028,456){\uP}
\put(1030,457){\uP}
\put(1033,458){\uP}
\put(1035,459){\uP}
\put(1037,460){\uP}
\put(1040,461){\uP}
\put(1042,462){\uP}
\put(1045,463){\uP}
\put(1047,464){\uP}
\put(1049,465){\uP}
\put(1051,466){\uP}
\put(1053,467){\uP}
\put(1055,468){\uP}
\put(1057,469){\uP}
\put(1059,470){\uP}
\put(1062,471){\uP}
\put(1064,472){\uP}
\put(1067,473){\uP}
\put(1069,474){\uP}
\put(1072,475){\uP}
\put(1074,476){\uP}
\put(1077,477){\uP}
\put(1079,478){\uP}
\put(1081,479){\uP}
\put(1084,480){\uP}
\put(1086,481){\uP}
\put(1089,482){\uP}
\put(1091,483){\uP}
\put(1093,484){\uP}
\put(1095,485){\uP}
\put(1097,486){\uP}
\put(1099,487){\uP}
\put(1101,488){\uP}
\put(1103,489){\uP}
\put(1106,490){\uP}
\put(1108,491){\uP}
\put(1111,492){\uP}
\put(1113,493){\uP}
\put(1116,494){\uP}
\put(1118,495){\uP}
\put(1121,496){\uP}
\put(1123,497){\uP}
\put(1125,498){\uP}
\put(1128,499){\uP}
\put(1130,500){\uP}
\put(1133,501){\uP}
\put(1135,502){\uP}
\put(1137,503){\uP}
\put(1139,504){\uP}
\put(1141,505){\uP}
\put(1143,506){\uP}
\put(1145,507){\uP}
\put(1147,508){\uP}
\put(1150,509){\uP}
\put(1152,510){\uP}
\put(1155,511){\uP}
\put(1157,512){\uP}
\put(1160,513){\uP}
\put(1162,514){\uP}
\put(1165,515){\uP}
\put(1167,516){\uP}
\put(1169,517){\uP}
\put(1172,518){\uP}
\put(1174,519){\uP}
\put(1177,520){\uP}
\put(1179,521){\uP}
\put(1181,522){\uP}
\put(1183,523){\uP}
\put(1185,524){\uP}
\put(1187,525){\uP}
\put(1189,526){\uP}
\put(1191,527){\uP}
\put(1194,528){\uP}
\put(1196,529){\uP}
\put(1199,530){\uP}
\put(1201,531){\uP}
\put(1204,532){\uP}
\put(1206,533){\uP}
\put(1209,534){\uP}
\put(1211,535){\uP}
\put(1213,536){\uP}
\put(1216,537){\uP}
\put(1218,538){\uP}
\put(1221,539){\uP}
\put(1223,540){\uP}
\put(1226,541){\uP}
\put(1228,542){\uP}
\put(1231,543){\uP}
\put(1232,544){\uP}
\put(1234,545){\uP}
\put(1236,546){\uP}
\put(1238,547){\uP}
\put(1240,548){\uP}
\put(1243,549){\uP}
\put(1245,550){\uP}
\put(1248,551){\uP}
\put(1250,552){\uP}
\put(1253,553){\uP}
\put(1255,554){\uP}
\put(1257,555){\uP}
\put(1260,556){\uP}
\put(1262,557){\uP}
\put(1265,558){\uP}
\put(1267,559){\uP}
\put(1270,560){\uP}
\put(1272,561){\uP}
\put(1275,562){\uP}
\put(1276,563){\uP}
\put(1278,564){\uP}
\put(1280,565){\uP}
\put(1282,566){\uP}
\put(1284,567){\uP}
\put(1286,568){\uP}
\put(1289,569){\uP}
\put(1291,570){\uP}
\put(1294,571){\uP}
\put(1296,572){\uP}
\put(1299,573){\uP}
\put(1301,574){\uP}
\put(1304,575){\uP}
\put(1306,576){\uP}
\put(1308,577){\uP}
\put(1311,578){\uP}
\put(1313,579){\uP}
\put(1316,580){\uP}
\put(1318,581){\uP}
\put(1320,582){\uP}
\put(1322,583){\uP}
\put(1324,584){\uP}
\put(1326,585){\uP}
\put(1328,586){\uP}
\put(1330,587){\uP}
\put(1333,588){\uP}
\put(1335,589){\uP}
\put(1338,590){\uP}
\put(1340,591){\uP}
\put(1343,592){\uP}
\put(1345,593){\uP}
\put(1348,594){\uP}
\put(1350,595){\uP}
\put(1352,596){\uP}
\put(1355,597){\uP}
\put(1357,598){\uP}
\put(1360,599){\uP}
\put(1362,600){\uP}
\put(1364,601){\uP}
\put(1366,602){\uP}
\put(1368,603){\uP}
\put(1370,604){\uP}
\put(1372,605){\uP}
\put(1374,606){\uP}
\put(1377,607){\uP}
\put(1379,608){\uP}
\put(1382,609){\uP}
\put(1384,610){\uP}
\put(1387,611){\uP}
\put(1389,612){\uP}
\put(1392,613){\uP}
\put(1394,614){\uP}
\put(1396,615){\uP}
\put(1399,616){\uP}
\put(1401,617){\uP}
\put(1404,618){\uP}
\put(1406,619){\uP}
\put(1408,620){\uP}
\put(1410,621){\uP}
\put(1412,622){\uP}
\put(1414,623){\uP}
\put(1416,624){\uP}
\end{picture}
\end{center}
\caption{Interface fluctuations with average density 0.55, $r=0.35$. The solid
line has slope 1/2.}
\label{var55}
\end{figure}
The error bars (three symbols are plotted for each measurement: the actual
value and that value shifted up or down by the error bound) represent
statistical error based on the approximate number of independent samples
selected from the steady state in each system. The line drawn through the
data points has a slope of 1/2, so that fluctuations grow like the square root
of the system size, indicating standard finite size behavior.
In figure~\ref{var35w-1} we plot the same quantities as in figure~\ref{var55},
but the average particle density is 0.5.
\begin{figure}[htb]
\begin{center}
% GNUPLOT: LaTeX picture
\setlength{\unitlength}{0.32pt}
\begin{picture}(1188,750)(212,44)
\newcommand{\mD}{\makebox(0,0){$\diamond$}}
\newcommand{\mC}{\makebox(0,0){$\circ$}}
\newcommand{\uP}{\usebox{\plotpoint}}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\put(850,25){\makebox(0,0){System Size}}
\put(264,133){\line(1,0){10}}
\put(1436,133){\line(-1,0){10}}
\put(264,168){\line(1,0){10}}
\put(1436,168){\line(-1,0){10}}
\put(264,197){\line(1,0){10}}
\put(1436,197){\line(-1,0){10}}
\put(264,222){\line(1,0){10}}
\put(1436,222){\line(-1,0){10}}
\put(264,245){\line(1,0){10}}
\put(1436,245){\line(-1,0){10}}
\put(264,265){\line(1,0){20}}
\put(1436,265){\line(-1,0){20}}
\put(245,265){\makebox(0,0)[r]{10}}
\put(264,397){\line(1,0){10}}
\put(1436,397){\line(-1,0){10}}
\put(264,474){\line(1,0){10}}
\put(1436,474){\line(-1,0){10}}
\put(264,529){\line(1,0){10}}
\put(1436,529){\line(-1,0){10}}
\put(264,571){\line(1,0){10}}
\put(1436,571){\line(-1,0){10}}
\put(264,606){\line(1,0){10}}
\put(1436,606){\line(-1,0){10}}
\put(264,635){\line(1,0){10}}
\put(1436,635){\line(-1,0){10}}
\put(264,660){\line(1,0){10}}
\put(1436,660){\line(-1,0){10}}
\put(264,683){\line(1,0){10}}
\put(1436,683){\line(-1,0){10}}
\put(264,703){\line(1,0){20}}
\put(1436,703){\line(-1,0){20}}
\put(245,703){\makebox(0,0)[r]{100}}
\put(264,113){\line(0,1){10}}
\put(264,760){\line(0,-1){10}}
\put(283,113){\line(0,1){20}}
\put(283,760){\line(0,-1){20}}
\put(283,78){\makebox(0,0){100}}
\put(409,113){\line(0,1){10}}
\put(409,760){\line(0,-1){10}}
\put(483,113){\line(0,1){10}}
\put(483,760){\line(0,-1){10}}
\put(535,113){\line(0,1){10}}
\put(535,760){\line(0,-1){10}}
\put(576,113){\line(0,1){10}}
\put(576,760){\line(0,-1){10}}
\put(609,113){\line(0,1){10}}
\put(609,760){\line(0,-1){10}}
\put(637,113){\line(0,1){10}}
\put(637,760){\line(0,-1){10}}
\put(661,113){\line(0,1){10}}
\put(661,760){\line(0,-1){10}}
\put(682,113){\line(0,1){10}}
\put(682,760){\line(0,-1){10}}
\put(701,113){\line(0,1){20}}
\put(701,760){\line(0,-1){20}}
\put(701,78){\makebox(0,0){1000}}
\put(827,113){\line(0,1){10}}
\put(827,760){\line(0,-1){10}}
\put(901,113){\line(0,1){10}}
\put(901,760){\line(0,-1){10}}
\put(953,113){\line(0,1){10}}
\put(953,760){\line(0,-1){10}}
\put(994,113){\line(0,1){10}}
\put(994,760){\line(0,-1){10}}
\put(1027,113){\line(0,1){10}}
\put(1027,760){\line(0,-1){10}}
\put(1055,113){\line(0,1){10}}
\put(1055,760){\line(0,-1){10}}
\put(1079,113){\line(0,1){10}}
\put(1079,760){\line(0,-1){10}}
\put(1101,113){\line(0,1){10}}
\put(1101,760){\line(0,-1){10}}
\put(1120,113){\line(0,1){20}}
\put(1120,760){\line(0,-1){20}}
\put(1120,78){\makebox(0,0){10000}}
\put(1246,113){\line(0,1){10}}
\put(1246,760){\line(0,-1){10}}
\put(1319,113){\line(0,1){10}}
\put(1319,760){\line(0,-1){10}}
\put(1372,113){\line(0,1){10}}
\put(1372,760){\line(0,-1){10}}
\put(1412,113){\line(0,1){10}}
\put(1412,760){\line(0,-1){10}}
\put(264,113){\line(1,0){1172}}
\put(1436,113){\line(0,1){647}}
\put(1436,760){\line(-1,0){1172}}
\put(264,760){\line(0,-1){647}}
\put(1260,725){\makebox(0,0)[r]{$r=0.35$}}
\put(1300,725){\mD}
\put(283,140){\mD}
\put(283,142){\mD}
\put(283,144){\mD}
\put(357,162){\mD}
\put(357,164){\mD}
\put(357,165){\mD}
\put(369,164){\mD}
\put(369,165){\mD}
\put(369,167){\mD}
\put(442,191){\mD}
\put(442,193){\mD}
\put(442,195){\mD}
\put(483,206){\mD}
\put(483,208){\mD}
\put(483,210){\mD}
\put(511,207){\mD}
\put(511,209){\mD}
\put(511,211){\mD}
\put(593,239){\mD}
\put(593,243){\mD}
\put(593,246){\mD}
\put(609,247){\mD}
\put(609,250){\mD}
\put(609,252){\mD}
\put(661,260){\mD}
\put(661,263){\mD}
\put(661,266){\mD}
\put(735,286){\mD}
\put(735,289){\mD}
\put(735,291){\mD}
\put(775,305){\mD}
\put(775,308){\mD}
\put(775,310){\mD}
\put(808,311){\mD}
\put(808,314){\mD}
\put(808,318){\mD}
\put(882,329){\mD}
\put(882,333){\mD}
\put(882,336){\mD}
\put(901,345){\mD}
\put(901,347){\mD}
\put(901,349){\mD}
\put(953,352){\mD}
\put(953,355){\mD}
\put(953,359){\mD}
\put(1027,388){\mD}
\put(1027,392){\mD}
\put(1027,395){\mD}
\put(1055,403){\mD}
\put(1055,405){\mD}
\put(1055,408){\mD}
\put(1120,408){\mD}
\put(1120,411){\mD}
\put(1120,414){\mD}
\put(1193,439){\mD}
\put(1193,442){\mD}
\put(1193,446){\mD}
\put(1205,444){\mD}
\put(1205,447){\mD}
\put(1205,451){\mD}
\put(1286,458){\mD}
\put(1286,463){\mD}
\put(1286,468){\mD}
\put(1331,493){\mD}
\put(1331,496){\mD}
\put(1331,500){\mD}
\put(1372,497){\mD}
\put(1372,501){\mD}
\put(1372,505){\mD}
\put(1405,500){\mD}
\put(1405,505){\mD}
\put(1405,511){\mD}
\put(1260,681){\makebox(0,0)[r]{$r=0.50$}}
\put(1300,681){\mC}
\put(344,317){\mC}
\put(344,321){\mC}
\put(344,325){\mC}
\put(450,348){\mC}
\put(450,353){\mC}
\put(450,359){\mC}
\put(535,379){\mC}
\put(535,383){\mC}
\put(535,386){\mC}
\put(576,396){\mC}
\put(576,405){\mC}
\put(576,414){\mC}
\put(661,421){\mC}
\put(661,424){\mC}
\put(661,427){\mC}
\put(701,453){\mC}
\put(701,462){\mC}
\put(701,472){\mC}
\put(827,476){\mC}
\put(827,485){\mC}
\put(827,494){\mC}
\put(901,503){\mC}
\put(901,508){\mC}
\put(901,513){\mC}
\put(953,543){\mC}
\put(953,550){\mC}
\put(953,557){\mC}
\put(1055,575){\mC}
\put(1055,580){\mC}
\put(1055,585){\mC}
\put(1120,582){\mC}
\put(1120,585){\mC}
\put(1120,587){\mC}
\put(1153,576){\mC}
\put(1153,581){\mC}
\put(1153,587){\mC}
\put(1246,634){\mC}
\put(1246,637){\mC}
\put(1246,640){\mC}
\put(1279,638){\mC}
\put(1279,643){\mC}
\put(1279,648){\mC}
\sbox{\plotpoint}{\rule[-.15pt]{1.75pt}{.30pt}}
\put(276,291){\uP}
\put(278,292){\uP}
\put(282,293){\uP}
\put(286,294){\uP}
\put(288,295){\uP}
\put(290,296){\uP}
\put(293,297){\uP}
\put(297,298){\uP}
\put(301,299){\uP}
\put(303,300){\uP}
\put(305,301){\uP}
\put(308,302){\uP}
\put(311,303){\uP}
\put(315,304){\uP}
\put(317,305){\uP}
\put(320,306){\uP}
\put(322,307){\uP}
\put(325,308){\uP}
\put(327,309){\uP}
\put(330,310){\uP}
\put(333,311){\uP}
\put(337,312){\uP}
\put(339,313){\uP}
\put(342,314){\uP}
\put(344,315){\uP}
\put(348,316){\uP}
\put(352,317){\uP}
\put(354,318){\uP}
\put(356,319){\uP}
\put(359,320){\uP}
\put(363,321){\uP}
\put(367,322){\uP}
\put(369,323){\uP}
\put(371,324){\uP}
\put(374,325){\uP}
\put(377,326){\uP}
\put(381,327){\uP}
\put(383,328){\uP}
\put(386,329){\uP}
\put(388,330){\uP}
\put(391,331){\uP}
\put(393,332){\uP}
\put(396,333){\uP}
\put(399,334){\uP}
\put(403,335){\uP}
\put(405,336){\uP}
\put(408,337){\uP}
\put(410,338){\uP}
\put(414,339){\uP}
\put(418,340){\uP}
\put(420,341){\uP}
\put(422,342){\uP}
\put(425,343){\uP}
\put(428,344){\uP}
\put(432,345){\uP}
\put(434,346){\uP}
\put(437,347){\uP}
\put(439,348){\uP}
\put(443,349){\uP}
\put(447,350){\uP}
\put(449,351){\uP}
\put(451,352){\uP}
\put(454,353){\uP}
\put(456,354){\uP}
\put(459,355){\uP}
\put(461,356){\uP}
\put(465,357){\uP}
\put(469,358){\uP}
\put(471,359){\uP}
\put(473,360){\uP}
\put(476,361){\uP}
\put(480,362){\uP}
\put(484,363){\uP}
\put(486,364){\uP}
\put(488,365){\uP}
\put(491,366){\uP}
\put(494,367){\uP}
\put(498,368){\uP}
\put(500,369){\uP}
\put(503,370){\uP}
\put(505,371){\uP}
\put(509,372){\uP}
\put(513,373){\uP}
\put(515,374){\uP}
\put(517,375){\uP}
\put(519,376){\uP}
\put(522,377){\uP}
\put(525,378){\uP}
\put(528,379){\uP}
\put(531,380){\uP}
\put(535,381){\uP}
\put(537,382){\uP}
\put(539,383){\uP}
\put(541,384){\uP}
\put(546,385){\uP}
\put(550,386){\uP}
\put(552,387){\uP}
\put(554,388){\uP}
\put(556,389){\uP}
\put(560,390){\uP}
\put(564,391){\uP}
\put(566,392){\uP}
\put(569,393){\uP}
\put(572,394){\uP}
\put(575,395){\uP}
\put(579,396){\uP}
\put(581,397){\uP}
\put(583,398){\uP}
\put(585,399){\uP}
\put(588,400){\uP}
\put(591,401){\uP}
\put(594,402){\uP}
\put(597,403){\uP}
\put(601,404){\uP}
\put(603,405){\uP}
\put(605,406){\uP}
\put(607,407){\uP}
\put(612,408){\uP}
\put(616,409){\uP}
\put(618,410){\uP}
\put(620,411){\uP}
\put(622,412){\uP}
\put(626,413){\uP}
\put(630,414){\uP}
\put(632,415){\uP}
\put(635,416){\uP}
\put(638,417){\uP}
\put(641,418){\uP}
\put(645,419){\uP}
\put(647,420){\uP}
\put(649,421){\uP}
\put(651,422){\uP}
\put(654,423){\uP}
\put(657,424){\uP}
\put(660,425){\uP}
\put(663,426){\uP}
\put(667,427){\uP}
\put(669,428){\uP}
\put(671,429){\uP}
\put(673,430){\uP}
\put(678,431){\uP}
\put(682,432){\uP}
\put(684,433){\uP}
\put(686,434){\uP}
\put(688,435){\uP}
\put(692,436){\uP}
\put(696,437){\uP}
\put(698,438){\uP}
\put(701,439){\uP}
\put(704,440){\uP}
\put(707,441){\uP}
\put(711,442){\uP}
\put(713,443){\uP}
\put(715,444){\uP}
\put(717,445){\uP}
\put(720,446){\uP}
\put(722,447){\uP}
\put(724,448){\uP}
\put(729,449){\uP}
\put(733,450){\uP}
\put(735,451){\uP}
\put(737,452){\uP}
\put(739,453){\uP}
\put(743,454){\uP}
\put(747,455){\uP}
\put(749,456){\uP}
\put(752,457){\uP}
\put(755,458){\uP}
\put(758,459){\uP}
\put(762,460){\uP}
\put(764,461){\uP}
\put(766,462){\uP}
\put(768,463){\uP}
\put(773,464){\uP}
\put(777,465){\uP}
\put(779,466){\uP}
\put(781,467){\uP}
\put(783,468){\uP}
\put(786,469){\uP}
\put(788,470){\uP}
\put(790,471){\uP}
\put(795,472){\uP}
\put(799,473){\uP}
\put(801,474){\uP}
\put(803,475){\uP}
\put(805,476){\uP}
\put(809,477){\uP}
\put(813,478){\uP}
\put(815,479){\uP}
\put(818,480){\uP}
\put(821,481){\uP}
\put(824,482){\uP}
\put(828,483){\uP}
\put(830,484){\uP}
\put(832,485){\uP}
\put(834,486){\uP}
\put(839,487){\uP}
\put(843,488){\uP}
\put(845,489){\uP}
\put(847,490){\uP}
\put(849,491){\uP}
\put(852,492){\uP}
\put(854,493){\uP}
\put(856,494){\uP}
\put(861,495){\uP}
\put(865,496){\uP}
\put(867,497){\uP}
\put(869,498){\uP}
\put(871,499){\uP}
\put(875,500){\uP}
\put(879,501){\uP}
\put(881,502){\uP}
\put(884,503){\uP}
\put(887,504){\uP}
\put(890,505){\uP}
\put(894,506){\uP}
\put(896,507){\uP}
\put(898,508){\uP}
\put(900,509){\uP}
\put(905,510){\uP}
\put(909,511){\uP}
\put(911,512){\uP}
\put(913,513){\uP}
\put(915,514){\uP}
\put(918,515){\uP}
\put(920,516){\uP}
\put(922,517){\uP}
\put(927,518){\uP}
\put(931,519){\uP}
\put(933,520){\uP}
\put(935,521){\uP}
\put(937,522){\uP}
\put(941,523){\uP}
\put(945,524){\uP}
\put(947,525){\uP}
\put(950,526){\uP}
\put(953,527){\uP}
\put(956,528){\uP}
\put(960,529){\uP}
\put(962,530){\uP}
\put(964,531){\uP}
\put(966,532){\uP}
\put(969,533){\uP}
\put(972,534){\uP}
\put(975,535){\uP}
\put(978,536){\uP}
\put(982,537){\uP}
\put(984,538){\uP}
\put(986,539){\uP}
\put(988,540){\uP}
\put(992,541){\uP}
\put(996,542){\uP}
\put(998,543){\uP}
\put(1001,544){\uP}
\put(1004,545){\uP}
\put(1007,546){\uP}
\put(1011,547){\uP}
\put(1013,548){\uP}
\put(1015,549){\uP}
\put(1017,550){\uP}
\put(1022,551){\uP}
\put(1026,552){\uP}
\put(1028,553){\uP}
\put(1030,554){\uP}
\put(1033,555){\uP}
\put(1035,556){\uP}
\put(1037,557){\uP}
\put(1040,558){\uP}
\put(1044,559){\uP}
\put(1048,560){\uP}
\put(1050,561){\uP}
\put(1052,562){\uP}
\put(1055,563){\uP}
\put(1058,564){\uP}
\put(1062,565){\uP}
\put(1064,566){\uP}
\put(1067,567){\uP}
\put(1069,568){\uP}
\put(1073,569){\uP}
\put(1077,570){\uP}
\put(1079,571){\uP}
\put(1081,572){\uP}
\put(1084,573){\uP}
\put(1088,574){\uP}
\put(1092,575){\uP}
\put(1094,576){\uP}
\put(1096,577){\uP}
\put(1099,578){\uP}
\put(1101,579){\uP}
\put(1103,580){\uP}
\put(1106,581){\uP}
\put(1110,582){\uP}
\put(1114,583){\uP}
\put(1116,584){\uP}
\put(1118,585){\uP}
\put(1121,586){\uP}
\put(1124,587){\uP}
\put(1128,588){\uP}
\put(1130,589){\uP}
\put(1133,590){\uP}
\put(1135,591){\uP}
\put(1139,592){\uP}
\put(1143,593){\uP}
\put(1145,594){\uP}
\put(1147,595){\uP}
\put(1150,596){\uP}
\put(1154,597){\uP}
\put(1158,598){\uP}
\put(1160,599){\uP}
\put(1162,600){\uP}
\put(1165,601){\uP}
\put(1167,602){\uP}
\put(1169,603){\uP}
\put(1172,604){\uP}
\put(1176,605){\uP}
\put(1180,606){\uP}
\put(1182,607){\uP}
\put(1184,608){\uP}
\put(1187,609){\uP}
\put(1190,610){\uP}
\put(1194,611){\uP}
\put(1196,612){\uP}
\put(1199,613){\uP}
\put(1201,614){\uP}
\put(1205,615){\uP}
\put(1209,616){\uP}
\put(1211,617){\uP}
\put(1213,618){\uP}
\put(1216,619){\uP}
\put(1220,620){\uP}
\put(1224,621){\uP}
\put(1226,622){\uP}
\put(1228,623){\uP}
\put(1231,624){\uP}
\put(1233,625){\uP}
\put(1235,626){\uP}
\put(1238,627){\uP}
\put(1242,628){\uP}
\put(1246,629){\uP}
\put(1248,630){\uP}
\put(1250,631){\uP}
\put(1253,632){\uP}
\put(1256,633){\uP}
\put(1260,634){\uP}
\put(1262,635){\uP}
\put(1265,636){\uP}
\put(1267,637){\uP}
\put(1271,638){\uP}
\put(1275,639){\uP}
\put(1277,640){\uP}
\put(1279,641){\uP}
\put(1282,642){\uP}
\put(1286,643){\uP}
\put(1290,644){\uP}
\put(1292,645){\uP}
\put(1294,646){\uP}
\put(1297,647){\uP}
\put(1299,648){\uP}
\put(1301,649){\uP}
\put(1304,650){\uP}
\put(1307,651){\uP}
\put(1311,652){\uP}
\put(1313,653){\uP}
\put(1316,654){\uP}
\put(1318,655){\uP}
\put(1322,656){\uP}
\put(1326,657){\uP}
\put(1328,658){\uP}
\put(1330,659){\uP}
\put(1333,660){\uP}
\put(1337,661){\uP}
\put(1341,662){\uP}
\put(1343,663){\uP}
\put(1345,664){\uP}
\put(1348,665){\uP}
\put(1351,666){\uP}
\put(1355,667){\uP}
\put(1357,668){\uP}
\put(1360,669){\uP}
\put(1362,670){\uP}
\put(1365,671){\uP}
\put(1367,672){\uP}
\put(1370,673){\uP}
\put(1373,674){\uP}
\put(1377,675){\uP}
\put(1379,676){\uP}
\put(1382,677){\uP}
\put(1384,678){\uP}
\put(1388,679){\uP}
\put(1392,680){\uP}
\put(1394,681){\uP}
\put(1396,682){\uP}
\put(1399,683){\uP}
\put(1403,684){\uP}
\put(1407,685){\uP}
\put(1409,686){\uP}
\put(1411,687){\uP}
\put(1414,688){\uP}
\put(1417,689){\uP}
\sbox{\plotpoint}{\rule[-0.15pt]{1.75pt}{0.30pt}}
\multiput(283,125)(4,1.33){6}{\uP}
\multiput(303,132)(4,1.33){6}{\uP}
\multiput(322,138)(4,1.33){6}{\uP}
\multiput(342,146)(4,1.33){6}{\uP}
\multiput(361,153)(4,1.33){6}{\uP}
\multiput(381,160)(4,1.33){6}{\uP}
\multiput(401,166)(4,1.33){6}{\uP}
\multiput(420,173)(4,1.33){6}{\uP}
\multiput(440,180)(4,1.33){6}{\uP}
\multiput(459,187)(4,1.33){6}{\uP}
\multiput(479,194)(4,1.33){6}{\uP}
\multiput(498,201)(4,1.33){6}{\uP}
\multiput(518,207)(4,1.33){6}{\uP}
\multiput(537,214)(4,1.33){6}{\uP}
\multiput(557,221)(4,1.33){6}{\uP}
\multiput(576,228)(4,1.33){6}{\uP}
\multiput(596,234)(4,1.33){6}{\uP}
\multiput(616,242)(4,1.33){6}{\uP}
\multiput(635,248)(4,1.33){6}{\uP}
\multiput(655,255)(4,1.33){6}{\uP}
\multiput(674,262)(4,1.33){6}{\uP}
\multiput(694,269)(4,1.33){6}{\uP}
\multiput(713,275)(4,1.33){6}{\uP}
\multiput(733,282)(4,1.33){6}{\uP}
\multiput(753,289)(4,1.33){6}{\uP}
\multiput(772,296)(4,1.33){6}{\uP}
\multiput(792,303)(4,1.33){6}{\uP}
\multiput(811,310)(4,1.33){6}{\uP}
\multiput(831,316)(4,1.33){6}{\uP}
\multiput(850,323)(4,1.33){6}{\uP}
\multiput(870,330)(4,1.33){6}{\uP}
\multiput(889,337)(4,1.33){6}{\uP}
\multiput(909,344)(4,1.33){6}{\uP}
\multiput(929,350)(4,1.33){6}{\uP}
\multiput(948,357)(4,1.33){6}{\uP}
\multiput(968,364)(4,1.33){6}{\uP}
\multiput(987,371)(4,1.33){6}{\uP}
\multiput(1007,378)(4,1.33){6}{\uP}
\multiput(1026,385)(4,1.33){6}{\uP}
\multiput(1046,391)(4,1.33){6}{\uP}
\multiput(1065,398)(4,1.33){6}{\uP}
\multiput(1085,405)(4,1.33){6}{\uP}
\multiput(1104,412)(4,1.33){6}{\uP}
\multiput(1124,419)(4,1.33){6}{\uP}
\multiput(1143,426)(4,1.33){6}{\uP}
\multiput(1163,432)(4,1.33){6}{\uP}
\multiput(1183,439)(4,1.33){6}{\uP}
\multiput(1202,446)(4,1.33){6}{\uP}
\multiput(1222,453)(4,1.33){6}{\uP}
\multiput(1241,459)(4,1.33){6}{\uP}
\multiput(1261,467)(4,1.33){6}{\uP}
\multiput(1280,473)(4,1.33){6}{\uP}
\multiput(1300,480)(4,1.33){6}{\uP}
\multiput(1320,487)(4,1.33){6}{\uP}
\multiput(1339,494)(4,1.33){6}{\uP}
\multiput(1359,501)(4,1.33){6}{\uP}
\multiput(1378,507)(4,1.33){6}{\uP}
\multiput(1398,514)(4,1.33){4}{\uP}
\end{picture}
\end{center}
\caption{Interface fluctuations with average density 0.5, $r=0.35, 0.5$. The
solid lines have slope 1/3.}
\label{var35w-1}
\end{figure}
We examined two sets of systems, with blocking rates $r=0.35, 0.5$. Here the
lines drawn through the data points have slope 1/3, so that fluctuations are
suppressed compared with systems whose densities are different from 0.5,
yielding growth only like the cube root of the system size. (Note that
changing $r$ does not affect this conclusion; the relevant variable is whether
or not the density is 0.5.)
The reduction in the fluctuations of the interface when the average density is
1/2 is clearly caused by some type of cancellation since other fluctuations in
the system still scale like $L^{1/2}$---in figure~\ref{var35h} we plot the
standard deviation of the number of particles found to the left of the
interface, {\em i.e.}\ between site~1 and the second class particle;
\begin{figure}[hbt]
\begin{center}
% GNUPLOT: LaTeX picture
\setlength{\unitlength}{0.30pt}
\begin{picture}(1080,730)(250,36)
\put(264,140){\line(1,0){10}}
\put(1436,140){\line(-1,0){10}}
\put(264,186){\line(1,0){10}}
\put(1436,186){\line(-1,0){10}}
\put(264,223){\line(1,0){10}}
\put(1436,223){\line(-1,0){10}}
\put(264,255){\line(1,0){10}}
\put(1436,255){\line(-1,0){10}}
\put(264,282){\line(1,0){10}}
\put(1436,282){\line(-1,0){10}}
\put(264,307){\line(1,0){10}}
\put(1436,307){\line(-1,0){10}}
\put(264,328){\line(1,0){20}}
\put(1436,328){\line(-1,0){20}}
\put(242,328){\makebox(0,0)[r]{10}}
\put(264,470){\line(1,0){10}}
\put(1436,470){\line(-1,0){10}}
\put(264,553){\line(1,0){10}}
\put(1436,553){\line(-1,0){10}}
\put(264,612){\line(1,0){10}}
\put(1436,612){\line(-1,0){10}}
\put(264,658){\line(1,0){10}}
\put(1436,658){\line(-1,0){10}}
\put(242,658){\makebox(0,0)[r]{50}}
\put(264,695){\line(1,0){10}}
\put(1436,695){\line(-1,0){10}}
\put(264,727){\line(1,0){10}}
\put(1436,727){\line(-1,0){10}}
\put(346,113){\line(0,1){10}}
\put(346,741){\line(0,-1){10}}
\put(424,113){\line(0,1){10}}
\put(424,741){\line(0,-1){10}}
\put(479,113){\line(0,1){10}}
\put(479,741){\line(0,-1){10}}
\put(521,113){\line(0,1){10}}
\put(521,741){\line(0,-1){10}}
\put(556,113){\line(0,1){10}}
\put(556,741){\line(0,-1){10}}
\put(586,113){\line(0,1){10}}
\put(586,741){\line(0,-1){10}}
\put(611,113){\line(0,1){10}}
\put(611,741){\line(0,-1){10}}
\put(634,113){\line(0,1){10}}
\put(634,741){\line(0,-1){10}}
\put(654,113){\line(0,1){20}}
\put(654,741){\line(0,-1){20}}
\put(654,78){\makebox(0,0){1000}}
\put(786,113){\line(0,1){10}}
\put(786,741){\line(0,-1){10}}
\put(864,113){\line(0,1){10}}
\put(864,741){\line(0,-1){10}}
\put(919,113){\line(0,1){10}}
\put(919,741){\line(0,-1){10}}
\put(961,113){\line(0,1){10}}
\put(961,741){\line(0,-1){10}}
\put(996,113){\line(0,1){10}}
\put(996,741){\line(0,-1){10}}
\put(1026,113){\line(0,1){10}}
\put(1026,741){\line(0,-1){10}}
\put(1051,113){\line(0,1){10}}
\put(1051,741){\line(0,-1){10}}
\put(1074,113){\line(0,1){10}}
\put(1074,741){\line(0,-1){10}}
\put(1094,113){\line(0,1){20}}
\put(1094,741){\line(0,-1){20}}
\put(1094,78){\makebox(0,0){10000}}
\put(1226,113){\line(0,1){10}}
\put(1226,741){\line(0,-1){10}}
\put(1304,113){\line(0,1){10}}
\put(1304,741){\line(0,-1){10}}
\put(1359,113){\line(0,1){10}}
\put(1359,741){\line(0,-1){10}}
\put(1401,113){\line(0,1){10}}
\put(1401,741){\line(0,-1){10}}
\put(1436,113){\line(0,1){10}}
\put(1436,741){\line(0,-1){10}}
\put(264,113){\line(1,0){1172}}
\put(1436,113){\line(0,1){628}}
\put(1436,741){\line(-1,0){1172}}
\put(264,741){\line(0,-1){628}}
\put(850,33){\makebox(0,0){System Size}}
\put(276,144){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(278,145){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(280,146){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(282,147){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(283,148){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(285,149){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(286,150){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(289,151){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(291,152){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(293,153){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(295,154){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(297,155){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(299,156){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(301,157){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(303,158){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(305,159){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(307,160){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(309,161){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(311,162){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(313,163){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(315,164){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(317,165){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(319,166){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(321,167){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(323,168){\rule[-0.2pt]{0.7pt}{0.4pt}}
\put(325,169){\rule[-0.2pt]{0.7pt}{0.4pt}}
\put(328,170){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(330,171){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(332,172){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(334,173){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(336,174){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(338,175){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(340,176){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(342,177){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(344,178){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(346,179){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(348,180){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(350,181){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(352,182){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(354,183){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(356,184){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(358,185){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(360,186){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(362,187){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(364,188){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(365,189){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(367,190){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(368,191){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(371,192){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(373,193){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(375,194){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(377,195){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(379,196){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(381,197){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(384,198){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(387,199){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(389,200){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(391,201){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(393,202){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(395,203){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(397,204){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(399,205){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(401,206){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(403,207){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(405,208){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(407,209){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(409,210){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(411,211){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(412,212){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(414,213){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(415,214){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(418,215){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(420,216){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(422,217){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(424,218){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(426,219){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(428,220){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(430,221){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(432,222){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(434,223){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(436,224){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(438,225){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(440,226){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(442,227){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(444,228){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(446,229){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(449,230){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(452,231){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(453,232){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(455,233){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(456,234){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(459,235){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(461,236){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(463,237){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(465,238){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(467,239){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(469,240){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(471,241){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(473,242){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(475,243){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(477,244){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(479,245){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(481,246){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(483,247){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(485,248){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(487,249){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(489,250){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(491,251){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(493,252){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(494,253){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(496,254){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(497,255){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(500,256){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(502,257){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(504,258){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(506,259){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(508,260){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(510,261){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(513,262){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(516,263){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(518,264){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(520,265){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(522,266){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(524,267){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(526,268){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(528,269){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(530,270){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(532,271){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(534,272){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(535,273){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(537,274){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(539,275){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(541,276){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(543,277){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(545,278){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(547,279){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(549,280){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(551,281){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(553,282){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(555,283){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(557,284){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(559,285){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(561,286){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(563,287){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(565,288){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(567,289){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(569,290){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(572,291){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(575,292){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(576,293){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(578,294){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(580,295){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(582,296){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(584,297){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(586,298){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(588,299){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(590,300){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(592,301){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(594,302){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(596,303){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(598,304){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(600,305){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(602,306){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(604,307){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(606,308){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(608,309){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(610,310){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(612,311){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(614,312){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(616,313){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(617,314){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(619,315){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(621,316){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(623,317){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(625,318){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(627,319){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(629,320){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(631,321){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(633,322){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(636,323){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(639,324){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(641,325){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(643,326){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(645,327){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(647,328){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(649,329){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(651,330){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(653,331){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(655,332){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(657,333){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(658,334){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(660,335){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(662,336){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(664,337){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(666,338){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(668,339){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(670,340){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(672,341){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(674,342){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(676,343){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(678,344){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(680,345){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(682,346){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(684,347){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(686,348){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(688,349){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(690,350){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(692,351){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(695,352){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(698,353){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(700,354){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(702,355){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(704,356){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(705,357){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(707,358){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(709,359){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(711,360){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(713,361){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(715,362){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(717,363){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(719,364){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(721,365){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(723,366){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(725,367){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(727,368){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(729,369){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(731,370){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(733,371){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(735,372){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(737,373){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(739,374){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(741,375){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(743,376){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(745,377){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(746,378){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(748,379){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(750,380){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(752,381){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(754,382){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(756,383){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(759,384){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(762,385){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(764,386){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(766,387){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(768,388){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(770,389){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(772,390){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(774,391){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(776,392){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(778,393){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(780,394){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(782,395){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(784,396){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(786,397){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(787,398){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(789,399){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(791,400){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(793,401){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(795,402){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(797,403){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(799,404){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(801,405){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(803,406){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(805,407){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(807,408){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(809,409){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(811,410){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(813,411){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(815,412){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(817,413){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(819,414){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(821,415){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(824,416){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(827,417){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(828,418){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(830,419){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(832,420){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(834,421){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(836,422){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(838,423){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(840,424){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(842,425){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(844,426){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(846,427){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(848,428){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(850,429){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(852,430){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(854,431){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(856,432){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(858,433){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(860,434){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(862,435){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(864,436){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(866,437){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(868,438){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(869,439){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(871,440){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(873,441){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(875,442){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(877,443){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(879,444){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(882,445){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(885,446){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(887,447){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(889,448){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(891,449){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(893,450){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(895,451){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(897,452){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(899,453){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(901,454){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(903,455){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(905,456){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(907,457){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(909,458){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(910,459){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(912,460){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(914,461){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(916,462){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(918,463){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(920,464){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(922,465){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(924,466){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(926,467){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(928,468){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(930,469){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(932,470){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(934,471){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(936,472){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(938,473){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(940,474){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(942,475){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(944,476){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(947,477){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(950,478){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(951,479){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(953,480){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(955,481){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(957,482){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(959,483){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(961,484){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(963,485){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(965,486){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(967,487){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(969,488){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(971,489){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(973,490){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(975,491){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(977,492){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(979,493){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(981,494){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(983,495){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(985,496){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(987,497){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(989,498){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(991,499){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(993,500){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(995,501){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(997,502){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(998,503){\rule[-0.2pt]{0.44pt}{0.4pt}}
\put(1000,504){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1002,505){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1005,506){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1008,507){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1010,508){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1012,509){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1014,510){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1016,511){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1018,512){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1020,513){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1022,514){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1024,515){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1026,516){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1028,517){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1030,518){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1032,519){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1034,520){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1036,521){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1038,522){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1039,523){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1041,524){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1042,525){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1045,526){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1047,527){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1049,528){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1051,529){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1053,530){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1055,531){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1057,532){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1059,533){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1061,534){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1063,535){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1065,536){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1067,537){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1070,538){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1073,539){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1075,540){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1077,541){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1079,542){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1080,543){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1082,544){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1083,545){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1086,546){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1088,547){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1090,548){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1092,549){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1094,550){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1096,551){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1098,552){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1100,553){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1102,554){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1104,555){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1106,556){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1108,557){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1110,558){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1112,559){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1114,560){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1116,561){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1118,562){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1120,563){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1121,564){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1123,565){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1124,566){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1127,567){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1129,568){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1131,569){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1134,570){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1137,571){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1139,572){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1141,573){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1143,574){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1145,575){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1147,576){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1149,577){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1151,578){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1153,579){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1155,580){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1157,581){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1159,582){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1161,583){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1162,584){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1164,585){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1165,586){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1168,587){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1170,588){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1172,589){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1174,590){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1176,591){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1178,592){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1180,593){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1182,594){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1184,595){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1186,596){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1188,597){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1190,598){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1193,599){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1196,600){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1198,601){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1200,602){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1202,603){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1203,604){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1205,605){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1206,606){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1209,607){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1211,608){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1213,609){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1215,610){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1217,611){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1219,612){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1221,613){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1223,614){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1225,615){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1227,616){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1229,617){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1231,618){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1233,619){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1235,620){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1237,621){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1239,622){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1241,623){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1243,624){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1244,625){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1246,626){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1247,627){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1250,628){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1252,629){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1254,630){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1257,631){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1260,632){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1262,633){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1264,634){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1266,635){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1268,636){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1270,637){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1272,638){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1274,639){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1276,640){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1278,641){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1280,642){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1282,643){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1284,644){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1285,645){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1287,646){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1288,647){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1291,648){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1293,649){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1295,650){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1297,651){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1299,652){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1301,653){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1303,654){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1305,655){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1307,656){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1309,657){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1311,658){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1313,659){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1316,660){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1319,661){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1321,662){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1323,663){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1325,664){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1327,665){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1329,666){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1331,667){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1332,668){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1334,669){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1335,670){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1338,671){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1340,672){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1342,673){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1344,674){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1346,675){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1348,676){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1350,677){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1352,678){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1354,679){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1356,680){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1358,681){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1360,682){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1362,683){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1364,684){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1366,685){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1368,686){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1370,687){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1372,688){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1373,689){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1375,690){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1376,691){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1380,692){\rule[-0.2pt]{0.77pt}{0.4pt}}
\put(1383,693){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1385,694){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1387,695){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1389,696){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1391,697){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1393,698){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1395,699){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1397,700){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1399,701){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1401,702){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1403,703){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1405,704){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1407,705){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1409,706){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1411,707){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1413,708){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1414,709){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1416,710){\rule[-0.2pt]{0.45pt}{0.4pt}}
\put(1417,711){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1420,712){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1422,713){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1424,714){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(1426,715){\rule[-0.2pt]{0.5pt}{0.4pt}}
\put(291,158){\makebox(0,0){$\diamond$}}
\put(291,160){\makebox(0,0){$\diamond$}}
\put(291,162){\makebox(0,0){$\diamond$}}
\put(424,218){\makebox(0,0){$\diamond$}}
\put(424,220){\makebox(0,0){$\diamond$}}
\put(424,222){\makebox(0,0){$\diamond$}}
\put(556,279){\makebox(0,0){$\diamond$}}
\put(556,282){\makebox(0,0){$\diamond$}}
\put(556,284){\makebox(0,0){$\diamond$}}
\put(731,360){\makebox(0,0){$\diamond$}}
\put(731,363){\makebox(0,0){$\diamond$}}
\put(731,366){\makebox(0,0){$\diamond$}}
\put(864,436){\makebox(0,0){$\diamond$}}
\put(864,439){\makebox(0,0){$\diamond$}}
\put(864,442){\makebox(0,0){$\diamond$}}
\put(1026,517){\makebox(0,0){$\diamond$}}
\put(1026,520){\makebox(0,0){$\diamond$}}
\put(1026,523){\makebox(0,0){$\diamond$}}
\put(1183,588){\makebox(0,0){$\diamond$}}
\put(1183,592){\makebox(0,0){$\diamond$}}
\put(1183,596){\makebox(0,0){$\diamond$}}
\put(1316,620){\makebox(0,0){$\diamond$}}
\put(1316,626){\makebox(0,0){$\diamond$}}
\put(1316,631){\makebox(0,0){$\diamond$}}
\put(1393,669){\makebox(0,0){$\diamond$}}
\put(1393,675){\makebox(0,0){$\diamond$}}
\put(1393,681){\makebox(0,0){$\diamond$}}
\end{picture}
\end{center}
\caption{Fluctuations in the number of particles to the left of the interface:
average density 0.5, $r=0.35$. The solid line, which is a best linear fit to
the data, has slope 0.46.}
\label{var35h}
\end{figure}
this data is consistent with scaling exponent 1/2 (and inconsistent with 1/3).
An explanation of this behavior is given in the next section.
\section{Analysis}
To make sense of our results, it is convenient to decompose the stochastic
nature of our system into two pieces: the random flow of particles through the
blockage, and the random dynamics everywhere else. We hypothesize that the
``blockage randomness'' is responsible for the $L^{1/2}$ behavior and that the
``dynamical randomness'' is responsible for the $L^{1/3}$ behavior.
A particle traversing the blockage can be thought of as two simultaneous
events: a particle being released at site~$1$ while a hole is released at
site~$L$. These two events are of course completely correlated, but the
motions of the particle and hole {\em after} release are nearly independent.
The position of the interface is determined by the {\em difference} between
the number of particles which arrive from the left and the number of holes
which come from the right. (It may be convenient for visualization purposes
to consider the situation where the density to the right of the shock is very
close to one and the density on the left is near zero.) If we neglect the
dynamical randomness---{\em i.e.}\ if we assume that particles in the low
density region move at their average velocity $1-{\rho_{\rm low}}$, and holes
in the high density region move at their average velocity $-{\rho_{\rm high}}
= -(1-{\rho_{\rm low}})$, and consider a shock in the center of the system,
then no matter what the nature of the creation of excitations at the blockage,
particles and holes will always impinge upon the shock in pairs: when a pair
of excitations is created, the hole and particle will travel with equal speeds
and opposite directions, meeting (and annihilating) when they have each
traveled a distance $L/2$ and reach the shock position. Since the ``blockage
randomness'' plays no role here, a system with density~0.5, and therefore
particle-hole symmetry (figure~\ref{var35w-1}), has fluctuations determined
only by the ``dynamical randomness''---the deviation of the particle
velocities from their average.
On the other hand, if the shock is not in the center of the system, the nature
of ``blockage randomness'' becomes relevant as the times for particles and
holes to reach the interface are no longer the same. For example, if the
average density is greater than 1/2 the (average) shock position will be less
than $L/2$, and particles which start at the blockage and travel to the right
will reach the shock before holes (starting at the same time and place)
traveling to the left---the arrival times of the particles no longer match
those of the holes because the distances they need traverse are different, and
the resulting motion of the shock position will depend upon exactly how the
excitations are created. So a system with density different from 0.5
(figure~\ref{var55}) and thus with an average shock position shifted from the
center will have fluctuations from the ``blockage randomness'' proportional to
the amount of the shift.
The $L^{1/2}$ scaling of the ``blockage randomness'' results from the rapid
decay of correlations in both time and space in the stationary state of the
ASEP, as well as the use of an independent random choice for each attempt to
cross the blockage. Thus the creation of one excitation is very nearly
independent from the creation of another. Therefore the fluctuation in the
number of excitations will scale as the square root of the average number.
There is a finite density of excitations, so their number is $O(L)$, yielding
fluctuations that scale like $O(L^{1/2})$.
In fact, if we neglect the dynamical randomness and treat particle flow
through the blockage as a Poisson process, we can determine through
straightforward analysis the variance of the shock position.
Each particle or hole added to the shock moves it by
$(\rho_{\rm high} -\rho_{\rm low})^{-1}$. Thus the shock is shifted by an
amount $|\rho_{\rm avg} - 1/2| L/(\rho_{\rm high} -\rho_{\rm low})$,
which is half the difference in path length between particles and holes.
Therefore the time differential is $2|\rho_{\rm avg} - 1/2| L\big/
\left(\rho_{\rm high}(\rho_{\rm high} - \rho_{\rm low})\right)$. The
probability of the formation of an excitation is $\rho_{\rm high}\rho_{\rm
low}$ so that the standard deviation of the shock position should be
\begin{equation}\label{STDSP}
(\rho_{\rm high} -\rho_{\rm low})^{-1}\left(
\frac{2|\rho_{\rm avg} - 1/2| L}{\rho_{\rm high}(\rho_{\rm high} -
\rho_{\rm low})}
\rho_{\rm high}\rho_{\rm low}\right)^{1/2} =
\left(\frac{2{\rho_{\rm low}} |\rho_{\rm avg} - 1/2| L}{({\rho_{\rm high}} -
{\rho_{\rm low}})^3}\right)^{1/2}.
\end{equation}
This calculation should be accurate for $r$ near zero but will be too large
otherwise, as the actual variance of the particle flow is less than that of a
Poisson process since we do have (small) memory effects. For the system of
figure~\ref{var55}, the measured behavior is $0.50L^{1/2}$ while
eq.~(\ref{STDSP}) yields $0.63L^{1/2}$.
What is the nature of the randomness in the dynamics? Recent work by van
Beijeren~\cite{vB} indicates that density fluctuations grow in time like
$t^{1/3}$, while since the distance is of order $L$ and the speed is $O(1)$
the time required for an excitation to reach the interface from the blockage
is of order $L$. Thus an initially nonrandom arrangement (or a random
arrangement with number particles = number holes) of excitations will have
fluctuations of order $L^{1/3}$ by the time it reaches the interface. We
confirm this by examining a system where we alter the behavior near the
blockage so that particles pass through the blockage in a near-uniform manner.
Of course we can not have a completely uniform current since the occupation
numbers of each site are limited to 0 and 1, but we try to make the flow as
uniform and non-random as possible: if $n(t)$ is the number of particles
already having passed through the blockage at time $t$, we specify the current
exactly by opening the blockage if $n(t) < Jt$ and closing it otherwise. If a
particle is immediately to the left of the blockage while it is open and the
site to the right is vacant, the particle jumps through the blockage. Thus we
attempt to schedule particle jumps at intervals of $1/J$; if a jump is not
possible because the appropriate occupancy conditions are not satisfied we
``make up'' that jump as soon as possible.
By altering the nature of the blockage, we alter the behavior of the shock
fluctuations. For this uniform-block model, the lack of ``blockage
randomness'' means that even without particle-hole symmetry, {\em i.e.}\ even
when the shock is displaced from the center we have fluctuations that scale
like $L^{1/3}$ for {\em both} the interface position {\em and} the number of
particles to the left of the interface---see figure~\ref{varu}. Thus we have
(partial) confirmation of our hypothesis.
\begin{figure}[htb]
\begin{center}
% GNUPLOT: LaTeX picture
\setlength{\unitlength}{0.32pt}
\begin{picture}(1290,800)(165,70)
\newcommand{\mD}{\makebox(0,0){$\diamond$}}
\newcommand{\mC}{\makebox(0,0){$\circ$}}
\newcommand{\uP}{\usebox{\plotpoint}}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\put(264,228){\line(1,0){10}}
\put(1436,228){\line(-1,0){10}}
\put(264,310){\line(1,0){10}}
\put(1436,310){\line(-1,0){10}}
\put(264,374){\line(1,0){10}}
\put(1436,374){\line(-1,0){10}}
\put(242,374){\makebox(0,0)[r]{5}}
\put(264,426){\line(1,0){10}}
\put(1436,426){\line(-1,0){10}}
\put(264,470){\line(1,0){10}}
\put(1436,470){\line(-1,0){10}}
\put(264,508){\line(1,0){10}}
\put(1436,508){\line(-1,0){10}}
\put(264,541){\line(1,0){10}}
\put(1436,541){\line(-1,0){10}}
\put(264,571){\line(1,0){20}}
\put(1436,571){\line(-1,0){20}}
\put(242,571){\makebox(0,0)[r]{10}}
\put(264,768){\line(1,0){10}}
\put(1436,768){\line(-1,0){10}}
\put(264,163){\line(0,1){10}}
\put(264,832){\line(0,-1){10}}
\put(367,163){\line(0,1){10}}
\put(367,832){\line(0,-1){10}}
\put(446,163){\line(0,1){10}}
\put(446,832){\line(0,-1){10}}
\put(446,118){\makebox(0,0){500}}
\put(511,163){\line(0,1){10}}
\put(511,832){\line(0,-1){10}}
\put(566,163){\line(0,1){10}}
\put(566,832){\line(0,-1){10}}
\put(614,163){\line(0,1){10}}
\put(614,832){\line(0,-1){10}}
\put(656,163){\line(0,1){10}}
\put(656,832){\line(0,-1){10}}
\put(694,163){\line(0,1){20}}
\put(694,832){\line(0,-1){20}}
\put(694,118){\makebox(0,0){1000}}
\put(941,163){\line(0,1){10}}
\put(941,832){\line(0,-1){10}}
\put(1086,163){\line(0,1){10}}
\put(1086,832){\line(0,-1){10}}
\put(1189,163){\line(0,1){10}}
\put(1189,832){\line(0,-1){10}}
\put(1268,163){\line(0,1){10}}
\put(1268,832){\line(0,-1){10}}
\put(1268,118){\makebox(0,0){5000}}
\put(1333,163){\line(0,1){10}}
\put(1333,832){\line(0,-1){10}}
\put(1388,163){\line(0,1){10}}
\put(1388,832){\line(0,-1){10}}
\put(1436,163){\line(0,1){10}}
\put(1436,832){\line(0,-1){10}}
\put(264,163){\line(1,0){1172}}
\put(1436,163){\line(0,1){669}}
\put(1436,832){\line(-1,0){1172}}
\put(264,832){\line(0,-1){669}}
\put(850,73){\makebox(0,0){System Size}}
\put(1290,777){\makebox(0,0)[r]{interface position std.~dev.}}
\put(1312,777){\mD}
\put(367,437){\mD}
\put(367,441){\mD}
\put(367,444){\mD}
\put(614,504){\mD}
\put(614,507){\mD}
\put(614,510){\mD}
\put(838,565){\mD}
\put(838,569){\mD}
\put(838,574){\mD}
\put(1333,695){\mD}
\put(1333,702){\mD}
\put(1333,708){\mD}
\put(367,247){\mC}
\put(367,251){\mC}
\put(367,254){\mC}
\put(614,317){\mC}
\put(614,320){\mC}
\put(614,322){\mC}
\put(838,376){\mC}
\put(838,381){\mC}
\put(838,386){\mC}
\put(1333,506){\mC}
\put(1333,512){\mC}
\put(1333,518){\mC}
\sbox{\plotpoint}{\rule[-0.125pt]{1.33pt}{.25pt}}
\put(1290,732){\makebox(0,0)[r]{particles before shock position std.~dev.}}
\put(1312,732){\mC}
\put(286,232){\uP}
\put(289,233){\uP}
\put(293,234){\uP}
\put(297,235){\uP}
\put(301,236){\uP}
\put(304,237){\uP}
\put(308,238){\uP}
\put(311,239){\uP}
\put(315,240){\uP}
\put(319,241){\uP}
\put(323,242){\uP}
\put(326,243){\uP}
\put(330,244){\uP}
\put(333,245){\uP}
\put(337,246){\uP}
\put(341,247){\uP}
\put(345,248){\uP}
\put(348,249){\uP}
\put(352,250){\uP}
\put(355,251){\uP}
\put(359,252){\uP}
\put(363,253){\uP}
\put(367,254){\uP}
\put(370,255){\uP}
\put(374,256){\uP}
\put(377,257){\uP}
\put(381,258){\uP}
\put(385,259){\uP}
\put(389,260){\rule[-0.125pt]{1.69pt}{.250pt}}
\put(396,261){\uP}
\put(399,262){\uP}
\put(403,263){\uP}
\put(407,264){\uP}
\put(411,265){\uP}
\put(414,266){\uP}
\put(418,267){\uP}
\put(421,268){\uP}
\put(425,269){\uP}
\put(428,270){\uP}
\put(432,271){\uP}
\put(436,272){\uP}
\put(440,273){\uP}
\put(443,274){\uP}
\put(447,275){\uP}
\put(450,276){\uP}
\put(454,277){\uP}
\put(458,278){\uP}
\put(462,279){\uP}
\put(465,280){\uP}
\put(469,281){\uP}
\put(472,282){\uP}
\put(476,283){\uP}
\put(480,284){\uP}
\put(484,285){\uP}
\put(487,286){\uP}
\put(491,287){\uP}
\put(494,288){\uP}
\put(498,289){\uP}
\put(502,290){\uP}
\put(506,291){\uP}
\put(509,292){\uP}
\put(513,293){\uP}
\put(516,294){\uP}
\put(520,295){\uP}
\put(524,296){\uP}
\put(528,297){\rule[-0.125pt]{1.69pt}{.250pt}}
\put(535,298){\uP}
\put(538,299){\uP}
\put(542,300){\uP}
\put(546,301){\uP}
\put(550,302){\uP}
\put(553,303){\uP}
\put(557,304){\uP}
\put(560,305){\uP}
\put(564,306){\uP}
\put(568,307){\uP}
\put(572,308){\uP}
\put(575,309){\uP}
\put(579,310){\uP}
\put(582,311){\uP}
\put(586,312){\uP}
\put(590,313){\uP}
\put(594,314){\uP}
\put(597,315){\uP}
\put(601,316){\uP}
\put(604,317){\uP}
\put(608,318){\uP}
\put(612,319){\uP}
\put(616,320){\uP}
\put(619,321){\uP}
\put(623,322){\uP}
\put(626,323){\uP}
\put(630,324){\uP}
\put(634,325){\uP}
\put(638,326){\uP}
\put(641,327){\uP}
\put(645,328){\uP}
\put(648,329){\uP}
\put(652,330){\uP}
\put(656,331){\uP}
\put(660,332){\uP}
\put(663,333){\uP}
\put(667,334){\rule[-0.125pt]{1.69pt}{.250pt}}
\put(674,335){\uP}
\put(678,336){\uP}
\put(682,337){\uP}
\put(685,338){\uP}
\put(689,339){\uP}
\put(692,340){\uP}
\put(696,341){\uP}
\put(700,342){\uP}
\put(704,343){\uP}
\put(707,344){\uP}
\put(711,345){\uP}
\put(714,346){\uP}
\put(718,347){\uP}
\put(721,348){\uP}
\put(725,349){\uP}
\put(729,350){\uP}
\put(733,351){\uP}
\put(736,352){\uP}
\put(740,353){\uP}
\put(743,354){\uP}
\put(747,355){\uP}
\put(751,356){\uP}
\put(755,357){\uP}
\put(758,358){\uP}
\put(762,359){\uP}
\put(765,360){\uP}
\put(769,361){\uP}
\put(773,362){\uP}
\put(777,363){\uP}
\put(780,364){\uP}
\put(784,365){\uP}
\put(787,366){\uP}
\put(791,367){\uP}
\put(795,368){\uP}
\put(799,369){\uP}
\put(802,370){\uP}
\put(806,371){\rule[-0.125pt]{1.69pt}{.25pt}}
\put(813,372){\uP}
\put(817,373){\uP}
\put(821,374){\uP}
\put(824,375){\uP}
\put(828,376){\uP}
\put(831,377){\uP}
\put(835,378){\uP}
\put(839,379){\uP}
\put(843,380){\uP}
\put(846,381){\uP}
\put(850,382){\uP}
\put(853,383){\uP}
\put(857,384){\uP}
\put(861,385){\uP}
\put(865,386){\uP}
\put(868,387){\uP}
\put(872,388){\uP}
\put(875,389){\uP}
\put(879,390){\uP}
\put(883,391){\uP}
\put(887,392){\uP}
\put(890,393){\uP}
\put(894,394){\uP}
\put(897,395){\uP}
\put(901,396){\uP}
\put(905,397){\uP}
\put(909,398){\uP}
\put(912,399){\uP}
\put(916,400){\uP}
\put(919,401){\uP}
\put(923,402){\uP}
\put(927,403){\uP}
\put(931,404){\uP}
\put(934,405){\uP}
\put(938,406){\uP}
\put(941,407){\uP}
\put(945,408){\uP}
\put(949,408.5){\uP}
\put(953,409){\uP}
\put(956,410){\uP}
\put(960,411){\uP}
\put(963,412){\uP}
\put(967,413){\uP}
\put(971,414){\uP}
\put(975,415){\uP}
\put(978,416){\uP}
\put(982,417){\uP}
\put(985,418){\uP}
\put(989,419){\uP}
\put(993,420){\uP}
\put(997,421){\uP}
\put(1000,422){\uP}
\put(1004,423){\uP}
\put(1007,424){\uP}
\put(1011,425){\uP}
\put(1014,426){\uP}
\put(1018,427){\uP}
\put(1022,428){\uP}
\put(1026,429){\uP}
\put(1029,430){\uP}
\put(1033,431){\uP}
\put(1036,432){\uP}
\put(1040,433){\uP}
\put(1044,434){\uP}
\put(1048,435){\uP}
\put(1051,436){\uP}
\put(1055,437){\uP}
\put(1058,438){\uP}
\put(1062,439){\uP}
\put(1066,440){\uP}
\put(1070,441){\uP}
\put(1073,442){\uP}
\put(1077,443){\uP}
\put(1080,444){\uP}
\put(1084,445){\uP}
\put(1088,445.5){\uP}
\put(1092,446){\uP}
\put(1095,447){\uP}
\put(1099,448){\uP}
\put(1102,449){\uP}
\put(1106,450){\uP}
\put(1110,451){\uP}
\put(1114,452){\uP}
\put(1117,453){\uP}
\put(1121,454){\uP}
\put(1124,455){\uP}
\put(1128,456){\uP}
\put(1132,457){\uP}
\put(1136,458){\uP}
\put(1139,459){\uP}
\put(1143,460){\uP}
\put(1146,461){\uP}
\put(1150,462){\uP}
\put(1154,463){\uP}
\put(1158,464){\uP}
\put(1161,465){\uP}
\put(1165,466){\uP}
\put(1168,467){\uP}
\put(1172,468){\uP}
\put(1176,469){\uP}
\put(1180,470){\uP}
\put(1183,471){\uP}
\put(1187,472){\uP}
\put(1190,473){\uP}
\put(1194,474){\uP}
\put(1198,475){\uP}
\put(1202,476){\uP}
\put(1205,477){\uP}
\put(1209,478){\uP}
\put(1212,479){\uP}
\put(1216,480){\uP}
\put(1220,481){\uP}
\put(1224,482){\rule[-0.125pt]{1.69pt}{.250pt}}
\put(1231,483){\uP}
\put(1234,484){\uP}
\put(1238,485){\uP}
\put(1242,486){\uP}
\put(1246,487){\uP}
\put(1249,488){\uP}
\put(1253,489){\uP}
\put(1256,490){\uP}
\put(1260,491){\uP}
\put(1264,492){\uP}
\put(1268,493){\uP}
\put(1271,494){\uP}
\put(1275,495){\uP}
\put(1278,496){\uP}
\put(1282,497){\uP}
\put(1286,498){\uP}
\put(1290,499){\uP}
\put(1293,500){\uP}
\put(1297,501){\uP}
\put(1300,502){\uP}
\put(1304,503){\uP}
\put(1307,504){\uP}
\put(1311,505){\uP}
\put(1315,506){\uP}
\put(1319,507){\uP}
\put(1322,508){\uP}
\put(1326,509){\uP}
\put(1329,510){\uP}
\put(1333,511){\uP}
\put(1337,512){\uP}
\put(1341,513){\uP}
\put(1344,514){\uP}
\put(1348,515){\uP}
\put(1351,516){\uP}
\put(1355,517){\uP}
\put(1359,518){\uP}
\put(1363,519){\rule[-0.125pt]{1.69pt}{.25pt}}
\put(1370,520){\uP}
\put(1373,521){\uP}
\put(1377,522){\uP}
\put(1381,523){\uP}
\put(1385,524){\uP}
\put(1388,525){\uP}
\put(1392,526){\uP}
\put(1395,527){\uP}
\put(1399,528){\uP}
\put(1403,529){\uP}
\put(1407,530){\uP}
\put(1410,531){\uP}
\put(1414,532){\uP}
\put(282,419){\uP}
\put(286,420){\uP}
\put(289,421){\uP}
\put(293,422){\uP}
\put(297,423){\uP}
\put(301,424){\uP}
\put(304,425){\uP}
\put(308,426){\uP}
\put(311,427){\uP}
\put(315,428){\uP}
\put(319,429){\uP}
\put(323,430){\uP}
\put(326,431){\uP}
\put(330,432){\uP}
\put(337,433){\uP}
\put(341,434){\uP}
\put(345,435){\uP}
\put(348,436){\uP}
\put(352,437){\uP}
\put(355,438){\uP}
\put(359,439){\uP}
\put(363,440){\uP}
\put(367,441){\uP}
\put(370,442){\uP}
\put(374,443){\uP}
\put(377,444){\uP}
\put(381,445){\uP}
\put(385,446){\uP}
\put(389,447){\uP}
\put(392,448){\uP}
\put(396,449){\uP}
\put(399,450){\uP}
\put(403,451){\uP}
\put(407,452){\uP}
\put(411,453){\uP}
\put(414,454){\uP}
\put(418,455){\uP}
\put(421,456){\uP}
\put(425,457){\uP}
\put(428,458){\uP}
\put(432,459){\uP}
\put(436,460){\uP}
\put(440,461){\uP}
\put(443,462){\uP}
\put(447,463){\uP}
\put(450,464){\uP}
\put(454,465){\uP}
\put(458,466){\uP}
\put(462,467){\uP}
\put(465,468){\uP}
\put(469,469){\uP}
\put(472.5,469.5){\uP}
\put(476,470){\uP}
\put(480,471){\uP}
\put(484,472){\uP}
\put(487,473){\uP}
\put(491,474){\uP}
\put(494,475){\uP}
\put(498,476){\uP}
\put(502,477){\uP}
\put(506,478){\uP}
\put(509,479){\uP}
\put(513,480){\uP}
\put(516,481){\uP}
\put(520,482){\uP}
\put(524,483){\uP}
\put(528,484){\uP}
\put(531,485){\uP}
\put(535,486){\uP}
\put(538,487){\uP}
\put(542,488){\uP}
\put(546,489){\uP}
\put(550,490){\uP}
\put(553,491){\uP}
\put(557,492){\uP}
\put(560,493){\uP}
\put(564,494){\uP}
\put(568,495){\uP}
\put(572,496){\uP}
\put(575,497){\uP}
\put(579,498){\uP}
\put(582,499){\uP}
\put(586,500){\uP}
\put(590,501){\uP}
\put(594,502){\uP}
\put(597,503){\uP}
\put(601,504){\uP}
\put(604,505){\uP}
\put(608,506){\uP}
\put(612,506.5){\uP}
\put(616,507){\uP}
\put(619,508){\uP}
\put(623,509){\uP}
\put(626,510){\uP}
\put(630,511){\uP}
\put(634,512){\uP}
\put(638,513){\uP}
\put(641,514){\uP}
\put(645,515){\uP}
\put(648,516){\uP}
\put(652,517){\uP}
\put(656,518){\uP}
\put(660,519){\uP}
\put(663,520){\uP}
\put(667,521){\uP}
\put(670,522){\uP}
\put(674,523){\uP}
\put(678,524){\uP}
\put(682,525){\uP}
\put(685,526){\uP}
\put(689,527){\uP}
\put(692,528){\uP}
\put(696,529){\uP}
\put(700,530){\uP}
\put(704,531){\uP}
\put(707,532){\uP}
\put(711,533){\uP}
\put(714,534){\uP}
\put(718,535){\uP}
\put(721,536){\uP}
\put(725,537){\uP}
\put(729,538){\uP}
\put(733,539){\uP}
\put(736,540){\uP}
\put(740,541){\uP}
\put(743.5,541.5){\uP}
\put(747,542){\uP}
\put(751,543){\uP}
\put(755,544){\uP}
\put(758,545){\uP}
\put(762,546){\uP}
\put(765,547){\uP}
\put(769,548){\uP}
\put(773,549){\uP}
\put(777,550){\uP}
\put(780,551){\uP}
\put(784,552){\uP}
\put(787,553){\uP}
\put(791,554){\uP}
\put(795,555){\uP}
\put(799,556){\uP}
\put(802,557){\uP}
\put(806,558){\uP}
\put(809,559){\uP}
\put(813,560){\uP}
\put(817,561){\uP}
\put(821,562){\uP}
\put(824,563){\uP}
\put(828,564){\uP}
\put(831,565){\uP}
\put(835,566){\uP}
\put(839,567){\uP}
\put(843,568){\uP}
\put(846,569){\uP}
\put(850,570){\uP}
\put(853,571){\uP}
\put(857,572){\uP}
\put(861,573){\uP}
\put(865,574){\uP}
\put(868,575){\uP}
\put(872,576){\uP}
\put(875,577){\uP}
\put(879,578){\uP}
\put(883,578.5){\uP}
\put(887,579){\uP}
\put(890,580){\uP}
\put(894,581){\uP}
\put(897,582){\uP}
\put(901,583){\uP}
\put(905,584){\uP}
\put(909,585){\uP}
\put(912,586){\uP}
\put(916,587){\uP}
\put(919,588){\uP}
\put(923,589){\uP}
\put(927,590){\uP}
\put(931,591){\uP}
\put(934,592){\uP}
\put(938,593){\uP}
\put(941,594){\uP}
\put(945,595){\uP}
\put(949,596){\uP}
\put(953,597){\uP}
\put(956,598){\uP}
\put(960,599){\uP}
\put(963,600){\uP}
\put(967,601){\uP}
\put(971,602){\uP}
\put(975,603){\uP}
\put(978,604){\uP}
\put(982,605){\uP}
\put(985,606){\uP}
\put(989,607){\uP}
\put(993,608){\uP}
\put(997,609){\uP}
\put(1000,610){\uP}
\put(1004,611){\uP}
\put(1007,612){\uP}
\put(1011,613){\uP}
\put(1014,614){\uP}
\put(1018,615){\uP}
\put(1022,615.5){\uP}
\put(1026,616){\uP}
\put(1029,617){\uP}
\put(1033,618){\uP}
\put(1036,619){\uP}
\put(1040,620){\uP}
\put(1044,621){\uP}
\put(1048,622){\uP}
\put(1051,623){\uP}
\put(1055,624){\uP}
\put(1058,625){\uP}
\put(1062,626){\uP}
\put(1066,627){\uP}
\put(1070,628){\uP}
\put(1073,629){\uP}
\put(1077,630){\uP}
\put(1080,631){\uP}
\put(1084,632){\uP}
\put(1088,633){\uP}
\put(1092,634){\uP}
\put(1095,635){\uP}
\put(1099,636){\uP}
\put(1102,637){\uP}
\put(1106,638){\uP}
\put(1110,639){\uP}
\put(1114,640){\uP}
\put(1117,641){\uP}
\put(1121,642){\uP}
\put(1124,643){\uP}
\put(1128,644){\uP}
\put(1132,645){\uP}
\put(1136,646){\uP}
\put(1139,647){\uP}
\put(1143,648){\uP}
\put(1146,649){\uP}
\put(1150,650){\uP}
\put(1154,651){\uP}
\put(1158,652){\uP}
\put(1161.5,652.5){\uP}
\put(1165,653){\uP}
\put(1168,654){\uP}
\put(1172,655){\uP}
\put(1176,656){\uP}
\put(1180,657){\uP}
\put(1183,658){\uP}
\put(1187,659){\uP}
\put(1190,660){\uP}
\put(1194,661){\uP}
\put(1198,662){\uP}
\put(1202,663){\uP}
\put(1205,664){\uP}
\put(1209,665){\uP}
\put(1212,666){\uP}
\put(1216,667){\uP}
\put(1220,668){\uP}
\put(1224,669){\uP}
\put(1227,670){\uP}
\put(1231,671){\uP}
\put(1234,672){\uP}
\put(1238,673){\uP}
\put(1242,674){\uP}
\put(1246,675){\uP}
\put(1249,676){\uP}
\put(1253,677){\uP}
\put(1256,678){\uP}
\put(1260,679){\uP}
\put(1264,680){\uP}
\put(1268,681){\uP}
\put(1271,682){\uP}
\put(1275,683){\uP}
\put(1278,684){\uP}
\put(1282,685){\uP}
\put(1286,686){\uP}
\put(1290,687){\uP}
\put(1293,688){\uP}
\put(1297,689){\uP}
\put(1300.5,689.5){\uP}
\put(1304,690){\uP}
\put(1307,691){\uP}
\put(1311,692){\uP}
\put(1315,693){\uP}
\put(1319,694){\uP}
\put(1322,695){\uP}
\put(1326,696){\uP}
\put(1329,697){\uP}
\put(1333,698){\uP}
\put(1337,699){\uP}
\put(1341,700){\uP}
\put(1344,701){\uP}
\put(1348,702){\uP}
\put(1351,703){\uP}
\put(1355,704){\uP}
\put(1359,705){\uP}
\put(1363,706){\uP}
\put(1366,707){\uP}
\put(1370,708){\uP}
\put(1373,709){\uP}
\put(1377,710){\uP}
\put(1381,711){\uP}
\put(1385,712){\uP}
\put(1388,713){\uP}
\put(1392,714){\uP}
\put(1395,715){\uP}
\put(1399,716){\uP}
\put(1403,717){\uP}
\put(1407,718){\uP}
\put(1410,719){\uP}
\put(1414,720){\uP}
\end{picture}
\end{center}
\caption{Fluctuations with uniform (non-random) flow through the block.
Average density=0.55. Solid lines have slope 1/3.}\label{varu}
\end{figure}
\section{Infinite System with Block}\label{infinite}
We have been considering particles evolving according to the ($p=1$) ASEP
dynamics on a circle of $L$ sites with a blockage parameterized by $r$. For
each such system with a fixed number of particles $N$ (or average density
$\rho_{\rm avg} \equiv N/L$) the stationary state is unique. The question
naturally arises, ``What happens to these states in the infinite volume limit
$L\rightarrow\infty$, $N\rightarrow\infty$, $N/L\rightarrow\rho$ fixed?'' To
make this question precise we have to specify the position of our domain of
observation relative to the blocked site as $L\rightarrow\infty$. This is
unlike the case in the finite system where except for labeling the state is
independent of the position of the blocked bond, since the domain of
observation corresponds to all $L$ sites.
Let us label the sites of our system (relative to the observation domain) from
$-L/2$ to $L/2$, and put the left end of the blocked bond at $K_L$. Then
if $K_L\rightarrow\pm\infty$ as $L\rightarrow\infty$ we are in the situation
of an infinite uniform system, and the stationary states are superpositions of
product states. The product states are of two kinds: for $0