HYDRODYNAMIC LIMITS FOR ONE-DIMENSIONAL PARTICLE SYSTEMS WITH MOVING BOUNDARIES 1

Transcription

1 The Annals of Pobabiliy 996, Vol. 4, o., HYDRODYAMIC LIMITS FOR OE-DIMESIOAL PARTICLE SYSTEMS WITH MOVIG BOUDARIES BY L. CHAYES AD G. SWIDLE Univesiy of Califonia a Los Angeles and Univesiy of Califonia a Sana Babaa We analyze a new class of one-dimensional ineacing paicle sysems feauing andom boundaies wih a andom moion ha is coupled o he local paicle densiy. We show ha he hydodynamic limiing behavio in hese sysems coesponds o he soluion of an appopiae Sefan Ž fee-bounday. equaion and descibe some applicaions of hese esuls.. Inoducion. Oveview and peliminaies. The sudy of ineacing paicle sysems wih consevaion laws fequenly evolves aound hydodynamic limis, in which he long-ime, lage-lengh scale behavio of a sysem is chaaceized by he evoluion of he local densiy of he conseved quaniy accoding o a paial diffeenial equaion. The ypical seup consiss of a paicle sysem defined on a closed laice wih peiodic bounday condiions o, pehaps, wih a fixed paicle densiy a he bounday. See, fo example, 3 and efeences heein fo an exensive discussion of hese opics. In each of hese cases, he dynamics of he paicle sysem is pescibed by he ansiion aes, he iniial condiion and, if elevan, he bounday condiions. In paicula, he inenal dynamics of he paicle sysems does no affec he behavio of he boundaies. In his pape we sudy ceain one-dimensional paicle sysems wih exclusion dynamics and he addiional feaue ha he egion in which exclusion dynamics occus is aleed by he dynamics iself. The wo basic examples can boh be egaded as cude micoscopic models of he dynamics of a liquidsolid sysem wih an ineface: he fis case coesponds o he meling of a solid and he second o he feezing of a supecooled liquid. To descibe he pocess of meling, conside he following paicle sysem wih configuaions in, 0, 4, whee,,...,, 4. Received Sepembe 994; evised June 995. Wo suppoed in pa by he SF unde Gans DMS Ž L.C.. and DMS Ž G.S... AMS 99 subjec classificaions. 60K35, 60H5. Key wods and phases. Paicle sysems, exclusion pocess, Sefan s equaion. 559

2 560 L. CHAYES AD G. SWIDLE We will denoe he configuaion a ime by and he sae of sie i a ime by Ž i.. As usual, we hin of sies i fo which Ž i. as being occupied by paicles and when Ž i. 0, he sie is vacan. When Ž i., he sae is quie diffeen fom eihe of he above. Le us egad hese sies as being occupied by land mines o Ž saic. anipaicles. The ansiion aes ae descibed as follows: any neaes-neighbo pai of sies in which one is occupied and he ohe is vacan exchange hei saes a uni ae. oe ha his ule efes only o he exchanges among 0 s and s, fo example, The ule fo anipaicles is ha any neighboing pai of sies in which one is occupied and he ohe is a mine will become a pai of vacan sies a uni ae: All ohe ansiions involving he anipaicles ae suppessed. Le us conside his pocess wih he iniial configuaion saisfying, if i 0,. 0Ž i. ½ 0 o, ohewise, in which all sies o he lef of ae anipaicles and all sies o he igh of 0 ae vacan o occupied. I is clea ha among he anipaicles, he fis ansiion does no occu unil he lefmos jumps ono he mine a he oigin. They annihilae, vacaing boh he sies a 0 and a. Subsequen annihilaion ansiions coninue o eode he egion of s while fuhe educing he numbe of paicles. Wih he above labeling, he bloc of s may be hough of as he cold esevoi of a meling bloc and he paicles as copuscles of hea. Alenaively, exchanging labels among 0 s and s, each is evealed o be a fozen pai of paicles ha decouple, a uni ae, wheneve space is available. In any case, as he bloc mels, moe oom is made available fo dynamics. Quie he opposie behavio is obseved a he waeice ineface of a supecooled liquid. Hee he bounday moves inwad, encoaching on he fluid egion. We model his sysem on he same space as befoe,, 0, 4, and, similaly, he ansiions among he 0 s and s ae given by he usual ules fo he simple exclusion pocess. Fuhemoe, he s ae again essenially saic, bu his ime, new s can be geneaed. Fis, we insill he ule ha any configuaions in which a neighbos a is fobidden. ŽThis saemen is ue, by fia, in he iniial configuaion and is dynamically enfoced heeafe.. The dynamics among s and s may now be descibed as follows: when a exchanges wih a 0 ha neighbos a, duing he couse of he exchange, he is ansfomed ino a. Explicily, he ansiions and occu a uni ae. These wo pocesses will now be defined pecisely. Fo any finie ineval, i is sufficien o specify he acion of he infiniesimal geneao,, on any eal-valued funcion on he configuaion space, 0, 4. Saing wih he fis Ž meling. pocess, le, 0, 4 denoe a paicle configu-

3 OE-DIMESIOAL PARTICLE SYSTEMS 56 aion and, fo i and j in, we define he possible i j paicle ansfe configuaions by. i; j Ž. Ž., i, j, Ž j., Ž j. and Ž i., i; j Ž j. ½ Ž j., ohewise, 0, Ž i. and Ž j., i; j Ž i. ½ Ž i., ohewise. If denoes he endpoins of and, we define he geneao Ý i; i i; i.3a f f f f f, i whee is a geneao fo descibing paicle ansfes o he endpoins ha will depend on he paiculas of he poblem. To disinguish he second Ž feezing. pocess fom he fis, we will use o denoe he paicle configuaion and L o denoe he geneao. The acion of he geneao is ohewise exacly he same as in 3a., bu hee he i j paicle ansfe configuaions ae given by 4. i; j Ž. Ž., i, j,, if Ž i., Ž j. 0 and Ž j.,, if Ž i., Ž j. 0 and Ž j. i; j Ž j. o Ž j., Ž j., ohewise, 0, Ž i. and Ž j., i; j Ž i. ½ Ž i., ohewise. To simplify he noaion in much of wha is o follow in lae secions, fo appopiaely smooh funcions gž ;. o, infomally, gž ;. ha depend explicily on ime as well as he insananeous configuaion, 0, 4, we will use o denoe he evoluion opeao: 3b. g Ž ;. lim E gž ;. g Ž ;.. 0 Similaly, fo he feezing poblem we will use he noaion Y. REMARK. Povided ha, he objecs and L ae clealy sufficien o specify he pocess. Fo a sysem defined on all of, moe wo would be equied Žcf., Chape, fo a discussion of hese maes., bu fo ou puposes, such a esul would hadly be woh he effo. Hydodynamic limis mos ofen involve ahe han saing on a sysem ha is infinie fom he ouse. Fuhemoe, as paicle sysems pe se, hese

4 56 L. CHAYES AD G. SWIDLE wo examples ae no so ineesing o sudy. In boh cases, he invaian measues would have a endency o be iviala bes, one of he invaian measues fo he simple exclusion pocess. As is well nown, he hydodynamic limi of he usual exclusion pocess is he Ž linea. hea equaion. Thus, i is no had o imagine ha he hydodynamic limi of he above pocesses ae Sefan equaions in which he diffusive evoluion govens boh he elaxaion of he inenal densiy as well as he moion of he fee bounday. The classical fom of he Ž one-sided. Sefan equaion fo he densiy Ž x,. and he fee bounday B is given by 5., x subjec o a possible Ž fixed. bounday condiion a, fo example, x x 0, an iniial condiion on B and an iniial condiion on he densiy: 6. BŽ 0. B 0, Ž x 0,. F Ž., Ž x, 0. Ž x., B x x We may assume ha all of he above is aing place fo 0 T whee, fo simpliciy, we have T. The bounday condiion a he moving bounday is, ypically, 7. BŽ., 0, and, finally, we aive a he so-called Sefan condiion ha elaes he evoluion of B o he flux of hough B. The mos ofen sudied vesion of he Sefan equaion Ž subjec o a myiad of genealizaions. has, as he Sefan condiion, db 8a. Ž BŽ.,.. d Clealy, his coesponds o he ouwad displacemen of he bounday Žmel- ing. and is he ype of limiing behavio ha one would expec fom he paicle sysem defined in. and 3.. The ohe possibiliy is db 8b. Ž BŽ.,. d Ž and is genealizaions. coesponding o an inwad displacemen Ž feezing. a he bounday. One would expec his so of limiing behavio fom he paicle sysem descibed in 5.. Of couse, since we have descibed he Sefan poblem in classical ems, all of he peceding equaions ae subjec o he sipulaion ha he vaious quaniies ae well defined; in pacice, one usually deals wih he wea foms of The sysems govened by he ype of Sefan condiion in 8a. ae fa moe acable han hose sysems coesponding o 8b.; he vas majoiy of

5 OE-DIMESIOAL PARTICLE SYSTEMS 563 he wo on Sefan s equaion concens he fis case and is genealizaions. The eason fo his discepancy can be undesood fom he pespecive of he enhalpy. Since his quaniy will ene diecly ino ou analysis, i is wohwhile o descibe he diffeences beween he wo sysems in hese ems. Fomally Ž o in he wea vesion. boh sysems may be expessed as a diffusion equaion fo an enhalpy funcion, až x,., wih a diffusion coefficien ha depends disconinuously on he value of a. In boh cases, we have až x,. Ž x,. fo x BŽ.; howeve, fo he sysem descibed in 8a., he elevan definiion is 9a. až x,. if x BŽ., while in he case of he sysem descibed in.8b, one defines 9b. až x,. if x BŽ.. In he fome case, he diffusion coefficien is a monoone nondeceasing funcion of he enhalpy while, in he lae case, i is no. Wih he benefi of his monooniciy, uniqueness of he Sefan sysemin quie some genealiy is immediae. Howeve, wihou his monooniciy, he poblem is fa moe challenging. ŽIn paicula, he sysem is bodeline ill-posed and has a definie poenial fo insabiliies.. In fac, he exising lieaue on he subjec uns ou o be insufficien fo ou puposes. Reuning he discussion o he paicle sysems, i is inuiively clea ha in he fis sysem, he enhalpy coesponds o iself while in he second, he enhalpy coesponds o. In boh cases, an appopiae Ž maingale. vesion of he wea fom of he elevan Sefan equaion is eadily deived. In he fome case, he desied hydodynamic limi is hen an immediae consequence of he esablished uniqueness esuls fo his equaion. ŽThe pecise noion of convegence will appea in he saemen of Theoem... In he lae case, geneal esuls on uniqueness ae no nown and hence mus be esablished hee. Even so, ou esuls equie he a pioi nowledge ha he bounday is a coninuous funcion of ime. In fac, coninuiy of he bounday is no a diec consequence of he exisence of a wea soluion o his vesion of Sefan s equaion. I heefoe mus be diecly esablished in he conex of he paicle sysem. This is somewha nonivial Žwhen one consides diffusive ime scaling. and consiues a majo poion of ou effos. Howeve, once hese ingediens have been assembled, a hydodynamic limi is eadily esablished fo his case as well. ŽThe pecise noion of convegence will appea in he saemen of Theoem Oganizaion. The oganizaion of his pape is as follows. In Secion, we analyze he poblem wih meling boundaies. Using he sandad deivaions in he heoy of hydodynamic limis, we eadily aive a a wea fom of he appopiae Sefan equaion. The well-nown uniqueness esuls fo his vesion of he equaion povide us wih he final sages of he hydodynamic limi in his case.

6 564 L. CHAYES AD G. SWIDLE In Secion 3, we sa he pocedue along he same lines Žwih he analysis of Secion allowing us o avoid mos of he calculaions.. Thus, any hydodynamic limi saisfies a wea vesion of he appopiae Sefan equaion. To maneuve ino a posiion whee uniqueness esuls can be bough o bea, coninuiy of he escaled boundaies is esablished in a seies of deailed agumens. Using he esuls fom he Appendix, he es of he hydodynamic limi is saighfowad. In Secion 4, we discuss an applicaion of a vaian of he sysem sudied in Secion 3: he dynamical disappeaance of a wo-dimensional dople ha is apped agains he cone of a sample. The appopiae exensions of he analysis in Secion 3 ae biefly pefomed. In he Appendix, we povide he missing ingediens fo he poofs of he esuls in Secions 3 and 4, namely, a poof of uniqueness fo he soluions of Sefan equaions wih inwadly moving boundaies unde he hypohesis of coninuous boundaies.. Analysis of a sysem wih a meling bounday. Conside he paicle sysem on wih bul dynamics as descibed in. and 3.. To complee he definiion of he sysem, we mus descibe he behavio a he endpoins. A he igh end, we will foce he sysem ino a pescibed Ž deeminisic. paicle densiy. Le R denoe a Ž piecewise. F coninuous funcion wih 0 R F. Fo he sie a and fo i, we define ; i as befoe. Le. ; Ž. Ž. fo, 0, if Ž., ; Ž. ½ Ž., ohewise, denoe he configuaion whee he sie a has jus shipped ou a paicle and le. ; Ž. Ž. fo, Ž., if Ž., ; Ž. ½ Ž. Ž., ohewise, denoe he configuaion whee he sie a has jus eceived one. Simila noions apply a he lef bounday Ž o, in geneal, o any ohe sie.. Fo he poblem a hand, we define 3. ; ; f RF f f RF f f ; ; f f f f. The esul of his acion, a he igh endpoin, will be o push he densiy owad R Ž.. Ž F A he lef end, fo compleeness, we have insalled a bounday condiion ha would enfoce zeo gadien a. eedless o say, his will be pacically ielevan: ou iniial configuaion will place

7 OE-DIMESIOAL PARTICLE SYSTEMS 565 deep inside a fozen bloc and he pocess iself ceases o be of inees jus when he fis paicle has made is way o his poin.. In he coninuum, a wea vesion of he Sefan equaion wih eceding boundaies is fomulaed as follows: suppose, wihou any significan loss of genealiy, ha he spaial domain is, and ha he iniial posiion of he bounday is BŽ We will assume ha a x, he densiy is fixed a which is piecewise smooh and saisfies 0 F F. Fo x 0, we will ae he iniial configuaion o be given by some Ž x. 0 which is also piecewise smooh and bounded beween 0 and. Le GŽ x,. denoe any smooh es funcion wih GŽ,. 0. Then, he enhalpy funcion až x,. saisfies 4a. whee H až x, s. GŽ x, s. až x, 0. GŽ x, 0. dx s G G s G H F 0 x 0 x H H a HŽ a. dx d Ž,. d, 0, if a 0, 4b. HŽ a. ½ a, if a 0. REMARK. Fomally, 4. is a nonlinea diffusion equaion wih he diffusion coefficien given by DŽ a. H Ž a.. The case of he single-phase Sefan poblem can also be eaed by he inoducion of an auxiliay densiy, nž x., which enes ino 4a. as he coefficien of g x x. We will see ha he fomulaion using nž x. is slighly moe convenien fo his secion, while in he nex secion i is acually necessay because of a spuious ambiguiy in he fomulaion of he poblem as a nonlinea diffusion equaion. The bounday, B, in eihe fomulaion is simply 4 5. B sup xaž x,., which, as i uns ou, is well defined fo each Žcf. he discussion in Poposiion... I also uns ou ha in he soluion o his sysem, he bounday moves coninuously o he lef unil some ime T which depends on he funcions and Ž x. F 0 when i his he poin x. In he pesen fomulaion, he soluion may be coninuedif F has been defined bu heeafe, i is an odinay diffusion poblem on, wih a zeogadien bounday condiion a x. Fo his paicle sysem, he local enhalpy is he value of a each sie and he auxiliay densiy is he posiive pa heeof. We fomally define 6a. a Ž. Ž. and 6b. n Ž. Ž..

8 566 L. CHAYES AD G. SWIDLE Fuhe, if y 0, is a eal numbe, we will wie 7. a Ž y. a Ž y. fo he y ha saisfies y and similaly fo n y. In his and in he lae secions, we will be ineesed in convegence o he hydodynamic limi via a sequence of sysems ha double in size a each sage. We hus sa wih some ha is abiay Ž 0 bu lage enough so ha all quaniies unde discussion mae sense. and we define, fo a posiive inege, 0. Of pimay inees will be he funcion 8a. a Ž x,. a Ž x. and 8b. n Ž x,. n Ž x., wih x and 0 T. Ou fis subsanive esul will concen he expeced values of hese quaniies. PROPOSITIO.. Le S 0 denoe any posiive ime and le F and Ž x. 0 be defined as above wih 0 S. Conside he paicle sysems on as descibed in., 3. and. 3. wih R given by Ž. F F. Suppose, fuhe, ha each ealizaion of he paicle sysem comes equipped wih an iniial paicle configuaion ha is, deeminisically, Ž. 0 fo 0 while, fo 0, he Ž. ae independen wih E Ž. 0 0 Ž.. Le a Ž x,. E a Ž x,. 0 denoe he expeced value of he enhalpy in he h paicle sysem. Then a Ž x,. až x,., wealy in L, whee až x,. is he Ž unique. soluion o he Sefan poblem descibed in 4.. REMARK. The fac ha he wea limi is uniquely deemined is of no immediae consequencei is jus a spin-off of he nown uniqueness esuls fo his sysem. Wha will acually be poved is ha he limi of any conveging subsequence of Ž a. is a wea soluion o he sysem descibed in 4.. Le g Ž. denoe any Ž deeminisic. funcion on which, fo all and, is diffeeniable wih espec o. Define ² g, a: by 9. ² g, a: g Ž. a Ž.. I is clea ha Ý g 0. ² g, a:, a ² g, a :, ; whee he noaion in.0 is defined analogously o ha in.9.

9 OE-DIMESIOAL PARTICLE SYSTEMS 567 In he ineio, acs in exacly he same fashion as he geneao fo he usual exclusion pocess; howeve, he behavio a he boundaies equies special consideaion. We define he micoscopic bounday as he locaion of he anipaicle ha is fahes o he igh:. b Ý Ž.,, whee Ž., indicaes he pesence of an anipaicle a he sie. The ules fo a Ž. ae easily discened fom he definiion of he pocess:. Ž i. a Ž. 0, b Ž.,. Ž ii. a b a b,. Ž iii. a b a b a b,. Ž iv. a Ž. a Ž. a Ž. a Ž. a Ž.,. Ž v. a Ž. R a Ž. a Ž.. We hus have F b, ² g, a: g Ž b. a Ž b. g Ž b. a Ž b. a Ž b. 3. Ý b g Ž. a Ž. gž. RF až. až.. Defining g Ž. 0 and pefoming he usual summaion by pas, we obain ² g, a: až. gž. 4. b Ý R F gž. gž., whee he dummy g Ž. is included fo he ease of fuue efeence. Since, fo b, a Ž. n Ž. and, fo b, n Ž. 0, i is seen ha he fis em on he igh-hand side of 4. can be expessed as ² n, g :. We can now conclude ha 5. ž ; / g m g, ² g, a: ² g, a:, a ² g, n: 0 H s ds 0 s H F 0 R g Ž. g Ž. ds

10 568 L. CHAYES AD G. SWIDLE is a maingale wih zeo expecaion. Le GŽ x,. be a smooh es funcion ha vanishes a x and define, fo s S, ž / s 6. gsž. G,. oice ha g is now defined on,..., 4 s. We may fomally exend a Ž. o his domain by defining a Ž. R Ž s. s s F and, fo example, a Ž. s 0. Using he noaion ž ž / / s M Ž G. m G, and ecalling he quaniies a and n fom he definiion pio o he saemen of his lemma, we find ha 7. H M Ž G. GŽ x,. a Ž x,. GŽ x, 0. a Ž x, 0. dx G H ds F Ž s. Ž, s. x 0 ž / ž / G G H dsh a Ž x, s. n Ž x, s. dx O. x 0 Aveaging ove boh sides of 7., we have, modulo ems of he ode of, he wea fom of he Sefan equaion. Hence, any convegen subse- quence conveges o a wea soluion of his sysem. The uniqueness of soluions o 4. is well esablished; an elegan deivaion can be found in 7. Fuhemoe, unde he saed iniial and bounday condiions, he soluion is classical and B is C fo posiive imes. REMARK. Any eade who has checed he deails of he calculaions leading o 4. will noice a fouious cancellaion of ems a he micoscopic bounday b. Indeed, had we defined he aes fo he swiches o be insead of uniy, hee would have been an unwaned em of he fom 8. u a Ž b. g Ž b. g Ž b.. oe ha, in he hydodynamic limi, his does no vanish due o powe couning alone: indeed, we ge one faco of in he ansiion fom m Ž g. o M G and anohe because we have on display, essenially, imes he gadien of G. Howeve, hee ae also of he ode of ime seps involved in he inegaion, so osensibly we could end up wih a faco of he ode of uniy. eveheless, unde diffusive scaling, his em exes no influence Ž povided ha 0.. Pesumably, his is due o he Ž a poseioi. fac ha he densiy a he moving boundayhee epesened by he em a Ž b. vanishes in he hydodynamic limi. We have no been able o

11 OE-DIMESIOAL PARTICLE SYSTEMS 569 implemen his diecly ino an agumen bu he following is unquesionably elaed. Le us assume, fo simpliciy, ha g Ž. is ime independen. Ž The effecs of ime dependence in g indeed fall o naive powe couning.. Obseve ha he unwaned em admis he expession 9. u Ý až. g Ž.. b Thus, in expecaion, he ime inegal of his quaniy is equal o he agumen of he geneao evaluaed a he endpoins: ž / H Ý s Ý 0 0 b b 0 0. E u ds E a Ž. g Ž. a Ž. g Ž.. Since he a s ae consans in boh he sums, he gadien may be summed and he igh-hand side is jus g evaluaed a he endpoins. Thus, a his sage, he unwaned em is of he ode of uniy and hen hee is he addiional faco of in he passage fom m o M Ž.. The final agumen of his secion equies only one moe se of calculaions. THEOREM.. Le a Ž x,. be as descibed in Poposiion.. Then, wih pobabiliy, a Ž x,. až x,., wealy in L, whee až x,. is he soluion o he Sefan poblem descibed in 4. wih he saed iniial and bounday condiions. PROOF. Fo any smooh G, we have esablished ha he andom vaiables M Ž G. have expeced values ha ae bounded by invese powes of. If he same can be esablished fo M, hen due o he apid gowh of he Ž., we have ha M Ž G. conveges o 0 wih pobabiliy. Bu his implies ha all wealy convegen subsequences of he andom a Ž x,. convege o he soluion až x,. which diecly implies he saed esul. We poceed wih he quadaic vaiaion calculaion which, excep fo he acion a he bounday, is faily sandad in he sudy of hydodynamic limis. Since M is of he fom W HW d, he quadaic vaiaion is equal o W W W. I is easily seen ha we may ignoe he consan Ž 0. em. Wiing. ² g, a: a Ž j. a Ž. g Ž j. g Ž., Ý j, i is found ha all ems involving ime deivaives of g dop ou of ² a, g: ² a, g: ² a, g :. Fuhemoe, fo any j and ha saisfy

12 570 L. CHAYES AD G. SWIDLE a Ž j. a Ž. a Ž j. a Ž. a Ž. a Ž j., we ge a cancellaion. This leaves only he diagonal and nea-diagonal ems: Ý M g a a a Ý g Ž. g Ž. a Ž. a Ž.. :, až. až. a Ž. a Ž. Ý g Ž. g Ž. a Ž. a Ž. :, až. až. a Ž. a Ž.. Afe a lenghy calculaion, wih special aenion o he bounday ems, he following esuls emege: he second and hid summaions of. Žwhich ae idenical afe eindexing. each lead o Ý b g Ž. g Ž. a Ž. a Ž. až b. gž b. gž b., while he fis summaion on he igh-hand side of. equals Ý b g Ž. g Ž. a Ž. a Ž. a b g b g b g RF a R F a. Combining hese wo lines, we finally aive a 3. Ý Ž. b M G a a g g až b. g Ž b. g Ž b. g Ž. R a Ž. R a Ž.. F F Inegaing he igh-hand side fom 0 o, we see ha EM Ž G. vanishes a leas as apidly as.

13 OE-DIMESIOAL PARTICLE SYSTEMS 57 REMARK. oe ha he conibuions fom he moving bounday and he saic bounday wee boh of he ode of in he lae case because g Ž.. Oddly enough, in he calculaion of he quadaic vaiaion, a diffeen jump ae a he moving bounday would only have geneaed a em of his ode and heefoe would no have equied a sepaae agumen as in he ema following Poposiion.. 3. Analysis of a sysem wih a feezing bounday. In his secion, we will ae up he analysis of he paicle sysem descibed in 4. and 3.. As fo he bounday geneaos, a x, we will use he same device as in he pevious secion cf.. and. and he elevan pas of 3.. We will always sa he pocess wih Ž. 0 ; hus, as can be seen fom he dynamics, no bounday ems will be necessay on his side. ŽIn he nex secion, we will discuss some poblems in which he moving boundaies ae placed on boh sides of ; in hese cases, hee will be essenially no need fo a bounday geneao.. In his case, he coninuum descipion pesens a few difficulies. If one aemps o descibe his sysem as a nonlinea diffusion poblem fo he enhalpy, i is clea ha his quaniy mus be defined accoding o 9b. wih an HŽ a. given by a, if a, Ž 3.. HŽ a. ½, if a. Howeve, his foces DŽ a. 0 wheneve a while, fom he classical descipion, hee is no eason o suppose ha he enhalpy does no exceed o he igh of he bounday. Even in he paicle sysem, a egion of a could epesen Ž mobile. paicles paced a uni densiy. This is no an insumounable poblem in ou paicula case because hese siuaions can only occu in he iniial condiion and will disappea he insan ha he pocess sas. Howeve, in closely elaed paicle modelsfo example, in which he bounday moves moe han one laice spacing fo each paicle ha i adsobshe same poblem will cop up and, pehaps, canno simply be defined away. Theefoe, in pinciple, we should wo wih he auxiliay densiy fom of he wea equaion which eads: Ž 3.. H až x, s. GŽ x, s. až x, 0. GŽ x, 0. dx s G G s G H F 0 x 0 x H H až x,. nž x,. dx d Ž,. d. oe ha, in geneal, i aes he combinaion of boh a and n o deemine he naue of he soluion. This is in shap conas o he wea equaion feaued in he pevious secion. As discussed peviously, fa less is nown abou he soluion o Ž 3.. han he soluion o 4.; in paicula, hee he bounday is no a pioi a well-defined objec. On he ohe hand, suppose hee is a soluion o Ž 3.. in

14 57 L. CHAYES AD G. SWIDLE which B defined, e.g., as he bounday of he egion whee nž x,. až x,. is nown o be a coninuous funcion. Then nž x,. saisfies he equaion Ž 3.3. H H 0 B BŽ 0. nž x,. GŽ x. dx Ž x. GŽ x. dx G G B H dsh nž x, s. H F Ž s. ds H GŽ x. dx, x x 0 B s 0 B 0 whee, fo simpliciy, we have assumed ha he es funcion is ime independen. Wih some mild esicions on Ž x. and Ž. 0 F, i is possible o show ha only one pai ŽnŽ x,., B. exiss ha saisfies his equaion. This is he subjec of he Appendix, which is no paiculaly sho and, fom he pespecive of paicle sysems, no paiculaly enlighening. Fo he paicle sysem unde sudy in his secion, i is saighfowad o show ha he wea Ž hydodynamic. limis saisfy Ž 3.. indeed, his is jus a ecapiulaion of he deivaions in Poposiion. and Theoem.. The ey ingedien in his secion is heefoe a poof of coninuiy of he bounday BŽ.. This is he subjec of Poposiion 3.3. Ou final esulhe analog of Theoem.is hen a faily saighfowad coollay o all he above menioned. PROPOSITIO 3.. Conside he paicle sysems on as descibed in 4. and 3. wih bounday geneao a as descibed in.,. and he elevan poion of 3.. Le and n Ž. Ž. a Ž. Ž.. Le us define funcions a Ž x,. and n Ž x,. as in 7. and 8. and le a and n denoe he expeced values of hese quaniies. Then Ž a, n. is a soluion o he Ž Sefan. equaion Ž 3.., whee Ž a, n. denoes any wea limi of Ž a, n.. Ž. Fuhemoe, wih pobabiliy, if a x,, n x, is any wea L limi of he Ž andom. sequence Ž a, n., hen Ža Ž x,., n Ž x,.. also saisfies his equaion. PROOF. We follow closely he pevious deivaions. Fo echnical convenience, hee and in he emainde of his secion, we will define he micoscopic bounday as he posiion of he hole ha is jus o he igh of he egion of fozen paicles: Ž 3.4. b Ý Ž Ž.,. so ha we may sill wie Ž b. L b. Mimicing exacly he seps in 9. o 6., we aive, unimpeded, a he analog of 7.. The ema following he poof of Poposiion. applies hee as well; hee, he unwaned bounday ems ae he fom Ž. g Ž b. g Ž b. Ž b. and can

15 OE-DIMESIOAL PARTICLE SYSTEMS 573 be handled by he same agumen. The saed esul fo Ž a, n. follows immediaely. The second poion of his poposiion is anohe quadaic vaiaion calculaion fo he analog of he maingale M Ž G.. Ž Hee we will use he same noaion fo he coesponding objec.. The esul of an idenical pocedue is exacly 3. wih he b as given by Ž 3.3. above. Since he ae so spase, his implies M Ž G. 0 wih pobabiliy. Le Ž. a, n denoe a single ealizaion and le a, n denoe a wea L subse- Ž quenial limi of a, n. oe ha, osensibly, a, n. is andom. Le G denoe a counable collecion of es funcions Žwih he appopiae bounday. condiion ha ae dense in L. Since, wih pobabiliy, fo all, M Ž G. 0 Ž along he subsequence, his implies a, n. is a wea soluion o he Sefan equaion in he sense of Ž 3... REMARK. Insofa as Poposiion 3. is concened, we can be faily cavalie abou he naue of he bounday and he iniial condiions. Fo he sae of Ž ulimaely. obaining classical esuls, we will assume ha Ž x. and 0 F ae piecewise coninuous. Hencefoh, we will also assume ha Ž x, 0. does no go o 0 oo fas a x o o oo fas a x. Explicily, we assume ha hee ae consans w and v 0 such ha and x w x 0 0Ž x. vž x.. Wih a ceain amoun of addiional labo, all of he above can be elaxedalhough some condiion is needed ha pevens Ž x. 0 in a neighbohood of x. We now aend o he behavio of he boundaies. The saing poin will be a lemma concening a quaniy ha also plays a cenal ole in he coninuum analysis of he Sefan equaion Ž in he Appendix. as well as in he applicaions. DEFIITIO. Le Ý a Ž. denoe he displacemen in he paicle configuaion. Obseve ha if he iniial configuaion has a Ž. 0, hen is exacly how much oal lefwad moion has occued in he paicle sysem. The coec Ž diffusive. scaling fo his objec is seen o be Ž. Ž.. Thus, we fuhe define.

16 574 L. CHAYES AD G. SWIDLE LEMMA 3.. Le be as defined above and le Q H Ž s. F 0 F ds. Then Ž., in he sup nom, wih pobabiliy, whee Ž 0. QF Ž., QF Ž 0., ½, Q F Ž. Ž 0.. REMARK. This is essenially a epea of he analysis in ; fo compleeness, a bief deivaion will be included. PROOF OF LEMMA 3.. The configuaion can be naually divided ino a numbe, K Ž., of disinc clumps of paicles. Howeve, we will adop he convenion ha if Ž., we will no coun he ighmos clump in ou calculaion of K. The quaniy changes by each ime a paicle goes fowad o bacwad, and such evens only ae place a he endpoins of he clumps o a he ighmos Ž non. clump of he sysem. Le denoe any configuaion in which Ž. Ž i.e., b. so hee ae sill some dynamics. Each clump epesens he chance o incease by one uni a uni ae. In addiion, if 0, a new paicle will be inoduced ino he sysem a ae RF and ohewise hee is he final uncouned clump ha allows fo one addiional oppouniy o incease Ž by one uni, a uni ae.. Evidenly, unde he condiion b, Ž 3.5a. a ae K Ž. Ž. R. Similaly, F Ž 3.5b. a ae K Ž. Ž R F.. The above may be expessed, succincly, as Ž 3.5c. L R. F b To emove he Ž unwaned. indicao in Ž 3.6., le us define supb 4 and Ž.,, Ž 3.6. D Ž. R Ž s. ds,. H F I is hus seen ha he quaniy D Ž Q Ž 0.. F is a maingale. Le us compue he vaiance. I is clea ha if, hen Y D D R. Fo, an easy calculaion shows Ž 3.7. Y Ž ;. R K Ž. Ž R. R. F F F Of couse, R F is wha we anicipae fo deeminisic moion. The final few ems involving R F ae insignifican, so he only em of poenial significance is K Ž.. We claim, howeve, ha in any configuaion, Ž K.Ž K.. In- deed, le p m, m,,..., K, denoe he displacemen of he paicle a he

17 OE-DIMESIOAL PARTICLE SYSTEMS 575 lef end of he mh clumpcouning fom he lef. I is clea ha o he igh of he mh clump, hee mus be a leas m holes. Thus, we have p Ž m. and hence Ý p Ž K.Ž K.. Using he bound m m m ' ' E D D 0 E D D 0 c, we can aveage and inegae 3.7 o obain 3 Ž 3.8. E D E Ž D. c, whee, in he above, c and c ae consans of ode uniy. Defining d Ž. D and so on, Ž 3.8. easily implies Ž 3.9. d dž., poinwise, wih pobabiliy. Le simulaion Ž 3.0. Ž 0. Ž s. ds. H 0 denoe he anicipaed end of he Choosing a ime geae han, say, and using Doob s inequaliy Žmodified suiably fo a coninuous-ime pocess ha, wih pobabiliy, has only a finie numbe of jumps., one obains 3 c Ž 3.. Pob sup d d. The above equaion eadily implies convegence in he sup nom, wih pobabiliy and, incidenally, ha. ex we show ha limiing boundaies poduced by he paicle sysem ae coninuous funcions of ime. PROPOSITIO 3.3. Le B denoe he andom bounday escaled so as o be of ode uniy and expessed as a funcion of escaled ime: B b. Then, fo any infinie sequence, hee is, wih pobabiliy, a fuhe Ž osensibly andom. subsequence such ha B conveges o some limi- ing Ž osensibly andom. funcion B ha is monoone and coninuous. PROOF. Ou saegy can be boen down ino hee seps. Fis, we mae use of Lemma 3. o show ha in any paicula ineval of ime, he oveall anspo o he lef is no uneasonably lage. ex we show ha, wih high pobabiliy, he paicle densiy a he bounday is bounded away fom uniy. In ou final sep, we combine he above wo ingediens: in essence, we allow all he available paicles o use all he available anspo o ceae as much bounday as possible in he allowed ime. Afe escaling, wha emeges is Ž excep fo mino deails. a saemen of Holde coninuiy wih exponen. F

18 576 L. CHAYES AD G. SWIDLE To eep he equaions easonable, we will shif eveyhing o he igh by unisso ha he pocess now aes place on he laice sies 0,,..., 4and we will assume ha he B bounday sas ou a 0. Le B Ž q. denoe any subsequence ha conveges poinwise o some BŽ q. fo all aional q in 0,. Obviously, BŽ q. is monoone. If we can esablish ha BŽ q. is coninuous, hen since each B is monoone we will have ha B BŽ. Le. denoe any posiive aional ha is no oo lage and ecall he quaniy v ha was defined in he ema following Poposiion 3.. We will show ha, wih pobabiliy, 3. B q B q ' o ' v BŽ q. holds fo all aional q s in 0,. such ha he lef-hand side is defined. By Ž counable. subaddiiviy, he above can be achieved by woing wih an abiay aional q in 0,.; hus, le q be any aional in 0,.. Recall he displacemen vaiable and he quaniy Q F fom he peceding lemma. We claim ha, wih pobabiliy, Ž 3.3. q q Q q Q q F F Ž holds fo all bu a finie numbe of s. The expeced value of he lef-hand side is he middle em wihou he ; he ighmos inequaliy comes fom he fac ha F.. Ou nex as is o pu a cap on he densiy in he immediae viciniy of he bounday. Le H be a numbe of ode uniy, he pecise value of which will be deemined lae; le h H and le ' b v T h, v, b h vh h. ž / Ou midange goal will be a demonsaion ha h Ý Ž. p Ž 3.4. P b p T Ž h, v, b. Ž H. exp Ž H., whee and 0 ae consans independen of b. We ema in passing ha if he above can be esablished independenly of, hen, by fixing q a his Ž escaled. ime, wih pobabiliy, he saed even occus fo only finiely many values of. Le us sa by poining ou ha Ž 3.5. b h b v v h vh h Ý ž / ž / b v T Ž h, v, b. h

19 OE-DIMESIOAL PARTICLE SYSTEMS 577 is, on he aveage, wha would be obseved fo a poblem wih a linea densiy pofile ha has uni densiy a and a slope of v. This is, of couse, by design; such a sysem will be consuced and used as a compaison. Explicily, le us conside he simple exclusion pocess on v v ½ ž,,..., v / ž v / 5 wih he lef-end bounday loced a zeo densiy and he igh end a uni densiy. Le 0 be he iniial configuaion whee, wihou loss of genealiy, we will se 0 Ž 0. and assume ha 0 Ž. 0, 0. Le denoe he configuaion a ime and le l, v denoe he configuaion saing a 0 evolving unde he usual ules fo he exclusion pocess on ŽŽ v. v.,..., 4 wih he afoemenioned bounday condiions. By coupling accoding o he scheme whee he laice sies Žahe han he paicles. ae encoded wih jump insucions, i is clea ha, fo any and a any ime, l, v 0 0. Hence, if E S and E l, v denoe expecaions in hese pocesses and F is any nondeceasing funcion of Ž all of. is agumens, i is clea ha 3.6 E 0 F E 0 Ž F.. S Thus, we have, pahwise, ha he even descibed in Ž 3.4. is moe liely in he Ž l, v. pocess han in he Sefan sysem egadless of wha b would have uned ou o be. Howeve, he quesion of he value of b may pove o be a nuisance, so we will cicumven his issue by being incedibly waseful. We wie l, v Ž 3.7. h Ý Ž. P 0 S b p T h, v, b p Ý h 0 S ½ Ý 5 p 4 P Ž b p. T Ž h, v, b. b b b Ý h Ý P 0 S b p T h, v, b b p Ý h Ý P 0 l, v b p T h, v, b. b p Of couse, fo any fixed b, we have Ž 3.8. h Ý P 0 l, v b p T h, v, b p ž h / Ý p exp T h, v, b E 0 l, v exp b p.

20 578 L. CHAYES AD G. SWIDLE We will esimae he final em in Ž 3.7. by using dualiy. The following agumen is based on esuls found in ; simila echniques ae used in 5. The dual model o he Ž l, v. sysem is also he exclusion pocess on ŽŽ v. v.,..., 4 wih he addiional feaue ha paicles ae absobed a he lef and igh boundaies. Le us denoe hese ficiious bounday poins Ž which can house an indefinie numbe of paicles. by l and. Saing wih j paicles in he dual model, le us label hese paicles,,..., j and use ˆ o denoe he locaion of he h paicle a ime. By convenion, if fo some s, he h paicle is absobed, fo example, a he igh, we will say ˆ. The dual elaionship be- ween hese models may be expessed as follows: we denoe by Ž. 0 he even ha, a ime, he h paicle is on a sie ha was occupied in he configuaion o is esing a 0. Then, fo any A ŽŽ v. v.,..., 4, A 0 ˆ A Ł Ł l, v l, v Ž 0. A Ž 3.9. E Ž. E, whee Eˆ A l, v denoes expecaion wih espec o he dual pocess saing wih iniial configuaion A. In he case ha A is a singleon, as, he igh-hand side is given by he pobabiliy ha a andom wal saing a his he igh side befoe i his he lefhis is he oigin of he linea densiy pofile in he saionay measue. Fo A, on he basis of 9 and 0, i uns ou ha he igh-hand side can be bounded by he expecaion of he same funcion wih espec o he measue associaed wih a sysem of independen paicles ha, in all ohe especs, behaves idenically o he ineacing sysem. This inequaliy can be deived by following, sep by sep, he deivaion in, Chape 8, Poposiion.7, modifying, when necessay, fo he pesence of bounday condiions. Thus, so fa, we may wie Ž 3.0. Ž. Ł m E E 0 ˆ F ; A l, v j l, v Ž 0. ma j ˆ m 4 Ł l, v Ž 0. m E, whee A,..., 4 j and he F signifies expecaion wih espec o he disibuion of he independen paicle pocess. We have nealy achieved he midange goal. Howeve, o ge he esimae in Ž 3.4., we need o aveage he igh-hand side of Ž 3.8. ove all iniial configuaions 0 and i appeas ha he individual facos in he final em in Ž 3.0. ae angled by he iniial configuaions. eveheless, we claim ha, afe his aveaging ove he iniial configuaionsin which hee ae

21 OE-DIMESIOAL PARTICLE SYSTEMS 579 no coelaions in he disibuion of paicleswha emeges is he poduc of he individual aveages. Explicily, we claim ha j j ˆ m 4 Ł l, v Ł 0 ; m m m Ž 3.. E Ž., whee ² : 0 denoes he aveage oveealizaions of he iniial configuaions in which he paicles ae independenly deliveed accoding o PŽ Ž Ž. and Ž. is he paicle densiy in he Ž l, v. sysem 0 aveaged ove ime and iniial condiions. I is, of couse, noed, by dualiy, ha ² 4 E : l, v. Le us pefom he configuaional aveage of he 0 0 igh-hand side of Ž 3.0. befoe we ae he aveage ove he ime evoluion. Woing in he ensemble of he j nonineacing paicles, we condiion on he locaions ˆ,..., ˆ j and aveage ove he manifesly independen iniial paicle densiies a hese sies. Pefoming he ime aveage, we hus obain Ł ; ˆ ˆ F ; A F ; A m 3. E l, v m E l, v 0, 0 ma ma 0 ˆ ˆ o l, especively. Obviously, he igh-hand side of Ž 3.. facos and he esuling ems ae of he saed fom. oice ha he iniial densiy is smalle han he linea pofile ha epesens he asympoic densiy: Ž. vž. 0. I heefoe follows ha fo all imes Ž. vž. Ž.. We finally aive a m m m whee if and equals o 0 if ; Ł E 0 l, v Ž.... Ž j. Ž 0 m. m Ž 3.3. j m v, 0 j Ł Ł ž / whee in he above i is assumed ha he poins,..., j ae disinc. An immediae consequence of Ž 3.3. is he exponenial esimae: ; h Ý m ž / Ł p ž / Ý E 0 l, v exp b p exp b p h p 0 Ž 3.4. h exp b p, whee e. Looing bac o Ž 3.5., we conclude Ž 3.5. h Ý P Ž b p. T Ž h, v, b. p ž / p v exp h exp Ž. T Ž h, v, b..

22 580 L. CHAYES AD G. SWIDLE The desied esul now follows easily: we eplace b by on he igh-hand side of Ž 3.5. and segue his bound ino Ž 3.7. wih an exa, healhy faco of o accoun fo he sum ove b. By choosing small enough, i is no had o see ha we aive a Ž Le us now mashal all of he facs a ou disposal: wih pobabiliy, wih he possible excepion of only finiely many s, a ime q, hee ae no ' ' moe han H v v B H v H paicles lying in he egion b b H'. Fuhe Ž again, wih pobabiliy, wih only. finiely many excepions, in he ime ineval beween q and Ž q., he oal amoun of lefwad displacemen ha aes place in he enie sysem is no moe han. An elemenay coupling agumen ells us ha, egadless of wha is o happen in he ime ineval beween q and Ž q., he wos case scenaio paicle configuaion a ime q consisen wih he above ' Ž. ' infomaion is ha all of hese H v B H v H paicles ha ae supposed o be in he egion b b H' ae as close as possible o he wall a b and all ohe paicles in he sysem ae lined up, a uni densiy, in he egion b H'. Unde hese cicumsances, he mos efficien use of he available displacemen is o always advance he leading paicle. Allowing, as a bound, all of he neaby paicles o be consumed in he fis insan of ime, each subsequen addiion o he Ž ' bounday mus now coss a gap of size v B H v H.. Thus, fo he oal ime span of, we have Ž 3.6. ' Ž. B q B q H v B H v H ' vž B.. ' H v H We sill have he choice of H a o disposalpovided ha we choose fom a counable se. If we choose he H ha opimizes he igh-hand side of Ž 3.6., we aive a a bound of he fom saed in Ž 3... REMARK. As discussed befoe, he peceding esimaes easily anslae ino a poof of Holde coninuiy wih index. Had we been able o do even he slighes amoun bee, mos of he Appendix would have been unneces- say: Holde coninuiy wih index fo he boundaies is jus he dividing line beween classical and nonclassical behavio fo soluions of he hea equaion. Howeve, afe a momen s eflecion on he above poof, i is seen ha a bee coninuiy esul would have equied he nowledge, in he conex of he paicle sysem, ha he densiy goes o 0 a he bounday. A diec poof of his fac Žwhich, in ligh of he coninuum esuls poved in he Appendix, indeed uns ou o be he case. has o his dae poved elusive. Howeve, i is no difficul o see, in hindsigh, ha hee is a genuine connecion beween he vanishing of he paicle densiy on he one hand and classical behavio on he ohe. Wha is supising o he auhos Žagain in '

23 OE-DIMESIOAL PARTICLE SYSTEMS 58. hindsigh is ha, in he conex of his poblem, his issue was seled in he coninuum ahe han in he paicle sysem. We ae now finally in a posiion whee we can ge o he sysem descibed in 3.3. COROLLARY. Le T and le ŽaŽ x,., nž x,.. denoe any soluion o Ž 3.. on, 0, T ha has emeged as a wea subsequenial limi of a sequence Ža Ž x,., n Ž x,.. coming fom he paicle sysem. Then, wih pobabiliy, hee is a coninuous, monoone funcion, BŽ., ha is he Sefan bounday fo hese densiies. In paicula, až x,. and nž x,. 0 Ž a.e.. if x B and až x,. nž x,. Ž a.e.. if B x. PROOF. Le až x,. and nž x,. denoe any wea subsequenial limis of he sequences a Ž x,. and n Ž x,. ha have been aen along he same subsequence, hee denoed by Ž.. Le B denoe he bounday ha is poduced fom a fuhe subsequence, Ž., as descibed in Poposiion 3.3. Le us pove, fo example, ha až x,. in he egion x, 0, 0, T x BŽ.4. I is sufficien o show ha if V is any closed se in his egion, hen Ž 3.7. až x,. dx d V, whee denoes odinay Lebesgue measue. Le ic funcion Ž indicao. fo he se V. Clealy, H V H H V Ž 3.8. až x,. dx d lim a Ž x,. dx d. V V denoe he chaaceis- Using he monooniciy of he B s and he fac ha, wih pobabiliy fo any finie Ž aional. collecion q,..., q in 0, T, Ž 3.. m holds wih B eplaced by B Ž excep pehaps fo finiely many values of., i is no had o show ha B B unifomly on 0, T. This implies ha, wih pobabil- iy, hee is some numbe R such ha, fo all R, 4 Ž 3.9. V Ž x,. 0, 0, T x B Ž.. Bu in he egion Ž x,. 0, 0, T x B Ž.4, we now ha and hus Ž 3.7. follows. Simila easoning can be used o demonsae he second popey. Thus, as fa as he paicle sysem is concened, we may now expess ou densiies in he language of Ž 3.. o Ž 3.3. as we please. The elevan esul fom he Appendix will be saed below fo convenience: THEOREM A.3. Conside he sysem in Ž 3.. wih he iniial condiions BŽ 0., Ž x. and bounday value Ž.. I is assumed ha boh Ž x. 0 F 0 and ae piecewise smooh, bounded above by and below by 0, and ha F

24 58 L. CHAYES AD G. SWIDLE wž x. Ž x. fo some finie consan w and ha Ž x. vž x. 0 0 fo some posiive consan v. Then hee is exacly one soluion o his poblem and, fuhemoe, his soluion is classical. REMARK. The classical naue of he soluion has no beaing on he cuen siuaion, bu i is woh noing ha on Ž0,. he bounday is a leas C. As such, away fom he endpoins, gadiens of he densiy exis and ae coninuous a he bounday. Thus, we may acually eve o 5., 6., 7. and 8b. o descibe his poblem. The poof of ou pincipal esul is now almos immediae. THEOREM 3.4. Conside he ineacing paicle sysem on Ž whee. 0 ha was descibed in Poposiion 3. wih bounday and iniial condiions ha ae consisen wih hose in he saemen of Theoem A.3. Ž. Then, wih pobabiliy, a x,, n x, conveges wealy, in L, o he unique Ž classical. soluion o Ž PROOF. Le ŽaŽ x,., nž x,.. denoe any limiing densiy as descibed in Poposiion 3.. Passing o a fuhe subsequence, if necessay, we see ha ŽaŽ x,., nž x,.. enjoys a coninuous Sefan bounday. Hence, he limiing paicle densiy acually saisfies Ž 3.3. which, by Theoem A.3, has a unique soluion. This is he desied esul. 4. An applicaion o wo-dimensional inefacial dynamics. Aside fom he sandad, classical inepeaion of he poblems eaed in he peceding secions, he one-dimensional exclusion pocess has a well-nown applicaion o he sudy of wo-dimensional inefaces. The ey obsevaion, due o Ros, is ha if 0 s ae idenified as hoizonal edges and s as veical edges of an ineface on, exclusion dynamics among he 0 s and s epesens a dynamic evoluion of his ineface. In sysems ha ae a pioi infinie, hese models wee analyzed some ime ago: he poblem of an infinie cone wih complee bias was eaed in and he geneal poblem of an infinie cone was solved in, Chape 8. Of couse, hee is no eason ha one canno solve, in he sense of hydodynamic limis, hese poblems on finie laices. Fo example, he exclusion pocess on wih fixed densiy bounday condiions a he endpoins epesens a suface ha is consained o have fixed slopes a is ends. Somewha less ealisically Žsince he ineface has o be a monoone funcion., one may conside his poblem wih peiodic bounday condiions. I uns ou ha hese poblems all epesen he T 0 limi of he usual Ž Gibbs sample. Glaube dynamics of he sochasic Ising model. Hee, he ineface is he bounday sepaaing egions of opposie spin ype. The above poblem wih peiodic bounday condiions as well as moe complicaed inefaces on a cylinde Ž sill a funcion, bu no necessaily monoone. was discussed in 3 in his conex. In paicula, he diffusive behavio ha is ineviable in he paicle sysem anslaes ino he moion by Ž modified.

25 OE-DIMESIOAL PARTICLE SYSTEMS 583 mean cuvaue. This dynamical phenomenon was pediced in 8 and leads o he well-nown Lifshiz law. Of consideable inees, hen, is he behavio of doples unde his so of dynamics. In hese cases, he genealizaion of Lemma 3. goes hough Ž wih some esicions on he iniial shape. and amouns o he saemen ha he volume of he dople deceases linealy wih ime. This is a wea fom of he Lifshiz law. In geneal, such poblems appea o be difficul o model as paicle sysems. Howeve, i is ou undesanding ha he mehods in 3 can easily be exended o cove hese cases, a esul ha is due o Spohn. owihsanding, he behavio of doples nea he edge of a sysem is sill of some inees. Suppose in he egion i i 04 hee is an Ising feomagne wih he usual neaes-neighbo ineacions. If I is he ineface a ime, he Gibbs sample Glaube dynamics a zeo empeaue dicaes ha if J is an ineface ha can be obained fom I by he flip of a single spin, I J a ae if he oal lengh of he ineface is peseved and a ae if he oal lengh of he ineface deceases. All ohe moves ae suppessed. If he ineface exends ino he wall a i 0, i is easily seen ha he inefacial ansiions in his viciniy ae ievesible. In paicula, if he ineface is a monoone Ž say noninceasing. funcion, he behavio a his end is modeled by jus he paicle sysem ha was descibed in Secion 3 i.e., 4.; he enhancemen of he ansiion ae a he bounday uns ou o be of no consequence. In hese sysems, he behavio of finie doples is of consideably geae inees han he behavio in a semiinfinie sysem. Unfounaely, he bounday condiions we have used a he non-sefan end of he sysem ae no paiculaly ealisic fo his so of applicaion. Fo he poblem of an Ising dople in he cone of he sample Žnow defined as he posiive quadan., a paicle-sysems appoach along he lines we have been discussing is applicable. Howeve, he dynamics mus now exhibi complee paiclehole symmey; in paicula, we mus exend he single-paicle space o include an addiional sae ha can bes be descibed as a fozen hole. The dynamics beween paicles, holes and fozen paicles is exacly as descibed in 3. and 4., while he dynamics beween paicles, holes and fozen holes is idenical afe a swiching of labels. Howeve, in his exended sysem, a majo casualy is noaion: assigning numbes o he saes Ž. hee is no paiculaly efficien. Since we will end up descibing his poblem using he enhalpy and auxiliay densiy, we may as well use his noaion fom he ouse. We hus have / až. Ž 4.. Ž., ž n Ž. wih a and n aing on he values 0 o. The pocess is defined by he acion of he geneao L on funcions f of he configuaions on : i, j i, j Ž 4.. L f f f f f Ž., Ý Ž. D F i ji

26 584 L. CHAYES AD G. SWIDLE whee, fis, he swiched configuaions always saisfy i, j i, j D F unless až i. nž i. and až j. nž j.. Fuhe, he diffusive swiches in Ž 4.. i, j ae limied by unless až i. nž i. and až j. nž j. D and ohewise hey ae defined by he usual Ž., if i, j, Ž 4.3. i, j D Ž. Ž j., if i, Ž i., if j. ex, he feezing swiches ae defined by Ž 4.4. Ž i. if Ž 4.4. Ž ii. F i, j Ž. Ž. if i, j ž / ž / i, j i, j 0 F Ž i. and F Ž j. 0 0 ž / ž / 0 a Ž i. a Ž i a i 0, a j and o. nž i. nž i. ž / while až j. nž j. and similaly wih he oles of i and j evesed, and if Ž 4.4. Ž iii. ž / ž / i, j 0 i, j F Ž i. and F Ž j. ž / ž / a Ž i. a Ž i a i, a j 0 and o. 0 nž i. nž i. ž / while až j. nž j. and similaly wih he oles of i and j evesed. In Ž 4.. we have acily included fomal ems involving Ž. and Ž. ; hese ae as- 0 Ž. sumed o ae he values ž / and ž /, espec- 0 ively; elsewhee in he iniial condiion we will be assuming ha a Ž. 0 n Ž. 0. We have also no defined swiches fo he siuaions whee boh he sies i and j have fozen neighbos. In ou paicula insance, only one such ansiion occus, and his is he move ha signals he end of he simulaion. The coninuum descipion of he pupoed limi is, classically, n n Ž 4.5., B x CŽ., x

27 OE-DIMESIOAL PARTICLE SYSTEMS 585 wih some iniial condiion nž x, 0., he bounday condiion a he moving boundaies given by nž B. 0 and nž C. and he Sefan condiions Ž 4.6. db nž BŽ.,., d dc nž CŽ.,.. d In he weaes fom, he Sefan equaion is exacly Ž 3.. wih he bounday em absen Ž and no bounday esicion on he es funcion.. Theefoe, he inemediae vesion is Ž 4.7. H H C CŽ 0. nž x,. GŽ x. dx nž x, 0. GŽ x. dx B BŽ 0. CŽ s. G G H H H H ž 0 BŽ s. x 0 CŽ s. x / n x, s dx ds dx ds B H BŽ 0. GŽ x. dx, whee G is any ime-independen es funcion of compac suppo. Unfounaely, fa less is nown abou he soluions o he above sysem han in he case wih a single bounday. In paicula, we can only show uniqueness unde he assumpion of complee symmey in he iniial condiion: Ž x. Ž x Howeve, his coves a case of pincipal concen, namely ha of a dople ha is iniially squae. Hencefoh, we will esic aenion o his case. Ou pimay esul of his secion will be poved along lines ha follow vey closely he pevious deivaions. THEOREM 4.. Conside he ineacing paicle sysem defined on descibed in Ž 4.. o Ž 4.4. wih 0 and iniial condiions coesponding o BŽ 0., CŽ 0. wih Ž x, 0. if x 0 and Ž x, 0. 0 if x 0. Le Ža Ž x,., n Ž x,.. denoe he quaniies defined naually fom Ž x,. as Ž. in.8. Then, wih pobabiliy, a x,, n x, conveges wealy, in L, o he unique soluion wih his iniial condiion. PROOF Ž Sech.. In his sysem, we define b exacly as in Ž 3.4. wih he obvious modificaion fo his secion s noaionand, similaly, Ý. 4.8 c Ž Ž., 0. Thus, explicily, he C-bounday is he locaion of he paicle ha is igh on he edge of he line of fozen 0 s. The fis sep is o deive he discee analog of Ž 3... Following he seps of Poposiion., we ge his equaion, in maingale fom fo he quaniies M Ž G. defined exacly as in 5. 7.

28 586 L. CHAYES AD G. SWIDLE Žwihou he R F ems and wih he new meanings assigned o he n s and a s., excep ha his ime, he unwaned lefove ems ae he fom Ž. gž b. gž b. až b. Ž. gž c. gž c. až c.. Howeve, hese ems ae dispensed wih by using exacly he agumen found in he ema following Poposiion.. Evidenly, he aveages conain he desied wealy conveging subsequences. A quadaic vaiaion calculaion along he lines of Poposiion. can be caied ou. Afe some effo, i can be checed ha he same sos of ems wih he expeced addiional lefoves caused by he pesence of he exa bounday emege along wih he desied esul, namely ha, wih pobabiliy, andom subsequences convege o wea soluions of he Sefan equaion wih wo boundaies. ex, he agumens of Poposiion 3.3 can be applied wih only a few modificaions. Fis, i uns ou ha Q F is eplaced by in all maes ha elae o he displacemen. Second, he agumen mus officially be pefomed fo each bounday individually Žalhough he sepaae agumens ae idenical.. Thid, focusing aenion on he lef half of he configuaion,,,..., 04 whee, wih pobabiliy, wih only finiely many excepions, he bounday B is locaed, he configuaions hee may be dominaed by a seup ha has uni densiy a he oigin and zeo densiy on he lef end. Finally, fo he case of his paicula iniial disibuion, he esuls of Theoem A.3 can be exended o he wo-bounday poblem Ž Poposiion A.5., and, puing hese ingediens ogehe Žas was done fo he single-bounday case., he desied esul is esablished. REMARK. Fom he limiing soluion, Ž BŽ., CŽ., nž x,.., he suface can be econsuced, paameically, by he fomulas Ž 4.9a. Y Ž s,. B QŽ s,., Ž 4.9b. X Ž s,. s B QŽ s,., s whee Q s, H Ž x,. BŽ. dx. This shape is achieved by he sochasic model, fo example, in he sense ha, fo a.e. in 0,., Ž X Ž s,., Y Ž s,.. ŽXŽ s,., YŽ s,... Moe efined noions of convegence in hese poblems will be pusued in he fuue. APPEDIX We will now poceed wih he analysis of he Sefan poblem ha is defined, classically, by 5., 6., 7. and 8b. o, in moe genealiy, by Ž In ode o simplify he fohcoming discussion, we will analyze his Ž. Ž poblem on 0, 0, whee is defined below.. Mos of wha is o follow will be he analysis of a easonably geneal single Ž moving. bounday poblem subjec o he Sefan condiion. We ae looing fo some B ha is