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Fg(,)g(the)h(righ)o(t-hand)e(side)h(of)h(\(1.3\))g(cannot)f (approac)o(h)e(a)j(de\014nite)f(function)30 2072 y(of)h Fe(\022)h Fg(as)f Fe(k)e Fd(!)f(1)p Fg(.)23 b(This)17 b(di\016cult)o(y)h(w)o(as)d(also) j(suggested)d(b)o(y)i(a)g(preliminary)f(observ)m(ation)h(based)30 2172 y(on)c(the)g(data)g(for)f Fe(k)k Fg(=)d(3)p Fe(;)8 b Fg(4)p Fe(;)g Fg(5)p Fe(;)g Fg(10,)14 b(that)f(the)g(times)g Fe(t)g Fg(from)e(whic)o(h)i(\(1.3\))g(applies)h(increase)f(rapidly)30 2272 y(with)k Fe(k)r Fg(.)130 2401 y(The)h(main)g(ob)s(jectiv)o(e)h(of)g(the) g(presen)o(t)e(w)o(ork)h(is)h(to)g(repair)f(this)h(de\014ciency)h(of)e(the)h (mean-)30 2501 y(\014eld)e(theory)g(in)g(1)p Fe(d)p Fg(.)22 b(Th)o(us,)15 b(in)i(Section)g(2)g(w)o(e)f(in)o(tro)q(duce)h(a)f(new)h(appro) o(ximation)e(sc)o(heme,)f(still)30 2600 y(mean-\014eld)e(in)j(nature,)e(whic) o(h)h(allo)o(ws)h(the)f(deriv)m(ation)i(of)e(impro)o(v)o(ed)e(rate)i (equations)g(whic)o(h)g(are)30 2700 y(v)m(alid)j(in)e(a)g(signi\014can)o(tly) h(larger)f(regime)f(and)g(whic)o(h)h(apply)h(in)f(all)i(the)e(limiting)h (cases.)21 b(V)l(arious)898 2820 y({)16 b(4)h({)p eop %%Page: 5 5 bop 30 110 a Fg(implications)22 b(of)e(this)h(new)f(mean-\014eld)f (description)h(will)j(b)q(e)e(detailed)h(in)f(Sections)f(3)h(and)e(4,)30 209 y(with)d(the)g(latter)h(dev)o(oted)e(to)h(the)g(con)o(tin)o(uum)d(limit.) 23 b(The)15 b(original)i(rate)f(equation)g(\(1.3\))g(turns)30 309 y(out)h(to)f(hold)h(only)g(for)702 538 y Fe(t)d Fd(\035)g Fg([\()p Fe(k)e Fd(\000)f Fg(1\))p Fe(R`)p Fg(])1036 518 y Fc(\000)p Fb(1)1090 538 y Ff(e)1115 518 y Fh(k)q Fc(\000)p Fb(1)1204 538 y Fe(;)570 b Fg(\(1)p Fe(:)p Fg(4\))30 688 y(whic)o(h)17 b(explains)g(its)h(failure)f(to)g(describ)q(e)g(the)g(con)o(tin)o(uum)d (limit)k Fe(k)d Fd(!)f(1)p Fg(.)130 817 y(As)g(with)i(an)o(y)e(mean-\014eld)f (appro)o(ximation,)h(\015uctuations)g(are)g(exp)q(ected)i(to)f(dominate)f (the)30 917 y(large-time)21 b(b)q(eha)o(vior)h(in)g(some)d(cases,)j(suc)o(h)e (as)h(for)g Fe(k)j Fg(=)e(2.)36 b(The)21 b(appropriate)g(criteria)h(are)30 1016 y(deriv)o(ed)g(and)g(discussed)g(in)h(Section)g(5.)39 b(Finally)l(,)26 b(Section)d(6)f(is)h(dev)o(oted)f(to)h(summa)o(ry)d(and)30 1116 y(discussion)15 b(of)h(those)f(new)g(asp)q(ects)g(that)h(m)o(ust)d(b)q (e)j(considered)f(in)h(order)e(to)i(extend)f(the)h(theory)30 1216 y(to)k Fe(d)g(>)f Fg(1,)i(whic)o(h,)g(ho)o(w)o(ev)o(er,)e(is)h(not)g (attempted)g(here;)h(see)f(earlier)h(remarks)c(in)k(this)f(section)30 1315 y(and)c(the)h(discussion)f(in)h(Section)g(6.)30 1544 y Fi(2.)i(F)-6 b(AST)18 b(DIFFUSION)h(AND)g(DIFFUSER)f(SP)-5 b(A)n(CE)130 1674 y Fg(It)24 b(is)g(con)o(v)o(enien)o(t)f(to)h(relate)h(our)e (system)f(to)i(a)g(lattice)i(mo)q(del)d(of)h(hard-core)e(particles)30 1773 y(\(monomers)o(\))15 b(of)j(length)g(single)g(lattice)i(spacing)d Fe(b)p Fg(,)h(eac)o(h)f(attempting)g(to)h(hop)f(with)h(rate)g Fe(H)h Fg(=)30 1873 y(4)p Fe(D)q(=b)143 1855 y Fb(2)188 1873 y Fg(p)q(er)i(unit)h(time)g(to)f(the)h(left)h(or)e(righ)o(t)h(neigh)o(b)q(or) f(lattice)j(site)e(\(rates)g Fe(H)q(=)p Fg(2\).)38 b(W)l(e)22 b(will)30 1973 y(use)17 b(the)g(term)f(\\di\013users")g(to)i(refer)f(to)g (these)g(particles.)26 b(Hopping)17 b(attempts)f(succeed)h(only)h(if)30 2072 y(the)i(hard-core)d(constrain)o(t)i(is)h(not)f(violated.)33 b(The)19 b(precise)g(dynamics)f(of)i(di\013users)f(is)g(in)h(fact)30 2172 y(not)g(of)f(in)o(terest)h(to)g(us)f(here.)30 b(The)19 b(only)i(prop)q(ert)o(y)d(w)o(e)i(will)i(use)d(is)h(that)g(for)f(large)h (times)f(the)30 2272 y(system)d(of)h(hard-core)e(particles)j(approac)o(hes)d (a)i(steady)g(state)h(in)f(whic)o(h)g(they)h(are)f(distributed)30 2371 y(statisticall)q(y)j(uniformly)c(along)h(the)f(line)i([10].)130 2501 y(Consider)12 b(no)o(w)g(a)i(large)f(length)h Fe(L)f Fg(in)g(our)g (original)h(dep)q(osition)g(mo)q(del.)21 b(It)13 b(is)h(con)o(v)o(enien)o(t)f (to)30 2600 y(tak)o(e)g Fe(L)f Fg(to)g(b)q(e)h(the)f(system)f(size,)j(in)f (this)g(section.)21 b(W)l(e)12 b(disregard)f(\014nite-size)j(e\013ects)f(b)o (y)e(implying)30 2700 y(the)h(thermo)q(dynamic)e(limit)j(\014rst,)f(in)h(all) g(the)g(\\in)o(tensiv)o(e")f(expressions.)20 b(The)12 b(length)g Fe(L)g Fg(con)o(tains)898 2820 y({)k(5)h({)p eop %%Page: 6 6 bop 30 110 a Fe(\022)q(L=`)15 b Fg(extended)f(ob)s(jects,)g(eac)o(h)g Fe(k)i Fg(lattice)h(spacings)d(long.)21 b(Disregarding)14 b(dep)q(osition,)i (di\013usion)30 209 y(in)h(this)g(system)e(is)h(the)h(same)d(as)i(in)h(a)f (single-site-di\013user)h(system)e(in)i(whic)o(h)f(all)i(the)f(extended)30 309 y(ob)s(jects)e(of)g(length)h Fe(`)d Fg(=)h Fe(k)r(b)h Fg(are)g(replaced)g (b)o(y)f(di\013user)h(particles)h(of)f(length)h Fe(b)p Fg(.)22 b(The)15 b(total)h(length)30 409 y(a)o(v)m(ailable)j(to)e(these)f (di\013users)g(is)509 640 y(\003)e(=)f(\(1)f Fd(\000)e Fe(\022)q Fg(\))p Fe(L)j Fg(+)d Fe(\022)q(L=k)16 b Fg(=)1032 570 y Fa(\022)1069 640 y Fg(1)10 b Fd(\000)h Fe(\022)i Fg(+)1247 607 y(1)p 1246 629 28 2 v 1246 674 a Fe(k)1279 640 y(\022)1304 570 y Fa(\023)1349 640 y Fe(L)h(:)377 b Fg(\(2)p Fe(:)p Fg(1\))30 807 y(The)16 b(densit)o(y)h(of)g(di\013users)f(\(p)q(er)g(site)i(of)f(the)g (di\013user-space)e(lattice\))k(is)f(th)o(us)681 1039 y Fe(\032)13 b Fg(=)779 1005 y Fe(\022)q(L=`)p 779 1027 105 2 v 791 1073 a Fg(\003)p Fe(=b)903 1039 y Fg(=)1054 983 y Fb(1)p 1053 991 23 2 v 1053 1019 a Fh(k)1081 1002 y Fe(\022)p 962 1027 231 2 v 962 1075 a Fg(1)e Fd(\000)f Fe(\022)j Fg(+)1140 1056 y Fb(1)p 1139 1064 23 2 v 1139 1093 a Fh(k)1167 1075 y Fe(\022)1225 1039 y(:)549 b Fg(\(2)p Fe(:)p Fg(2\))130 1206 y(The)18 b(gap)f(size)i(b)q (et)o(w)o(een)f(ev)o(ery)g(pair)h(of)f(neigh)o(b)q(oring)g(di\013users)f(can) h(b)q(e)h Fe(mb)p Fg(,)f(where)g Fe(m)e Fg(=)30 1305 y(0)p Fe(;)8 b Fg(1)p Fe(;)g(:)g(:)g(:)p Fg(.)43 b(Note)24 b(that)g Fe(m)h Fg(=)g(0)f(corresp)q(onds)d(to)j(con)o(tact,)h(while)g(the)f(upp)q(er) e(limit)j(is)f(tak)o(en)30 1405 y(to)c(b)q(e)g(in\014nit)o(y)h(due)f(to)g (our)f(disregard)g(of)h(\014nite-)p Fe(L)g Fg(e\013ects.)32 b(Di\013usion)21 b(mak)o(es)d(particle)j(lo)q(ca-)30 1505 y(tions)f (uncorrelated.)32 b(In)20 b(terms)e(of)i(the)g(gap)g(sizes,)h(the)f (appropriate)f(statemen)o(t)g(is)h(that)h(the)30 1604 y(\(normalized\))16 b(probabilit)o(y)i(to)f(\014nd)f(gap)g(size)h Fe(m)g Fg(is)708 1836 y(Prob)8 b(\()p Fe(m)p Fg(\))13 b(=)h Fe(\032)p Fg(\(1)d Fd(\000)g Fe(\032)p Fg(\))1147 1815 y Fh(m)1199 1836 y Fe(:)575 b Fg(\(2)p Fe(:)p Fg(3\))130 2003 y(Of)14 b(course,)f(the)h(result)h(\(2.3\)) f(applies)h(only)g(for)f(large)g(times.)21 b(Ho)o(w)o(ev)o(er,)13 b(w)o(e)h(are)g(prop)q(osing)30 2102 y(to)g(use)f(\(2.3\))g(as)g(the)h(appro) o(ximate,)e(fast-di\013usion,)i(mean-\014eld)e(form)f(of)j(the)f(gap)g(size)h (distribu-)30 2202 y(tion)g(for)f(the)h(actual)h(dep)q(osition)f(mo)q(del.)21 b(The)13 b(implications)i(of)e(this)h(appro)o(ximate)e(assumption)30 2301 y(will)20 b(b)q(e)e(w)o(ork)o(ed)f(out)g(in)i(this)f(and)f(the)h(next)g (t)o(w)o(o)g(sections.)26 b(W)l(e)17 b(will)k(then)c(pro)q(ceed)h(to)g(deriv) o(e)30 2401 y(the)c(conditions)h(for)f(its)i(asymptotic)d(v)m(alidit)o(y)l(,) 18 b(in)d(Section)g(5.)21 b(Let)15 b(us)e(p)q(oin)o(t)i(out,)g(ho)o(w)o(ev)o (er,)e(that)30 2501 y(in)i(the)f(opp)q(osite)g(extreme)g(of)g(no)f (di\013usion)i(at)f(all,)i(exact)f(results)f(for)f(the)i(gap)e(size)i (distribution)30 2600 y(can)j(b)q(e)g(deriv)o(ed)f([3])h(although)g(they)g(w) o(ere)f(analyzed)i(in)f(detail)i(only)f(in)f(the)g(con)o(tin)o(uum)e(limit)30 2700 y Fe(k)f Fd(!)f(1)p Fg(.)22 b(These)16 b(results)g(will)j(b)q(e)e (describ)q(ed)g(in)g(Section)g(4.)898 2820 y({)f(6)h({)p eop %%Page: 7 7 bop 130 110 a Fg(W)l(e)14 b(no)o(w)g(return)f(to)h(the)h(original)h (full-line)h(problem,)c(and)g(assume)f(that)j(\(appro)o(ximately\))30 209 y(the)h(gap)f(sizes)h(are)f(distributed)h(according)g(to)g(\(2.3\),)g (where)f(the)h(time-dep)q(endence)e(en)o(ters)h(via)30 309 y(the)21 b Fe(\022)q Fg(-dep)q(endence)f(implied)h(b)o(y)f(\(2.2\).)34 b(The)21 b(total)h(n)o(um)o(ber)c(of)j(gaps)e(\(including)j Fe(m)f Fg(=)f(0\))h(is)30 409 y(equal)e(to)g(the)g(n)o(um)o(b)q(er)c(of)k(ob) s(jects.)28 b(Ho)o(w)o(ev)o(er,)18 b(only)h(gaps)f(of)h(size)g Fe(m)e Fd(\025)g Fe(k)j Fg(o\013er)e(\()p Fe(m)13 b Fd(\000)f Fe(k)i Fg(+)e(1\))30 508 y(distinct)20 b(lo)q(cations)f(for)f(dep)q(osition.) 28 b(Since)19 b(the)f(dep)q(osition)h(attempt)f(rate)g(p)q(er)g(lattice)i (site)g(is)30 608 y Fe(Rb)p Fg(,)d(the)g(o)o(v)o(erall)g(a)o(v)o(erage)f (rate)g(of)h(successful)f(dep)q(ositions)i(in)f Fe(L)p Fg(,)f(p)q(er)g(unit)i (time,)e(is)317 837 y Fe(!)g Fg(=)e(\()p Fe(Rb)p Fg(\)\()p Fe(\022)q(L=`)p Fg(\))690 775 y Fc(1)674 790 y Fa(X)664 897 y Fh(m)p Fb(=)p Fh(k)754 837 y Fg(\()p Fe(m)d Fd(\000)g Fe(k)i Fg(+)e(1\))p Fe(\032)p Fg(\(1)g Fd(\000)g Fe(\032)p Fg(\))1187 817 y Fh(m)1239 837 y Fg(=)1297 803 y Fe(LR\022)q Fg(\(1)h Fd(\000)f Fe(\032)p Fg(\))1544 785 y Fh(k)p 1297 826 272 2 v 1406 871 a Fe(k)r(\032)1589 837 y(:)185 b Fg(\(2)p Fe(:)p Fg(4\))30 1005 y(In)16 b(eac)o(h)g(dep)q(osition,)i(the)f(co)o(v)o(erage)e (is)i(incremen)o(ted)f(b)o(y)g Fe(`=L)p Fg(.)21 b(Th)o(us,)15 b(w)o(e)i(obtain)g(the)f(relation)752 1200 y Fe(d\022)p 752 1222 51 2 v 755 1268 a(dt)823 1234 y Fg(=)881 1200 y Fe(`R\022)q Fg(\(1)d Fd(\000)d Fe(\032)p Fg(\))1115 1182 y Fh(k)p 881 1222 259 2 v 984 1268 a Fe(k)r(\032)1160 1234 y(:)614 b Fg(\(2)p Fe(:)p Fg(5\))30 1394 y(Finally)l(,)18 b(w)o(e)f(use)f(\(2.2\))h(and)e(in)o (tro)q(duce)i(the)g(dimensionless)f(time)g(v)m(ariable)856 1623 y Fe(T)21 b Fd(\021)14 b Fe(R`t)g(;)724 b Fg(\(2)p Fe(:)p Fg(6\))30 1783 y(to)17 b(write)g(the)g(equation)h(for)e(the)h(co)o(v)o(erage) e(in)i(the)g(fast-di\013usion)g(appro)o(ximation,)699 1978 y Fe(d\022)p 693 2001 62 2 v 693 2046 a(dT)775 2012 y Fg(=)922 1978 y(\(1)12 b Fd(\000)f Fe(\022)q Fg(\))1071 1960 y Fh(k)p 834 2001 352 2 v 834 2020 a Fa(\000)857 2061 y Fg(1)g Fd(\000)f Fe(\022)j Fg(+)1035 2041 y Fb(1)p 1034 2049 23 2 v 1034 2078 a Fh(k)1062 2061 y Fe(\022)1087 2020 y Fa(\001)1110 2031 y Fh(k)q Fc(\000)p Fb(1)1219 2012 y Fe(:)555 b Fg(\(2)p Fe(:)p Fg(7\))30 2272 y Fi(3.)19 b(LA)-5 b(TTICE)19 b(DEPOSITION)f(WITH)h(F)-6 b(AST)18 b(DIFFUSION)130 2401 y Fg(In)g(this)i(section)g(w)o(e)e(consider)h (implications)h(of)f(the)g(relation)i(\(2.7\))e(for)g(\014xed)f Fe(k)r Fg(,)h(i.e.,)h(for)30 2501 y(lattice)d(dep)q(osition.)22 b(Some)12 b(conclusions)j(are)e(v)m(alid)j(also)f(in)f(the)h(limit)g Fe(k)h Fd(!)d(1)h Fg(whic)o(h,)h(ho)o(w)o(ev)o(er,)30 2600 y(will)20 b(b)q(e)e(considered)f(separately)h(in)g(the)f(next)i(section.)25 b(W)l(e)18 b(assume)d Fe(k)i Fg(=)e(2)p Fe(;)8 b Fg(3)p Fe(;)g Fg(4)p Fe(;)g(:)g(:)g(:)18 b Fg(since)g(the)30 2700 y(case)e Fe(k)g Fg(=)d(1)k(is)g(trivial.)25 b(In)16 b(fact,)h(\(2.7\))g(is)g(exact)g (for)g Fe(k)e Fg(=)f(1.)898 2820 y({)i(7)h({)p eop %%Page: 8 8 bop 130 110 a Fg(The)17 b(curv)o(e)h Fe(\022)q Fg(\()p Fe(T)7 b Fg(\))20 b(obtained)e(b)o(y)f(in)o(tegrating)i(\(2.7\))g(with)g(the)f (initial)j(condition)e Fe(\022)q Fg(\(0\))f(=)e(0)30 209 y(pro)o(vides)g(the) h(upp)q(er)f(b)q(ound)g(\(corresp)q(onding)g(to)i Fe(D)e Fg(=)e Fd(1)p Fg(\))j(for)g(an)o(y)f(dep)q(osition)i(pro)q(cess)e(with)30 309 y(\014xed)k(di\013usion)g(rate)h(\(recall)h(that)e(the)g(dep)q(osition)i (rate)e(w)o(as)f(absorb)q(ed)g(in)i(the)f(time)g(scale\);)30 409 y(see)15 b(\(2.6\).)21 b(The)15 b(exact)g(solution)h(for)e Fe(D)h Fg(=)f(0,)h(a)o(v)m(ailable)i(as)d(quadrature)f([3],)i(pro)o(vides)f (the)g(lo)o(w)o(er)30 508 y(b)q(ound.)36 b(The)22 b(t)o(w)o(o)f(curv)o(es)g (are)g(in)i(fact)f(quite)h(close)g(for)e(small)h(and)f(in)o(termediate)g (times)g(as)30 608 y(illustrated)d(in)g(Figure)e(1)g(for)g Fe(k)g Fg(=)d(5.)130 737 y(F)l(or)18 b(large)j(times,)f(ho)o(w)o(ev)o(er,)f (the)h(b)q(eha)o(vior)g(is)h(quite)g(di\013eren)o(t.)32 b(The)20 b Fe(D)h Fg(=)f(0)f(curv)o(e)h(ap-)30 837 y(proac)o(hes)12 b(the)h(jamming)e(v)m(alue)k Fe(<)e 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