Physica D. Survival probability for the stadium billiard. Carl P. Dettmann, Orestis Georgiou. a b s t r a c t

Transcription

1 Physica D Conens liss available a ScienceDiec Physica D jounal homepage: Suvival pobabiliy fo he sadium billiad Cal P. Demann, Oesis Geogiou School of Mahemaics, Univesiy of Bisol, Unied Kingdom a i c l e i n f o a b s a c Aicle hisoy: Received 6 Januay 2009 Received in evised fom 17 July 2009 Acceped 22 Sepembe 2009 Available online 6 Ocobe 2009 Communicaed by T. Saue Keywods: Sadium billiad Open billiads Escape ae Powe-law decay We conside he open sadium billiad, consising of wo semicicles joined by paallel saigh sides wih one hole siuaed somewhee on one of he sides. Due o he hypebolic naue of he sadium billiad, he iniial decay of ajecoies, due o loss hough he hole, appeas exponenial. Howeve, some ajecoies bouncing ball obis pesis and suvive fo long imes and heefoe fom he main conibuion o he suvival pobabiliy funcion a long imes. Using boh numeical and analyical mehods, we concu wih pevious sudies ha he long-ime suvival pobabiliy fo a easonably small hole dops like Consan ime 1 ; hee we obain an explici expession fo he Consan Elsevie B.V. All ighs eseved. 1. Inoducion Chaoic billiad heoy was inoduced by Yakov Sinai in 1970 [1]. Since hen i has developed o become a well esablished heoy fo dynamical sysems. A billiad is a dynamical sysem in which a paicle alenaes beween moion in a saigh line and specula eflecions fom he domain s bounday. The sequence of eflecions is descibed by he billiad map which compleely chaaceizes he moion of he paicle, hence billiads have hei boundaies as a naual Poincaé secion. Billiad sysems ae convenien models fo many physical phenomena, fo example whee one o moe paicles move inside a conaine and collide wih is walls. An excellen and compehensive mahemaical inoducion o chaoic billiad heoy can be found in he book of Chenov and Makaian [2]. In he ealy 80 s, mahemaicians suggesed invesigaing open sysems, sysems wih holes o leakages, as a means of geneaing ansien chaos [3], eieving infomaion fom disibuions [4], and deducing facs abou he equivalen closed sysems. The key disibuions of inees classically ae he escape pobabiliy densiy p e, which is given by he ajecoies ha leave he billiad a ime, whee 0, R, and also he suvival pobabiliy of obis P up o ime, given some iniial pobabiliy measue, ypically he equilibium measue defined in secion II below. These wo ae elaed by P = p e d. Such invesigaions have Coesponding auho. Tel.: addesses: Cal.Demann@bisol.ac.uk C.P. Demann, maxog@bisol.ac.uk O. Geogiou. naually been exended o billiad sysems as well. Links beween billiads and geomeical acousics [5 9], quanum chaos [10,11], conolling chaos [12 14], aom opics [15], hydodynamical flows [16 21], asonomy [22,23] and cosmology [24], have been esablished in he conex of open dynamical billiads. Fuhemoe, i has become appaen ove he pas few yeas, ha he subjec of open billiads and hei disibuions povide a pahway owads undesanding chaos and may even open doos o old, bu no so fogoen poblems such as he Riemann hypohesis [25,26]. The sadium billiad see Fig. 1 is a seemingly simple dynamical sysem, inoduced by Leonid Bunimovich in 1974 [27]. The billiad s bounday consiss of wo paallel saigh lines and wo semicicula acs. I was lae poven by him o be egodic, o be mixing, o have he Kolmogoov popey [28], and in 1996 by Chenov and Haskell o have he Benoulli popey [29]. I has been descibed as a sysem wih fully developed chaos [30]. Is enopy has been numeically esimaed in [31] and heoeically in [32]. The sadium billiad is a special case of a chaoic billiad. Being consuced fom wo fully inegable billiad segmens, he cicle and he ecangle, i is emakable ha he sysem emains compleely chaoic no mae how sho is paallel segmens ae. I is a limiing case of he lage se of Hamilonian sysems, which Bunimovich efes o as mushooms [33] wih cleanly divided phase-space aeas, egula and chaoic. The sadium is he fully chaoic limi of he naual mushoom billiad, while he cicle is he fully egula limi. If he paallel segmens ae of lengh 2a say, whee a > 0, and a R, hen he Lyapunov exponen λa 0 in boh limiing cases of a 0 and a. Also, i is well known ha he defocusing mechanism, which is one of he wo souces of chaos in billiads [34] he ohe being he dispesing mechanism, chaaceisic of all Bunimovich ype billiads, equies a > 0 in /$ see fon mae 2009 Elsevie B.V. All ighs eseved. doi: /j.physd

2 2396 C.P. Demann, O. Geogiou / Physica D ode fo any wave-fon o defocus and heefoe exhibi hypeboliciy. Wojkowski in 1986 [35] claified much of he mechanism behind his hypebolic behavio. The vey exisence of he paallel segmens of he bounday is also he souce of he inemien behavio found in he sadium billiad. They allow fo he exisence of a se of maginally sable peiodic obis of zeo measue bu indeed of gea impoance. They ae he main eason why he sadium is no unifomly hypebolic. Also, hough i is classically and quanum mechanically egodic, i does no have he popey of unique egodiciy [36,37]. This means ha no all eigenfuncions ae unifomly disibued and heefoe his causes scaing [38]. This is due o he exisence of he so called bouncing ball obis, someimes called sicky obis. Semi-classically, hey have caused much ouble in he eamen of he sysem as explained in gea deail by Tanne [39] since hey affec he sabiliy of peiodic obis close o hem bu do no conibue o individual eigenvalues in he specum of he sadium. Lai-Sang Young s infinie Makov exension consucion called a Young owe in 1998 [40,41] iggeed a seies of igoous mahemaical poofs concening he long ime saisical popeies of he sadium billiad. In 2004 Makaian [42] poved ha asympoically he billiad map in he sadium has a polynomial decay of coelaions of ode log n 2 n 1 hee n is he numbe of ieaions of he billiad map. This mehod was hen simplified and genealized by Chenov and Zhang in 2005 [43] o include fo example he dive-bel sadium whee he saigh segmens ae no longe paallel. Bálin and Gouëzel [44] in 2006, used his mehod o pove ha he Bikhoff sums of a sufficienly smooh geneic obsevable wih zeo mean in he sadium, saisfy a non-sandad limi heoem whee is convegence o a Gaussian disibuion equies a n log n nomalizaion. In 2008, Chenov and Zhang shapen hei pevious esimae by emoving he log n faco [45] and Bálin and Melboune show ha hese elaions hold fo obsevables smooh in he flow diecion as well his excludes posiion and velociy [46]. These esuls fo he ae of decay of coelaions can, a leas heuisically, be ansfeed ino he conex of he open sadium o addess poblems such as escape aes and suvival pobabiliies. Theefoe, even hough he sadium billiad has been obseved o exhibi song chaoic popeies fo sho imes such as appoximae exponenial decays of he escape imes disibuion and decays of coelaions of iniial condiions boh numeically and expeimenally, i has also been shown o expeience a cossove a longe imes, owads an asympoic powe-law behavio [47 49]. Hence, he sadium billiad is an example of a ansien chaoic sysem which exhibis inemiency. Inemiency descibed in moe deail in Secion 2, in open sysems is a elaively new subjec and is inceasingly being discussed and eseached in he conex of non-linea Hamilonian sysems. These invesigaions end o focus on he asympoic behavio of disibuions and ofen use he sadium as one of hei main examples [50,51]. In fac, i has ecenly been suspeced ha he vey long egula flighs pesen in he expanded sadium ae he eason why numeically, a leas, he momens of displacemen divege fom he Gaussian [52]. I is aguably an ideal model fo sudying he influence of almos egula dynamics nea maginally sable boundaies boh heoeically [39] and numeically [53]. Amsead Hun and O [54] have caied ou a deailed invesigaion ino he asympoic sadium dynamics and have shown ha P Cons fo long imes bu do no calculae he Consan. In his pape we sudy he escape fom he sadium billiad, we explicily calculae he measue of he se of obis causing he asympoic powe-law decay, and obain an analyic expession fo he longe imes suvival pobabiliy funcion. In conas wih Ref. [25], we do no assume o equie ha he hole is vanishingly Fig. 1. The Se-Up of he sadium billiad. small and in conas o Ref. [54], we do no use a pobabilisic descipion of he dynamics. The pape is oganized as follows. In Secion 2 we se he sage and sae he easoning as well as he main ideas of his pape and we also se up ou poblem and define all he vaiables and ses equied. In Secions 3 and 4 we conside he wo main ses of iniial condiions which conibue o he suvival pobabiliy a long imes while in Secion 5 we inoduce and explain he deails of he appoximaion mehod used. Finally, in Secion 6 we pesen ou numeical esuls fom compue simulaions and compae wih he analyical ones. Conclusions and discussions appea a he end, in Secion 7, whee fuue wok is also discussed. 2. The main ideas and se-up Conside an open sadium as shown in Fig. 1. A classical nonineacing paicle of uni mass and uni speed expeiences elasic collisions on Ɣ, he bounday of he billiad able. The lengh of he paallel sides is 2a, while he adius of he cicula segmen is. A hole of size ɛ is punched ono one of he saigh segmens of Ɣ wih x coodinaes x h 1, h 2, h 2 = h 1 + ɛ. x is he posiion coodinae which we only need o ake values along he saigh segmens, x [ a, a]. We define he inwad poining nomal veco ˆn which defines he angle θ made by he efleced paicle and ˆn. The angle θ π/2, π/2 and is posiive in he clockwise sense fom ˆn. The billiad flow conseves he phase volume and he coesponding invaian equilibium measue along he saigh segmens is dµ = C 1 cos θdθdx, whee C = π/2 π/2 cos θdθ Ɣ d = 2 Ɣ = 2 4a + 2π is he canonical pobabiliy measue peseved by he billiad map on he billiad bounday. dµ is also he disibuion of iniial condiions. Fo he pupose of his pape, no paameizaion of he posiion coodinae is needed along he cicula segmens of he billiad. Finally, if he paicle his he hole, i will escape; as peviously noed we ae ineesed in he long ime behavio of he suvival pobabiliy. As discussed in he Inoducion, he sadium is hypebolic as a esul of he defocusing mechanism. Howeve i is no unifomly hypebolic, as hee exis a small se of paabolic, non-isolaed peiodic obis [55] called bouncing ball obis which is of zeo measue. I has been obseved ha obis in he chaoic egion of he phase space which ae close o hese maginally sable peiodic obis show almos egula behavio fo finie imes. As a esul, he decay hough he hole appeas exponenial fo sho imes, followed by an algebaic ail [56]. A heoy fo explaining he inemediae ime ansiion fom exponenial o powe-law decay is ha of inemiency and was fis inoduced in 1979 by Manneville and Pomeau in hei sudy of he Loenz sysem [57,58]. Usually, inemiency signifies a small, finie ime Lyapunov exponen fo unsable peiodic obis appoaching egula egions in phase space [59]. Tajecoies almos angen o he cicula acs whispeing galley obis, o olling obis ae inemien wih

3 C.P. Demann, O. Geogiou / Physica D espec o he collision map, bu no wih espec o he flow. This is because hey ae of a bounded oal pah lengh and ae sufficienly unsable [39,46]. Nea bouncing ball obis on he conay ae no of bounded oal pah lengh and ae geneally believed o be he main cause of he inemiency of he flow, heefoe exclusively deemining he asympoic ail [49,54] of he suvival pobabiliy P and heefoe play a significan ole in he sadium s dynamics. These obis ae chaaceized by small angles θ nea veical ha emain small fo elaively long peiods of ime. Chenov and Makaian descibe hem in hei book [2] as obis wih a lage numbe of nonessenial collisions. Semi-classically, i has been suggesed ha an island of sabiliy suounds his maginally sable family. Is bounday depends explicily on h and he measue of his island shinks o zeo compaed wih he oal volume in he semiclassical limi h 0 [39]. Having noed, following [49,54], ha he se of obis ha suvive fo long imes is conained in he nea bouncing ball obis, wih small angles θ and posiion on he saigh segmens, we now caegoize hese obis ino wo simple families: obis iniially moving owads he hole, and obis iniially moving away fom he hole. We would like o idenify he se of obis fom hese wo families which do no escape unil a given ime. An impoan esul by Lee, ha daes back o 1988 [60], saes ha he angle of a nea bouncing ball obi in he sadium emains small even afe a eflecion wih a semicicula segmen. In fac, as will be explained in Secion 4, small angles can change by a mos a faco of 3 afe being efleced off he cuved billiad bounday. Theefoe, given a sufficienly lage ime consain, suviving obis ae esiced o angles insufficien o jump ove he hole, even afe a eflecion on he cicula pa of he billiad bounday. In his way we idenify he suviving obis as membes of ime dependen, monoonically shinking subses of he wo families of obis defined above. The measue of hese subses ends o zeo as. These wo subses ae consideed in deail in Secions 3 and 4 below and ae used o calculae, o leading ode, he sadium s suvival pobabiliy funcion fo long imes P see Eq Case I: Moving owads he hole We sa by consideing ajecoies iniially on he igh of he hole wih x h 2, a moving owads i. These ajecoies will pove o be only a pa of he suvival pobabiliy funcion fo long imes. Howeve, hey ae essenial in ode o consuc a complee and accuae expession fo he asympoic limi of he full suvival pobabiliy funcion. To ensue ha such ajecoies will escape when hey each he hole s viciniy, hey mus saisfy he following condiion: ɛ θ < acan, 1 4 hence hey will definiely no jump ove he hole. The se of iniial condiions x, θ fo ajecoies which will escape in exacly ime saisfies: sin θ δ4 an θ x h 2 a h 2, 2 whee 0 < δ < 1. Fo long imes δ4 an θ, which is he hoizonal disance fom he edge of he hole o whee he paicle exis he billiad, will shink o zeo 1/ as he se of suviving ajecoies is limied o nea veical angles. Hence we dop his nonsignifican em in wha follows. Noice ha we will be using physical ime fo ou calculaions, bu he equaions ae se up as if consideing a map beween p N collisions wih = 2p cos θ. This way, we do no need o define equaions fo he billiad map, which in any case would jus be descibed by he usual eflecion map. Fom Eqs. 1 and 2 we can deduce ha he angles mus saisfy: { ɛ a h2 } θ < min acan, acsin + O1/ The second em in 3 is he dominan one fo long imes. This leads o he following inegal fo he conseved measue of he billiad map: I = 2 C acsin a h 2 0 +O1/ 2 a h 2 + sin θ+o1/ cos θdx dθ, 4 whee he subscip sands fo igh and C was defined in Secion 2. We ae inegaing ove he se of iniial condiions, on he igh of he hole, which will no escape unil ime. Hence we ae consideing escape imes geae han o equal o. We have also dopped he modulus sign fom θ and muliplied he whole expession by a faco of 2, due o he veical symmey of he poblem. This simplifies o 2 a h2 I = C + O1/ 2. 5 This esul is valid fo ajecoies saisfying: a h2 ɛ acsin + O1/ 2 < acan, 4 ha is 8a ɛ, 6 since he supemum of a h 2 is 2a. We coninue wih his calculaion by adding he analogous conibuion I l fom he small angle ajecoies saing on he lef of he hole wih x a, h 1 moving owads i. This opeaion can easily be calculaed fom Eq. 5 by simply sending h 1 h 2 and h 2 h 1. 2 a + h1 I l = C + O1/ 2. Adding he wo inegals gives he measue of all iniial condiions moving owads he hole ha suvive unil ime : 2 2 a + h1 a h2 I +l = C + C + O1/ 2. 7 Hence, pa of he canonical Suvival Pobabiliy funcion due o he nonessenial obis iniially appoaching he hole fom eihe side, fo long imes saisfying condiion 6 is: 2 a + h1 + 2 a h 2 P 1 = 24a + 2π + O1/ 2. 8 This expession is essenially a sum of conibuions fom wo families of bouncing ball obis, each popoional o he squae of he available lengh. 4. Case II: Moving away fom he hole Numeical simulaions confim ha P 1 in Eq. 8 is indeed no he full expession fo he long ime suvival pobabiliy funcion of he open sadium billiad. The se of obis accouned fo in he pevious secion is only a facion of all he maginally sable peiodic obis discussed in he Inoducion of his pape. In his secion we will be consideing obis iniially moving away fom he hole, so ha hey expeience a eflecion pocess when hey collide wih he igh semicicula end. We only conside he igh semicicula end, as we shall lae use he symmey of he sadium o see wha happens a he ohe one. If he iniial angles ae small, hen he final angles, afe being efleced a he wings of he sadium, will emain small and heefoe suvive fo long imes and accoun fo he emaining se of obis and build up he long ime suvival pobabiliy. In his secion we will invesigae and

4 2398 C.P. Demann, O. Geogiou / Physica D Fig. 2. The wo possible scenaios of eflecion fom he semicicles, whee d 1 = a x 1 = a x i 2nθ i > 0. idenify exacly he iniial condiions which suvive fo long imes. As in he pevious secion, his invesigaion is based on he assumpion ha Eq. 1 povides an uppe bound on he magniude of angles consideed, heefoe ensuing ha he obis consideed ae indeed bouncing ball obis ha suvive fo long imes. Thoughou his and he following secion we will be using x i, θ i, whee x i = x 1 2nθ i, as he coodinaes of ou iniial condiions which lie on he igh of he hole, x i h 2, a, and move away fom i. Due o he sadium s symmey, we only need conside he case θ i > 0. We will use x 1, θ 1 o indicae he posiion and angle of a ajecoy igh afe is final collision on a fla segmen, while sill moving away fom he hole. Theefoe, he nex collision of such a ajecoy will be on he igh semi-cicula segmen of he billiad. This helps o disinguish beween he iniial condiions and hei ansfomed final values. Noice ha θ i = θ 1. We begin by fomulaing he ime o escape Tx i, θ i = 2n + 2m + D f, 9 cos θ i cos θ f whee n and m ae he especive numbes of non essenial collisions befoe and afe he eflecion pocess on he igh semicicula end, and ae defined as: a xi n = 2 an θ i δ i a h2 m = 2 an θ f δ f, wih 0 < δ i,f < 1, and f = 3, 4. We chose hese indices fo f 3 and 4 o indicae he numbe of collisions compising he eflecion pocess as he angles change wo o hee imes especively. D f is he ime aken fo he eflecion pocess a he semicicula end, and is bounded by: 1 4 < D 3 < 2 6 < D 4 < cos θ i cos θ cos θ i cos θ 4 The numbe of eflecions on he cuved bounday alenaes beween wo scenaios, as shown in Fig. 2, he case wih one collision on he semicicle and he case wih wo collisions depending on he iniial condiions of he ajecoy x i and θ i. Specifically, θ f can be found and defined by he use of small angle appoximaions as: θ 3 = 2d 1 θ 4 = 4d 1 3θ i < 0, 10 5θ i > 0 11 see Refs. [54,60], whee d 1 = a x 1 = a x i 2nθ i > 0, is he hoizonal disance beween x = x 1 and x = a as indicaed in Fig. 2. Taking only he leading ode ems of θ 1 and θ f fom ou expessions is jusified by he fac ha in long ime scenaios, unsable peiodic obis have vey small angles befoe and afe being efleced, acing ou nea veical ajecoies. Using he above infomaion, Eq. 9 can be expessed in he following way: Tx i, θ i = a x i θ i + a h 2 θ f + f, 12 whee f = D f 2 δ i cos θ i + δ f. cos θ f Befoe we coninue, i is essenial o find he boundaies of validiy fo he funcions of θ f. These will define he geomey of he wo scenaios. We ask he quesion: When do we see one and when wo collisions a he semicicula ends? This is answeed by consideing a numbe of inequaliies. The fis and mos obvious one is θ 1 d 1 which equies he nex collision o be on he 2 cicula segmen. We also noe ha θ 1 = 3d 1 is he ansiion line 4 beween θ 3 and θ 4 and is he case whee he efleced paicle will hi exacly he poin x = a, whee he cicula segmen mees he saigh segmen. Finally, if we ae also o saisfy condiion 1 we mus fom wo moe inequaliies: Eq. 10 gives θ i < 2d 1 3 θ i < 4d 1 5 acan + 3 and Eq. 11 gives acan 5 ɛ 4 ɛ 4, These inequaliies enclose a small aea in he x i θ 1 plane, he plane of iniial condiions. Noice ha d 1 is a funcion of x i bu also depends on n, he numbe of collisions befoe he eflecion pocess, which if no equal o zeo inoduces an exa θ 1 em. This means ha inequaliies 13 and 14 have o be solved fo θ 1 fo evey n = 0, 1, 2, 3... This is done and shown in Fig. 3, up o and including n = 2, whee z is aken o be equal o acanɛ/4. These boundaies of validiy define he se of obis which escape afe being efleced a he igh semicicula segmen of he billiad. I is no immediaely clea fom Fig. 3, bu he peaks of hese spikes ae of he same heigh θ 1 = 3z. Howeve, we noe ha he se of obis ha suvive up o ime T, whee T is lage, is no idenical o he fome se. To moivae wha is o follow we have a look momenaily o Fig. 4, which on is lef panel shows a numeical simulaion which

5 C.P. Demann, O. Geogiou / Physica D Fig. 3. Colo online The se of iniial condiions iniially on he igh of he hole, which collide and eflec on he igh semicicula segmen of he sadium and do no coss ove he hole ae defined by he boundaies of validiy. These ae shown hee fo n = 0, 1, 2. The doed diagonal lines ae given by θ i = a x i /2n + 1 and sepaae he plane ino he elaive aeas of n. The op iangle of each spike dak blue is fo f = 3 one collision on semicicle while he boom ligh blue is fo f = 4 wo collisions on semicicle. They ae sepaaed by he saigh lines given by θ i = 3a x i /23n + 2. The emaining wo ses of lines which define he spikes ae given by he soluions of Eqs. 13 and 14. Noice ha all spikes have he same maximum heigh of 3z = 3 acanɛ/4. Fig. 4. Lef: Numeical simulaion idenifying he se of iniial condiions iniially moving away fom he hole ha suvive unil ime = 50. Righ: Aea enclosed by he hypebolas fo imes = 50, fo n = 0, 1, 2 and 3. The line unning hough he middle of each spike and he diagonal lines sepaaing hem ae as descibed in Fig. 3. Noice ha unlike Fig. 3, he heigh of each spike is diffeen. This is because he spikes ae fomed by segmens of he ime dependen hypebolae defined in Eqs. 15 and 16. This is fuhe explained in he ex. The paamees used fo boh lef and igh figues ae: a = 2, = 1, ɛ = 0.2, h 1 = ɛ, h 2 = 0. The ageemen of he wo figues indicaes ha = 50 is sufficienly lage. idenifies he se of iniial condiions x i, θ i which suvive unil ime = 50. Noice ha he peaks of he spikes gow in heigh as we move fom igh o lef, moving away fom he end of he fla segmen a x = a, and heefoe in a sense inceasing he coun of pe-eflecion collisions n. To find he suvival pobabiliy funcion, P 2, of hese ajecoies we mus now conside Tx i, θ i as he paamee. Hence we can eaange Eq. 12 accoding o 10 and 11, o fom wo expessions which depend no only on x i, θ i and n, bu also on he ime : 2d1 f 3 x i, θ i, a x i 3θ i a h 2 θ i 2d θ i θ i = d1 f 4 x i, θ i, a x i 5θ i + a h 2 θ i 4d θ i θ i = The above expessions ae conic secions as hey ae quadaic in boh x i and θ i and descibe hypebolas in he plane of iniial condiions. This is no immediaely obvious because of he facos x i and θ i which ae hidden in he d 1 em. The wo hypebolas, 15 and 16, appoach each ohe as an effec of inceasing he ime and il and shif disconinuously when inceasing n. We noice ha if we impose hese hypebolas ono he boundaies of validiy we found ealie see Fig. 3, we ae essenially imposing a ime consain on he se of iniial condiions which will suvive up o ime. Thei effec will be fo small n o eode he aea enclosed by he inequaliies, heefoe shapening hem, and fo lage n o hicken hem fom eihe side, effecively shaping hem ino a seies of spikes, allowing fo lage and lage values of θ i as we incease n fom lef o igh. This effec can be seen on he igh of Fig. 4, whee he boundaies of validiy fo n = 0, 1, 2 and 3 ae eoded by he hypebolas which ae ime dependen, causing each spike o gow alle as we move away fom he edge of he saigh segmen of he billiad. I can be easily seen ha he picues in Fig. 4 ae almos idenical o a vey small eo, confiming ha we ae measuing he coec se of iniial condiions and heefoe he obis hey descibe. The hickening effec is a consequence of he suvival pobabiliy funcion s se up. Tajecoies which fall jus ouside of he aea enclosed by he boundaies of validiy, bu fo lage enough n, ae apped beween he wo hypebolas and will no evenually escape hough he hole, i.e. hey will jump ove i, bu hey will sill suvive unil he given ime. We noice ha fo finie, n is also finie. The key of he elaion beween and n lies in he conic secion equaions Solving hem fo n we discove ha fo n = and 4 n = especively, he conic secions ae no longe hypebolas bu un ino negaive paabolas. Hence, we can define

6 2400 C.P. Demann, O. Geogiou / Physica D Fig. 5. Colo online The cones which make up he polygons which appoximae he hypebolas, fo any n, ae defined by he coodinaes of poins A, B, C and he coesponding coodinae of he endpoin of he saigh segmen in his case a, 0. The x i and θ i coodinaes of hese cones ae clealy maked by he doed lines. The colo coding is simila o ha of Fig. 3. The geen/dashed veical line indicaes he hypoheic posiion of he closes edge of he hole in his case x i = h 2 which acs and changes he shape of he spike by defining wo new cones A 1 and A 2 insead of A. he maximum numbe of pe-eflecion collisions fo finie ime as: { N max = min, } = 17 fo lage, whee he lowe squae backes h ae defined as he inege pa of h also known as he floo funcion. Acually, as we shall find ou in he nex secion, his em N max is neve eached in pacice see N 3 in Eq. 22. I seems ha we have he aea of inees well defined and bounded. We hus need o inegae ove he aea of each spike and hen sum hem all up o N max fo any given. Muliplying he esul by a faco of 2 veical symmey, would evenually give he measue of he obis iniially on he igh of he hole moving away fom i. Inegaing hypebolas and hen summing hei enclosed aeas is a lenghy and unpleasan pocess. This calculaion has been done numeically and an accuae esul has been obained successfully and is pesened in Secion 7. Howeve, his calculaion can only be caied ou numeically, as an analyical esul is in ou opinion impossible o obain. Theefoe we shall pesen a simple appoximaion mehod fo P 2, which is analyically acable, bu sill accuae o leading ode in 1/. 5. Appoximaing hypebolas In his secion we will ake Eqs. 15 and 16 and ague ha fo lage enough imes, he secions of he hypebolas ha ae of inees can be simply and accuaely descibed by saigh lines. We can visually confim his fom Fig. 4; howeve, fuhe invesigaions have shown ha he disance beween he foci of each hypebola conveges o zeo fase han he lenghs of he inegaion limis on θ i = a x i /2n and θ i = a x i /2n+1 as. In fac, fo any n we find ha he coesponding aes ae 1.5 and 1. This in un shows ha he eo made by appoximaing hypebolas by saigh lines, afe inegaing and summing ove n, is sill negligible wih espec o he leading em 2 ahe han 1. This appoximaion mehod will lae be veified by fuhe numeical simulaions whee we calculae he eo beween he appoximae soluion and he numeical inegaion esul. In his secion we will use τ = wihou any subscip o avoid unnecessay confusion given ha fo long imes, will compleely vanish fom ou esuls. We conside he same iniial condiions as in he pevious secion. Fo he fis sep of he appoximaion mehod, we mus find he coodinaes whee Eqs. 15 and 16 mee wih θ i = 3a x i. We also need o find he coodinaes whee Eq. 16 mees wih θ i = a x i 21+n and finally Eq n wih θ i = a x i. These hee poins along wih a, 0 define he fou 2n cones of a quadilaeal in phase space shown in Fig. 5. Hee ae hei coodinaes: A = a + 6a h n 9a h 2, 4 + 6n 3τ 3τ 4 + 6n B = a + 2a h 2n 32n τ, a h 2, 3τ 2n C = a + 2a h 21 + n n τ, a h 2 3τ 21 + n The nex sep is o fom fou equaions, one fo each side of he quadilaeal, by using hese coodinaes fo n 0. Afewads, elemenay inegaion mehods in x i ae used o poduce explici funcions fo he aea of each spike wih τ and n as he only paamees: Aea 1 = a h 2 2 2n τ 21 + n τ. 18 Befoe we inse Eq. 18 ino a sum, we mus figue ou he uppe limi of n fo which his expession is valid. We do his by finding he smalles inege value of n, fo which he x i coodinae of poin A is smalle han h 2, and call i N 1. This is because, fo inceasing values of n, all he cones A, B, C shif o he lef causing he spike o il and sech, bu when n N 1, poin A is no longe a valid coodinae. A close look a he siuaion eveals ha poin A will spli ino wo poins, A 1 and A 2 say, boh siuaed on he line x i = h 2. Fo n > N 1, he quadilaeal is eplaced by a penagon as he spike s peak poin A ovesho he viciniy of he hole s locaion. This pocess is bes descibed diagammaically in Fig. 5 whee he fou cones A, B, C and a, 0 ae eplaced by A 1, A 2, B, C and a, 0 which ae hen fed back o he inegaion mehod o poduce a new expession descibing he aea of he uncaed spike. The same will happen o all cones, and diffeen combinaions of hem will be necessay o poduce he coesponding aea funcions. In ohe wods, as he x i coodinaes of cones A, C and B, in his ode, oveshoos he hole s locaion a x i = h 2, as we incease n, a new aea funcion Aea j, j = 1 4 via inegaion, wih a new expiaion numbe summaion limi via equaion solving in n, will be equied. This pocess poduces hee moe aea funcions, and heefoe fou summaions, each wih diffeen limis: Aea 2 =, a h 2 2 [ 1152n τ 114τ 2 + 9τ 3 τ + 192n τ + 4n τ 2 ] 762τ + 4n τ 2 45τ τ n2 + 3n nτ 2n τ [ 21 + n τ 9τ + 4n 4 + 6n 3τ], 19

7 C.P. Demann, O. Geogiou / Physica D a h 2 32n n ττ whee Aea 3 = [ ], 20 9/u 3 z 41 + nτ 2n 9τ + 4n 4 + 6n 3τ 1 =, 24 Aea 4 = a h + 3/u n + n z 2 =, 24 31/u Having all he puzzle pieces a hand, we fom an expession fo z 3 =, he invaian measue of all he iniial condiions moving away 1/u + 3 fom he hole fom he igh. The sum ove he aeas of quadilaeals Aea 1 added o he sum of penagons Aea 2 and anohe z 4 =, sum of quadilaeals Aea 3 and finally he infinie sum of iangles, Aea 4, gives: n=0 N 3 N 1 N 2 Aea Righ = Aea 1 + Aea 2 + Aea 3 + n=n 1 n=n 2 whee N 1 =, N 2 = N 3 =. 3 3 n=n 3 Aea 4, 22, and All hese sums, excep he second one, wee simplified as follows by Mahemaica v.6, by allowing he summaion limis o acquie hei non-inege values. This is allowed since he uppe limi N l l = 1, 2, 3, and heefoe losing o gaining a em fom he end of each summaion effecively makes no diffeence whasoeve fo long imes. N 1 n=0 Aea 1 = a h τ, 23 2τ9τ N 3 32a h τ + 3τ 2 Aea 3 = n=n 2 + 3τ τ τ 2 72τ, 24 a h22 Aea 4 =. 4N n=n The simplificaion of he second sum ha of Aea 2 equies a moe lenghy and icky pocess, as i can no be simplified explicily by any convenional means. This is so, no only because Aea 2 has he mos complicaed of he fou expessions, bu also because is sum coves he lages ange ove n N 2 N 1 /4. Theefoe, fo lage, n is neve small. By using a subsiuion of he fom u = 1/, assuming u o be small fo lage and hen subsiuing n = s/u, whee s is of O1, befoe expanding Aea 2 ino a powe seies effecively incopoaes he effec of lage n ino he leading ode em of he seies. We ge: Aea 2 = α k u k k=0 a h s + 32s 2 2 u 2 = + Ou 32 s 3, s 2 and Ψ k s ae polygamma funcions. The polygamma funcion of ode k is defined as he k + 1h deivaive of he logaihm of he gamma funcion: Ψ k z = dk+1 ln Ɣz. dz k+1 Founaely he polygamma funcions ae of he fom z = a + c, bu whee a, b and c ae consans, and can be expanded as a Taylo seies o leading ode as follows: Ψ 0 a bu + c = lna/b ln u + Ou, Ψ k 2 a bu k bu + c = 1 k 1 k 1! + Ou k+1. a Subsiuing hese expessions ino Eq. 27 will simplify he expession damaically, finally leaving us wih he desied esul. We subsiue = 1/u back in o ge: N 2 Aea 2 = a h22 9 ln n=n 1 + O1/ In ligh of Eqs and 28, we can now simplify 22 o fis ode o ge: Aea Righ = a h ln O1/ To find an expession fo Aea Lef we mus use he same appoach used in Secion 3 o calculae I l. This gives: Aea Toal = 2Aea Righ + Aea Lef, Aea Toal = 3 ln a + h a h O1/ Dividing by C, he oal measue of he billiad map, we can obain an appoximae esul fo he long ime suvival pobabiliy of all iniial condiions iniially moving away fom he hole: 3 ln a + h a h 2 2 P 2 = + O1/ a + 2π 6. Resul and numeical simulaion we evese he subsiuion, and simplify he sum o obain: N 2 Aea 2 = a h 2 2 n=n 1 12u Ψ 0 z 1 Ψ 0 z 2 + Ψ 0 z 3 Ψ 0 z Ψ 1 z 1 Ψ 1 z 2 + Ψ 1 z 3 + Ψ 1 z I emains o add he pobabiliy measue of he wo ypes of ajecoies o obain he asympoic limi of he suvival pobabiliy funcion: P s = P 1 + P 2, whee he subscip s sands fo he saigh lines we have appoximaed he hypebolas wih. This gives: 3 ln a + h a h 2 2 P s = + O1/ 2, 32 44a + 2π

8 2402 C.P. Demann, O. Geogiou / Physica D P P d P s P s P h P s P h D/ Fig. 6. Colou online Lef: Plo compaing he suvival pobabiliy P s ligh geen/hoizonal line found by Eq. 32, wih he numeical suvival pobabiliy P d dak blue/cuve, found by diec numeical simulaion, boh muliplied by he ime. Righ: The diffeence beween equaion P s and P h which is found by numeically inegaing hypebolas and summing he elaive aeas unde he spikes ceaed, decays as D/ 2. D is he coefficien of he second ode em in Eq. 33. which is valid only fo ajecoies saisfying 6, i.e. sufficienly lage. In he lef panel of Fig. 6, we compae P s Eq. 32 wih P d which is obained by a diec numeical simulaion using Mahemaica v6., consising of 1.5 million iniial condiions disibued accoding o he invaian measue of he billiad map. We see ha P s gives a good pedicion of he numeical suvival pobabiliy fo long imes P d. We have esed his esul wih ohe values of he paamees: a,, h 1, h 2 as well. Wha is even moe impoan howeve is ha Eq. 32 is found o be an asympoic fomula. This is shown in he igh panel of Fig. 6, whee we have ploed he P s P h a egula inevals of ime, and fied i o an invese ime cuve D/ 2, whee D is some consan. Hee, P h is he esul obained by numeically summing ove he aeas of each spike see Fig. 4, found by he inegaed diffeence of θ 3 x i,, n and θ 4 x i,, n which define he hypebolas in he x i θ i plane. We find ha he P s P h fis pefecly ino D/ 2, whee D needs o be calculaed by a numeical fi. Thus we confim ha he appoximaion chosen in Secion 5 was jusifiable fom he asympoic convegence o he inegal P h. D is simply he coefficien of he second ode em in: 3 ln a + h a h 2 2 P = + D 44a + 2π + 2 o1/ Conclusion and discussion In his pape, we have invesigaed he open sadium billiad and managed o deive, using phase space mehods, an expession consising of he wo main conibuions see Eqs. 8 and 31 o he long ime asympoic ail of he suvival pobabiliy funcion of he sadium billiad. Boh expessions ae o leading ode in. The second one Eq. 31 is an appoximae esul which conveges o he ue esul as 2 which means he eos ae O1/ 2 and hence do no appea in he simple closed fom of Eq. 32. The expession has been confimed hough numeical simulaion see Fig. 6 Lef. In oal, we confim ha he suvival pobabiliy of he sadium fo long imes goes as Consan, and we find ha he Consan depends quadaically on he lenghs of he paallel segmens of he billiad on eihe side of he hole and hence he size of he hole as well as is posiion on one of he saigh segmens of he bounday see Eq. 32. In he conex of sadia, hee is a vaiey of possible shapes fo which one can obseve simila popeies. In his pape we only consideed he sandad sadium ha is a consucion of wo paallel saigh lines and wo complee semicicula acs. I is also possible o consuc diffeen egodic sadia by using cicula acs of lenghs less han π o by using ellipical acs [61]. In boh cases we expec egodiciy and an iniial song decay of he suvival pobabiliy followed by an asympoic powe law decay a longe imes, povided ha he paallel saigh sides ae sill pesen, and ha he dynamics emain defocusing [62]. Hence, simila mehods used in he pesen pape should be applicable o some vaiaions of he sadium geomey as explained above. A his poin, we would like o commen on he ln 3 em, which fis appeaed in Eq. 28. Simila ems whee found in he wok of boh Bálin and Gouëzel [44] and Amsead s e al. [54] as well as seveal ohe papes elaing o he sadium s bouncing ball obis and is long ime dynamics. I appeas, ha he ln 3 em is a diec consequence of he geomey of ou sadium billiad. Moe specifically, he cicula cuvaue of he bounday nea he saigh segmens, leads o a efleced final angle θ f θ i /3, 3 θ i, if θ i is small enough; his follows fom Eqs. 10 and 11. Hence we popose ha any bounded change in he cuvaue of he focusing segmens of he billiad, such ha he bounday emains C 1 smooh, would change he dynamics quaniaively bu no qualiaively i.e. P Cons. Fuhe wok on his subjec may include eseaching he open sadium wih holes on he cicula segmens. Such an example is expeced o behave vey similaly o he case descibed in he pesen pape, as is numeically shown in [53]. This is because he ajecoies which dominae and suvive fo long imes, again will be chaaceised by small nea veical angles. Thei collisions will mainly be wih he saigh segmens of he billiad, bu also on vey sho segmens of he semicicula acs. Wha is obviously diffeen in such a siuaion is ha he numbe of collisions wih he semicicula acs is no esiced o only one, as was he case hee. This fac will complicae he dynamics subsanially. Theefoe one migh pefe o choose a pobabilisic appoach o such a poblem, as suggesed in Amsead s e al. [54], ahe han an analyic one. One migh also be ineesed in addessing his fom a diffeen pespecive, such as a semiclassical appoach o even a quanum mechanical one, and hopefully obain some so of coespondence beween esuls. Acknowledgemens We would like o hank Uzy Smilansky fo helpful discussions, he anonymous efeees fo hei helpful suggesions and OG s EPSRC Docoal Taining Accoun numbe SB1715. Refeences [1] Ya.G. Sinai, Dynamical sysems wih elasic eflecions, Egodic popeies of dispesing billiads, Mah. Suv [2] N. Chenov, R. Makaian, Chaoic Billiads, in: Mahemaical Suveys and Monogaphs, vol. 127, AMS, 2006.

9 C.P. Demann, O. Geogiou / Physica D [3] G. Pianigiani, J.A. Yoke, Expanding maps on ses which ae almos invaian: Decay and chaos, Tans. Ame. Mah. Soc [4] J. Schneide, T. Tél, Z. Neufeld, Dynamics of leaking Hamilonian sysems, Phys. Rev. E [5] W. Baue, G.F. Besch, Decay of odeed and chaoic sysems, Phys. Rev. Le [6] O. Legand, D. Sonee, Facal se of ecuen obis in billiads, Euophys. Le [7] O. Legand, D. Sonee, Coase-gained popeies of he chaoic ajecoies in he sadium, Physica D [8] O. Legand, D. Sonee, Fis eun, ansien chaos, and decay in chaoic sysems, Phys. Rev. Le [9] F. Moessagne, O. Legand, D. Sonee, Tansien chaos in oom acousics, Chaos [10] E. Doon, U. Smilansky, Chaoic specoscopy, Phys Rev. Le [11] E. Doon, U. Smilansky, Chaoic specoscopy, Chaos [12] V. Paa, N. Pavin, Buss in aveage lifeime of ansiens fo chaoic logisic map wih a hole, Phys. Rev. E [13] V. Paa, H. Buljan, Buss in he chaoic ajecoy lifeimes peceding conolled peiodic moion, Phys. Rev. E [14] H. Buljan, V. Paa, Many-hole ineacions and he aveage lifeimes of chaoic ansiens ha pecede conolled peiodic moion, Phys. Rev. E [15] V. Milne, J.L. Hanssen, W.C. Campbell, M.G. Raizen, Opical billiads fo aoms, Phys. Rev. Le [16] R.T. Pieehumbe, On ace micosucue in he lage-eddy dominaed egime, Chaos Solions Facals [17] Z. Neufeld, P. Haynes, G. Picad, The effec of focing on he spaial sucue and speca of chaoically adveced passive scalas, Phys. Fluids [18] J. Schneide, T. Tél, Z. Neufeld, Dynamics of leaking Hamilonian sysems, Phys. Rev. E [19] I. Tuval, J. Schneide, O. Pio, T. Tél, Opening up facal sucues of heedimensional flows via leaking, Euophys. Le [20] J. Schneide, V. Fenandez, E. Henandez-Gacia, Leaking mehod appoach o suface anspo in he Medieanean Sea fom a numeical ocean model, J. Maine Sys [21] J. Schneide, J. Schmalzl, T. Tél, Lagangian avenues of anspo in he Eah s manle, Chaos [22] J. Nagle, Cash es fo he Copenhagen poblem, Phys. Rev. E [23] J. Nagle, Cash es fo he esiced hee-body poblem, Phys. Rev. E [24] A.E. Moe, P.S. Leelie, Mixmase chaos, Phys. Le. A [25] L. Bunimovich, C. Demann, Open cicula billiads and he Riemann hypohesis, Phys. Rev. Le [26] L. Bunimovich, C. Demann, Peeping a chaos: Nondesucive monioing of chaoic sysems by measuing long-ime escape aes, Euophys. Le [27] L.A. Bunimovich, The egodic popeies of ceain billiads, Func. Anal. Appl [28] L.A. Bunimovich, On egodic popeies of nowhee dispesing billiads, Comm. Mah. Phys [29] N. Chenov, C. Haskell, Nonunifomly Hypebolic K-sysems ae Benoulli, Egodic Theoy and Dynamical Sysems, vol. 16, 1996, pp [30] S. Blehe, E. O, C. Gebogi, Roues o chaoic scaeing, Phys. Rev. Le [31] G. Benein, J.M. Selcyn, Numeical expeimens on he fee moion of a poin mass moving in a plane convex egion: Sochasic ansiion and enopy, Phys. Rev. A [32] P.V. Elyuin, Kolmogoov enopy of billiad sadiums, Dokl. Akad. Nauk SSSR [33] L.A. Bunimovich, Mushooms and ohe billiads wih divided phase space, Chaos [34] L.A. Bunimovich, Condiions of sochasiciy of wo dimensional billiads, Chaos [35] M. Wojkowski, Pinciples fo he design of billiads wih nonvanishing Lyapunov exponens, Comm. Mah. Phys [36] A. Bäcke, F. Seine, Quanum chaos and quanum egodiciy, in: B. Fiedle Ed., Egodic Theoy, Analalysis and Efficien Simulaion of Dynamical Sysems, Spinge-Velag, Belin, Heidelbeg, 2001, pp [37] A. Hassell, L. Hillaie, Egodic billiads ha ae no quanum unique egodic. pe-pin, axiv: v3 [mah.ap], [38] G. Calo, E. Vegini, P. Lusembeg, Sca funcions in he Bunimovich sadium billiad, J. Phys. A: Mah. Gen [39] G. Tanne, How chaoic is he sadium billiad? A semi-classical analysis, J. Phys. A: Mah. Gen [40] L.S. Young, Saisical popeies of dynamical sysems wih some hypeboliciy, Ann. of Mah [41] L.S. Young, Recuence imes and aes of mixing, Isael J. Mah [42] R. Makaian, Billiads wih Polynomial decay of coelaions, Egodic Theoy Dynam. Sysems [43] N. Chenov, H.K. Zhang, Billiads wih polynomial mixing aes, Nonlineaiy [44] P. Bálin, S. Gouëzel, Limi heoems in he sadium billiad, Comm. Mah. Phys [45] N. Chenov, H.K. Zhang, Impoved esimaes fo coelaions in billiads, Comm. Mah. Phys [46] P. Bálin, I. Melboune, Decay of coelaions and invaiance pinciples fo dispesing billiads wih cusps, and elaed plana billiad flows, J. Sa. Phys [47] B. Chiikov, Poincaé ecuences in micoon and he global ciical sucue. Pe-pin, axiv:nlin/ v1 [nlin.cd], [48] G. Zaslavsky, Facional kineics, and anomalous anspo, Phys. Rep [49] F. Vivaldi, G. Casai, I. Guanei, Oigin of long-ime ails in songly chaoic sysems, Phys. Rev. Le [50] A.S. Pikovsky, Escape exponen fo ansien chaos and chaoic scaeing in non-hypebolic Hamilonian sysems, Phys. A: Mah. Gen [51] E.G. Almann, Tamás Tél, Poincaé ecuences and ansien chaos in leaked sysems, Phys. Rev. E [52] M. Coubage, M. Edelman, S.M. Sabei Fahi, G.M. Zaslavsky, Poblem of anspo in billiads wih infinie hoizon, Phys. Rev. E [53] R.S. Dumon, P. Bume, Decay of a chaoic dynamical sysem, Chem. Phys. Le [54] D.N. Amsead, B.R. Hun, E. O, Powe-law decay and self-simila disibuions in sadium-ype billiads, Physica D [55] M.V. Bey, Regula and iegula moion, in: S. Jona Ed., Topics in Nonlinea Mechanics, in: Am. Ins. Ph. Conf. Poc., vol. 46, 1978, pp [56] H. Al, H.D. Gaf, H.L. Haney, R. Hoffebe, H. Rehfeld, A. Riche, P. Schad, Decay of classical chaoic sysems: The case of he Bunimovich sadium, Phys. Rev. E [57] P. Mannevillea, Y. Pomeau, Inemiency and he Loenz model, Phys. Le. A [58] P. Mannevillea, Y. Pomeau, Inemien ansiion o ubulence in dissipaive dynamical sysems, Comm. Mah. Phys [59] H.G. Schuse, W. Jus, Deeminisic Chaos: An Inoducion, 44h ed., Wiley John and Sons, [60] K.C. Lee, Long-ime ails in a chaoic sysem, Phys. Rev. Le [61] R. Makaian, S.O. Kamphos, S.P. Cavalho, Chaoic popeies of he ellipical sadium, Comm. Mah. Phys [62] L.A. Bunimovich, G. Del Magno, Semi-focusing Billiads: Egodiciy, in: Egodic Theoy and Dynamical Sysems, vol. 28, Cambidge Univesiy Pess, 2008, pp