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5th Warsaw School of Statistical Physics - ebook/pdf
5th Warsaw School of Statistical Physics - ebook/pdf
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Wydawca: Wydawnictwa Uniwersytetu Warszawskiego Język publikacji: polski
ISBN: 978-83-235-1739-9 Data wydania:
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Kategoria: ebooki >> naukowe i akademickie >> fizyka
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Zbiór materiałów konferencyjnych ze spotkania w Kazimierzu Dolnym, które odbyło się w dniach 22-29.06.2013 r. Organizatorami tych międzynarodowych konferencji jest Instytut Fizyki Teoretycznej UW oraz Fundacja Pro Physica. Publikacja w języku angielskim.

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5th Warsaw School of Statistical Physics 22-29 June 2013 Kazimierz Dolny, Poland 5 t h W a r s a w S c h o o l o f S t a t i s t i c a l P h y s i c s 2 2 - 2 9 J u n e 2 0 1 3 konferencja 5. roztrzelony indd.1 1 konferencja 5. roztrzelony indd.1 1 ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 2014-03-20 09:51:15 2014-03-20 09:51:15 5h Waaw S h f Saii a hyi  ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 5h Waaw S h f Saii a hyi  aziiez Dy ad 22 (cid:21) 29  e 2013 B. Ci h ki . aiókwki ad . iae ki Edi gaized by he i e f Theei a hyi  Uiveiy f Waaw ad  hyi a F dai ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== Cve deig aazya a zkiewi z Cve hgah Bgda Ci h ki hgah f he e  e ad ai ia Agiezka B dek Te hi a edi Ewa Szy zak © Cyigh by Wydawi wa Uiweye Wazawkieg Wazawa 2014 SB 978 83 235 1561 6 Waaw Uiveiy e Sae Deae: e. 48 22 55 31 333 e ai: dz.hadwy w.ed . ee Bkh: h://www.w w./kiegaia ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== Fewd Fwig he evi   h whi h have ake a e i aziiez evey w yea i e 2005 he i e f Theei a hyi  f he Uiveiy f Waaw gaized he 5h Waaw S h f Saii a hyi   e 22h  29h 2013. The ga f he  h wa eeiay ed f ix  e edig  vai  aea f eea h i he (cid:28)ed f aii a hyi . Six diig ihed  iei eeed edaggi a eie f e  e bigig a ea exaai f bai heei a idea ad e  agig f he eea h. The e  e wee a eded by hD  de gad ae eea he ad a by e exeie ed  iei ieeed i geig a  aied wih a ew (cid:28)ed. The ee v e ai he ex f he  e. We ae gaef  he ivied eake f hei wiige  ake hei e  e e eady f  bi ai. We d he he v e wi be ef  y  he ai ia f he  h b  a  a he ieeed i he e devee f idea i aii a hyi .  i a a ea e  a kwedge a he idivid a ad gaizai ied veeaf  wh ib ed  he  e f he  h. S iei(cid:28) gaizig Ciee: Bgda Ci h ki aek aiókwki aªaw iae ki F f he ifai ab  he S h ee: h://www.f w.ed ./∼ wh/ ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== S Fa y f hyi  Uiveiy f Waaw iiy f S ie e ad ighe Ed ai  a gaizig Ciee Bgda Ci h ki Chai aweª ak b zyk Se eay i Szy zak a Wdªwki ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== Ce 1  e iibi  Saii a e hai  f he S hai avie(cid:21)Ske E ai ad Gehi T b e e . . . . . . . . 3 Feddy B he Ceae adii ad T  Tagaife 2 The Fibe B de de . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Aex ae e C. ee ad S ahi adha 3 Dyai  f Chaged ai e Diei . . . . . . . . . . . . . . . . . . . 83 Gehad ägee 4 S fa e Tei: F F daea i ie  Ai ai i i id ad i Sid . . . . . . . . . . . . . . . . . . . . . . . . . 149 Yve ea 5 ayig wih abe: S  a ad Thedyai eie f ad Shee Sye . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Adé Sa 6 Ti  i he aheai a hyi  f Cd Be Gae . . . . . . 299 akb Ygva a Bkz zai ga C e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 e Aba  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 i f ai ia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1  e iibi  Saii a e hai  f he S hai avie(cid:21)Ske E ai ad Gehi T b e e Feddy B he Ceae adii ad T  Tagaife Aba  Tw dieia ad gehyi a  b e (cid:29)w have he ey  ef gaize ad eae age  ae hee je ad vi e. Thi i f ia e e f he a j  ee f he dyai  f Eah ahee. Fwig  age iiia iigh baed  j gaed wk by aheai ia ad hyi i hi f daea hyi a  e ha f d e exaai i he faewk f aii a e hai . A ia e iiiaed wey yea ag ha bee he  dy f he e iibi  aii a e hai  f he 2D E e ad he eaed  ai gehi de he ie Rbe Seia hey. Rea gehyi a ad exeiea (cid:29)w ae hweve diiaive ad aiaied by exea f e. Thee e  e f   e e heei a devee f he aii a e hai  f he  e iibi  i ai. The gee have bee a hieved ig  f (cid:28)ed hey ah iega ad ia  e iibi  aii a e hai  age deviai  hai aveagig. The ai f hee e  e i  bie(cid:29)y id e he heei a ae  f hi ga i he ie ex: he 2D  hai E e  avie Ske e ai ad he  ai gehi e ai. We eview ah iega eeeai f  hai  ee age deviai f aii babiiie a i iiizai ia hey f geea e hai a ye f ed by ad f e. We wi ay hi faewk i de  edi  e iibi  ad  e iibi  hae aii f he 2D E e avie Ske ad  ai gehi dyai  ad  edi  he ae f ae aii bewee w aa  i i ai f (cid:28) de hae aii. iei hey f ye wih g age iea i bh wih ad wih   hai exea f e ae exaied. Baed  hi kiei hey we edi   e iibi  hae aii ad di  hei e e exeiea beva i ad  ei a i ai. Eve if he de we have ideed  fa ae  ie a adei de he Feddy B he abaie de hyi e É e ae S ¯ie e de y e CRS 46 aée daie 69007 y Fa e eai: Feddy.B hee y.f ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 3 4 Feddy B he Ceae adii ad T  Tagaife exe ed eeva e f he aa he i he f  e f Eah ahee ad iae dyai  i bie(cid:29)y di ed. 1 d i 1.1 Sef gaizai f w dieia ad gehyi a (cid:29)w Ahei ad  eai (cid:29)w ae hee dieia 3D b  ae gy d iaed by he Cii f e aiy baa ed by e e gadie gehi baa e. The  b e e ha deve i  h (cid:29)w i aed gehi  b e e. de de ibig gehi  b e e have he ae ye f addiia ivaia a he f he w dieia 2D E e e ai. A a e e e eegy (cid:29)w ba kwad ad he ai hee i he fai f age  ae hee   e je y e ad ai y e. e  h exae i he fai f  ie Gea Red S Fig. 1. Fig. 1: i  e f  ie Gea Fig. 2: Zay aveaged ve iy (cid:28)e i he Red S a age  ae vex e hee f  ie. The (cid:29)w i ga i aed bewee bad f a ied i aeaig g je. hei je. h  ey f ASA: h://h j a.j.aa.gv/ aag /A00014. The aagy bewee 2D  b e e ad gehyi a  b e e i f he e haized by he heei a iiaiy bewee he 2D E e e ai de ibig 2D (cid:29)w ad he ayeed  ai gehi  haw wae de de ibig he age  ae f gehi  b e e: bh ae a e ai f a  aa  aiy by a  divege (cid:29)w evig a i(cid:28)ie  be f ivai a. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 5 The fai f age  ae hee   e i a fa iaig be ad a eeia a f he dyai  f Eah ahee ad  ea. Thi i he ai ivai f eig  a hey f he ef gaizai f 2D  b e e. 1.2 Saii a e hai  f he ef gaizai f w dieia ad gehyi a (cid:29)w: age e iibi   e Ay  b e e be ivve a h ge  be f degee f feed  ed via ex iea iea i. The ai f ay hey f  b e e i  dead he aii a eie f he ve iy (cid:28)ed.  i h  exeey eig  aa k hee be f a aii a e hai  i f view. Saii a e hai  i ideed a vey wef e f heei a  ha aw   ed e he exiy f a ye dw  a few hedyai aaee. A a exae he  e f hae aii aw   de ibe dai hage f he whe ye whe a few exea aaee ae haged. Saii a e hai  i he ai heei a aa h we deve i hee e  e.   eed i exaiig ay f he heea a iaed wih w dieia  b e e [13℄. Thi ay ee  iig a (cid:28) a i i a  beief ha aii a e hai  i   ef i hadig  b e e be. The ea f hi beief i ha   b e e be ae iii ay fa f e iibi . F ia e he fwad eegy a ade i hee dieia  b e e ivve a (cid:28)ie eegy diiai  ae hw a he vi iy aa  diia i ee f ia e age iighf ideai f he  evai f eegy by he hee dieia E e e ai [28℄. A a e  f hi (cid:28)ie eegy (cid:29) x hee dieia  b e (cid:29)w a be ideed e  e e iibi  diib i. By a w dieia  b e e de   (cid:27)e f he aa  diiai f he eegy  e iibi  aii a e hai   e  e i ibi  aii a e hai  ake ee whe a (cid:29) xe ae ee. The (cid:28) ae  e e iibi  aii a e hai  idea  exai he ef gaizai f w dieia  b e e dae f age wk i 1949 [51℄ ee [28℄ f a eview f age ib i   b e e hey. age wked wih he i vex de a de ha de ibe he dyai  f i g a i vi e (cid:28) ed by d evi ad whi h ed  a e ia a f  i f he 2D E e e ai. The e iibi  aii a e hai  f he i vex de ha a g ad vey ieeig hiy wih wdef ie e f aheai a a hievee [1; 18; 21; 26; 27; 37; 39; 51℄. The geeaizai f age idea  he 2D E e e ai wih a i   vi iy (cid:28)ed akig i a   a ivaia ha bee ed i he begiig f he 1990 [45; 57; 58; 60℄ eadig  he ie(cid:21)Rbe(cid:21)Seia hey RS hey. The RS hey i de he evi  age hey ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 6 Feddy B he Ceae adii ad T  Tagaife ad deeie wihi whi h ii he hey wi give eeva edi i ad e . The RS hey dea wih he i  ai a ivaia ea e.  edi  ha  i  i ae vi iy (cid:28)ed  eae i a ige a ae  vi iy (cid:28)ed bai ay have he ae age  ae ve iy (cid:28)ed. Thi exai why e h d exe  he (cid:29)w  ef gaize i hi e iibi  a ae. Thi e iibi  a ae i haa eized by he axiizai f a ey wih e ai eaed  dyai  ivaia. The ai f e i 3 i  ke h he deivai f hi vaiaia be whi h i he bai f he hey. The ai ai  he Gea Red S f  ie wi be bie(cid:29)y  aized. Thee w i i e a vey bief veview f e iibi  aii a e hai . ve he a (cid:28)fee yea he RS e iibi  hey ha bee aied  ef y  a age a f be f bh he w dieia E e ad  ai gehi e ai. Thi i de ay ieeig ai ai  h a he edi i f hae aii i di(cid:27)ee ex a de f he Gea Red S ad he via vi e ad de f  ea vi e ad je. A deaied de ii f he aii a e hai  f 2D ad gehyi a (cid:29)w he y ad f hee gehyi a ai ai i eeed i he eview [13℄. de eview  bk [40; 42; 64℄ give a vey ieeig eeay viewi eig aiy he hey ad abay exeie. The e by Y. ea [54℄ give a a vey ieeig e  he ea why he w dieia E e e ai by a wih  he e iibi  aa h f ai a (cid:28)ed hey de   (cid:27)e f he Rayeigh ea aadx bai ay he fa  ha a ai a (cid:28)ed ha a i(cid:28)ie hea aa iy. Thi i i f he di ed i [13℄. Fiay we e ha e iibi  aii a e hai  f w die ia ad gehyi a (cid:29)w i i a vey a ive  b je  wih ay ib i d ig he a few yea [8; 24; 33(cid:21)35; 48; 49; 55; 66; 70; 71; 73℄ ay f he by bigh y g  iei. A fa a e iibi  aii a e hai  i  eed he ai f hee e  e i j   exai he bai f ie(cid:21)Rbe(cid:21)Seia hey exai hw   e he ey f a ae ad h  hei babiiy h gh he e f age deviai hey. We di  hee i i e i 3 a a eve whi h i a eeeay a ibe. 1.3  e iibi  aii a e hai  f he ef gaizai f w dieia ad gehyi a (cid:29)w: aii a e hai  ad dyai   f a a  b e (cid:29)w ae  feey evvig hey ae ahe  ay f ed ad diiaed. The i aii ay aiay egie we i  h gh exea f e baa e eegy diiai  aveage.  he ii f vey a f e ad diiai aed  evaive e f he dy ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 7 ai  i i exe ed  (cid:28)d a g eai bewee hee  e iibi  (cid:29)w ad e f he ae edi ed by e iibi  aii a e hai .  de  give a e ie eaig  hi geea idea ad  dea wih fa f e iibi  i ai i i eeia  deve a he  e iibi  aii a e hai  f he 2D E e 2D avie Ske ad bai  ai gehi e ai. A we di  bew hi ha bee he  b je  f e e key adva e i he ai ai f aii a e hai    b e (cid:29)w. Thi i a  ay he ai  b je  f hee e  e. We ee w  e iibi  aii a e hai  aa he: he (cid:28) dea wih  e iibi  (cid:28) de hae aii ad he  ai f ai i ae ig age deviai ad he e d i a kiei hey aa h  he edi i f he age  ae (cid:29)w. 1.3.1 Saii a e hai  f ah i hae a e ad  e iibi  biabe  b e (cid:29)w ay  b e (cid:29)w a evve ad ef gaize wad w  e vey di(cid:27)e e ae.  e f hee ye he aii bewee w f  h ae ae ae ad   eaivey aidy. Exae i de he Eah agei (cid:28)ed eve a ve gegi a ie ae  i agei (cid:28)ed evea i D exeie e.g. he V    Sdi  VS  b e dya i Fig. 3 [3℄ Rayeigh Béad ve i e [17; 20; 50; 65℄ 2D  b e e [10; 41; 63℄ ee Fig. 4 3D (cid:29)w [56℄ ad f  ea ad ahei (cid:29)w [62; 72℄. The deadig f hee aii i a exeey di(cid:30)  be d e  he age  be f degee f feed age eaai f ie ae ad he  e iibi  a e f hee (cid:29)w. weve f f ed diiaed  b e ye i i  ea hw  de(cid:28)e he e f aa  f he dyai . Ah gh i he ii f weak f ig ad diiai e w d exe  ha he e f aa  w d vege  he e f he deeiii e ai.  he ae f he 2D E e e ai e iibi  aii a e hai  i he f f he e iibi  ie Rbe(cid:21) Seia hey aw f he edi i e f aa  f he dyai . They ae a  be f he eady ae f he 2D E e e ai he e iibi  aii a e hai  give a (cid:28) aia awe  he  ei f aa . eve i ai f he 2D avie Ske e ai i he weak f e ad diiai ii hwed ha he dyai  a  ay  eae e iey e  he e f he 2D E e e ai aa  [10℄. eeigy he ae i ai hwed adi  e iibi  hae aii whee he ye ae y wi hed bewee w aaey abe eady ae e ig i a ee hage i he a  i behavi ee (cid:28)g e 4. f he f e ad diiai ae weak he hee aii ae a  ay exeey ae  ig  a ie ae  h ge ha he dyai a ie ae.   h i ai whe he  b e (cid:29)w wi he a ad ie f e ye f aa   ahe a heei a ai i   e he aii ae.  i a fe he ae ha  aii ah f e aa   ahe  eae  a ige ah he a a a ai i   e hi  babe ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 8 Feddy B he Ceae adii ad T  Tagaife Fig. 3: Fig e ake f [3℄ hwig ad aii bewee ea abe ieai f he agei (cid:28)ed i a exeiea  b e dya. The ai azi ha e f he agei (cid:28)ed i hw i ed. Fig. 4: Fig e ake f [10℄ hwig ae aii i aed by he F ie e f he age y de bewee w age  ae aa  f he eidi 2D avie Ske e ai. The ye ed he a jiy f i ie e  he vex die ad aae (cid:29)w (cid:28)g ai. ah.  de a hieve he ai we wi e a ah iega eeeai f he aii babiiie ad  dy i ei ai a ii i a ayi exai whee he a aaee i he e ha deeie bh he f e ad diiai ai de.  hi ii if hi ei ai a aa h i eeva e exe  a age deviai e  iia  he e baied h gh he Feidi Weze hey[30℄.  de  i ae i a edaggi a way he geea aa h we wi ea i hee e  e he ai a ae f he ae de ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 9   ai f he aii ae f a ai e i a d be we eia. We wi geeaize he di i  a e f agevi dyai  ha i de he w dieia E e ad ai Gehi agevi dyai  ad (cid:28)ay we wi di  aia e  f he w dieia avie Ske e ai whe deaied baa e i  ai(cid:28)ed. The di(cid:27)ee i ae di ed i e i 4.4. 1.3.2 iei hey f za je e exae f ae  eege e f age  ae hee   e i ge hyi a (cid:29)w i he fai f za ea we je. The  i  e f  ie efe y i ae hi fa : he  fa e (cid:29)w i eay gaized i aae aeaig za je a hw i (cid:28)g e 2 wih a he ee e f gia ad vey abe vi e  h a he Gea Red S. S h age  ae fea e ae  e had wy diiaed aiy d e  a age  ae fi i e hai ad  he he had aiaied by he a  ae  b e e h gh Reyd ee. The ai e hai i h  a afe f eegy f he f ig  ae d e  bai ad ba ii iabiiie  he  b e  ae ad i he  ae f he je. A ia i i hi heegy i he fa  ha he  b e (cid:29)  ai ae f vey weak ai de aed  he ai de f he za je ad ha hey evve  h fae. Thi ea ha he yi a ie  ae f ad ve i ad hea f he (cid:29)  ai by he je i  h ae ha he yi a ie  ae f fai  diiai f he whe je. Thi ie  ae eaai i a vey e i(cid:28) ey f he gehyi a age  ae   e.  hi  b e ex he deadig f je fai e ie aveag ig   he e(cid:27)e  f aid  b e degee f feed i de  de ibe he w ev i f he je   e. S h a ak a exae f  e i  ay exeey had  ef f  b e (cid:29)w. Uig he ie  ae eaai eied eaie we ve ha i a be efed exi iy i hi be. Thi aa h aed a kiei hey by aagy wih iia aa he i he aii a e hai  f ye wih g age iea i i eeed i e i 5. 1.4 A eay aa h f aii a e hai : age deviai hey age wa he (cid:28)  ide a aii a e hai  exaai f w dieia  b e (cid:29)w [51℄. A he ie he wa  iei(cid:28) ay a ive age ade a age  be f de iive ib i  aii a e hai  hey:  i f he 2D ig de e i iy eai ib i  he a ii a e hai  f ee ye ad  b e e ad  . Si e ha ie he heei a aa he f eaig aii a e hai  be have bee  ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 10 Feddy B he Ceae adii ad T  Tagaife eey eewed. e f he ai hage ha bee he e f he ag age f age deviai hey f e ha 30 yea. F ia e e e e  i he deadig f e iibi  aii a e hai  be vig (cid:29)  ai hee age e i iy eai geeaized fa f e iibi  ad i deaig wih  e iibi  aii a e hai  be ae a eaed  age deviai hey. eeigy a we di  i hee e  e he  e ed by age i hi 1949 ae [51℄ i de  dead he ef gaizai f w dieia (cid:29)w ed a few de ade ae  e f he (cid:28) ai ai f age deviai hey  e iibi  aii a e hai  be. The hey f age deviai dea wih he ayi behavi f he ex eia de ay f he babiiie f ae  exee eve. The a iaed iiig aaee i  ay ake  be he  be f bevai he  be f a i e b  a be he aaee  h a vaihig ie  he eea e f a hei a ea i  age ie. age deviai hey a be ideed a geeaizai f he ea ii hee wih he e(cid:28)ee f i dig ifai ab  he behavi f he ai f he babiiy deiy. The ai e  f age deviai hey i he age deviai i ie a e  de ibig he eadig ayi behavi f he ai  age deviai f he babiiy diib i i he ii N → ∞. F ia e he age deviai i ie f a ad vaiabe XN i lim N→∞− 1 N log[P (XN = x)] = I(x), 1 whee P i he babiiy deiy f he ad vaiabe XN  ad I(x) i aed i=1 xi  whee xi ae ideede idei ay diib ed ad vaiabe he I(x) i give by Cae hee. he ae f  i. F ia e if XN = (1/N )PN Beide he ai ai de ibed i he evi  e i he ai f hee e  e i  exai ad deive he ii ay age deviai e  f he e i ibi  aii a e hai  f he w dieia E e ad  ai gehi e ai e iibi  ad f he 2D avie Ske   ai gehi e a i wih  hai f e  e iibi . The age deviai e  f he e iibi  ae e i 3 i deived h gh a geeaizai f Sav hee ad ead  a f a f he babiiy f a ae f he i  ai a ea e. The age deviai e  f he  e iibi  ae e i 4.4 ae deived h gh ei ai a ii i ah iega  e ivaey he Feidi Weze faewk ad ead  he eva ai f aii ah ad aii babiiie f biabe  b e (cid:29)w e   e iibi  hae aii. 1.5 gaizai f he e  e  e i 2 we ae he e ai f i ad hei evai aw.  e i 3 we   i  ai a ivaia ea e f he 2D E e e ai ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 11 ad di  he ey axiizai be i edi ig he  baby eady ae  he 2D E e e ai.  e i 4.4 we di  age devia i f  e iibi  be ad i ae hi ig a ie a adei exae he be f  ai f aii ae f he ae be fwed by he ai ai  he 2D avie Ske e ai. Fiay i e i 5 we di  he kiei hey f za je f he bai  ai gehi dyai . 2 The 2D E e bai ai Gehi  ad  hai avie(cid:21)Ske e ai 2.1 E ai f i The ai f hi e i i  ee he ie de ha de ibe w dieia ad gehyi a  b e (cid:29)w: he w dieia avie Ske e ai ad he bai e ai wih  hai f ig.  he ii whe f e ad diiai g  ze he w dieia avie Ske e ai e d e  he w dieia E e e ai. We de ibe he evai aw f hee e ai ad hei i(cid:29) e e  he dyai . The eview [13℄ give a vey bief id i  gehyi a (cid:29) id dyai  ad he  ai gehi de. A e ee id i i f d i exbk f gehyi a (cid:29) id dyai  [53; 68℄. We ae ieeed i he  e iibi  dyai  a iaed  he w dieia  hai ay f ed bai e ai a aed bai ai Gehi e ai: ∂q ∂t + v [q − h] ∇q = −αω + ν∆ω + √2αη, v = ez × ∇ψ, q = ω + h(y) = ∆ψ + h, 2 3 whee ω  v ad ψ ae ee ivey he vi iy he  divege ve iy ad he eaf  i. F ii iy i hee e  e we ide he dyai   a eidi dai D = [0, 2δπ) × [0, 2π) wih ae  ai δ . The ψ i eidi wih he f he dii RD dr ψ = 0. q i he eia vi iy ad h i a give gahy f  i wihRD dr h = 0. F h = 0 he bai e ai ed e  he 2D avie Ske e ai. The iea fi i e −αω de age  ae diiai. We ide  dieia e ai whee a yi a eegy i f de 1 ee [13℄  h ha ν i he ivee f he Reyd  be ad α i he ivee f a Reyd  be baed  he age  ae fi i. We a e ha he Reyd  be aify ν ≪ α ≪ 1.  he ii f weak f e ad diiai limα→0 limν→0  he 2D avie Ske e ai vege  he w dieia E e e ai f (cid:28)ie ie b  he ye f f ig ad diiai deeie  whi h e f aa  ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 12 Feddy B he Ceae adii ad T  Tagaife he dyai  evve  ve a vey g ie. The  f he f ig η(x, t) i a whie i ie Ga ia (cid:28)ed de(cid:28)ed by hη(x, t)η(x′, t′)i = C(x − x′)δ(t − t′) whee C i he eai f  i f a  hai ay hgee  ie. The w dieia E e e ai h = 0)  he ieia bai e a i h 6= 0 ae give by E. 2 wih f e ad diiai e  ze α = ν = 0. 2.2 Cevai aw f he ieia dyai  The kiei eegy f he (cid:29)w i give by E[q] = 1 2ZD dr v2 = 1 2ZD dr (∇ψ)2 = − 1 2ZD dr (q − h)ψ, 4 whee he a e aiy i baied wih a iegai by a. The kiei eegy i eved f he dyai  f he w dieia E e ad ieia bai e ai i.e. dE/dt = 0. Thee e ai a eve a i(cid:28)ie  be f f  ia aed Caii. They ae eaed  he degeeae   e f he i(cid:28)ie dieia aiia ye ad a be ded a ivaia aiig f ehe hee [61℄. Thee f  ia ae f he f s(q) dr, 5 Cs[q] =ZD Γ =ZD whee s i ay  (cid:30) iey eg a f  i. We e ha  a d by eidi dai he a i ai q dr, 6 i e eaiy e a  ze: Γ = 0. The i(cid:28)ie  be f eved  aiie ae eibe f he e ai havig a i(cid:28)ie  i   e f eady ae ee e i 2 i [13℄. Ay f he i(cid:28)ie  be f eady ae f he 2D E e  ieia bai e ai aify v ∇q = 0. F ia e if hee i a f  ia eai bewee he eia vi iy ad he eaf  i i.e. q = ∆ψ = f (ψ) whee f i ay i   f  i he ig 2 e eaiy he k ha v ∇q = 0. hyi ay hee ae ae ia be a e e f he a  a aa  f he dyai . Thee i a a g eii a ad  ei a evide e ha a ex ev i f he w dieia E e e ai ead  f he ie  aa  ha ae eady ae f he e ai. The e i(cid:28) f  i f ha i ea hed afe a ex ev i a be edi ed i eai i ai ig e iibi  aii a e hai a ag e eeed i he ex e i ee [13℄ f e deai. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 13 2.3 The evai f he vi iy diib i The w dieia E e ad ieia bai e ai eve he di ib i f eia vi iy i.e. he a aea f a e i(cid:28) eia vi iy eve e i eved. A we exai w he evai f he eia vi iy diib i i e ivae  he evai f a Caii. We (cid:28) ve ha he eia vi iy diib i i eved a a  e e e f Caii evai aw. We ide he e ia a f Caii 5: C(σ) =ZD H(−q + σ) dr, 7 whee H(  ) i he eaviide e f  i. The f  i C(σ) e  he aea  ied by a eia vi iy eve ae  e a  σ . C(σ) i a ivaia f ay σ ad heefe ay deivaive f C(σ) i a eved. Theefe he diib i f vi iy de(cid:28)ed a D(σ) = C′(σ) whee he ie dee a deivai wih ee   σ  i a eved by he dyai . The exei D(σ)dσ i he aea  ied by he vi iy eve i he age σ ≤ q ≤ σ + dσ . eve ay Caii a be wie i he f Cf [q] =ZD dσ f (σ) D(σ). The evai f a Caii E. 5 i heefe e ivae  he eva i f D(σ). The evai f he diib i f vi iy eve a ve abve a a be ded f he e ai f i. We (cid:28)d ha Dq/Dt = 0 hwig ha he va e f he eia vi iy (cid:28)ed ae agagia a e. Thi ea ha he va e f q ae aed h gh he  divege ve iy (cid:28)ed h  keeig he diib i  haged. F w  we ei   eve  a K eve vi iy diib i. We ake hi hi e f edaggi a ea b  a geeaizai f he di i f ex e i  a i   vi iy diib i i aighfwad. The K eve vi iy diib i i de(cid:28)ed a D(σ) = Akδ(σ − σk), 8 KXk=1 whee Ak dee he aea  ied by he vi iy va e σk . The aea Ak ae k=1 Ak = |D|. eve he ai  abiay hei   i he a aeaPK 6 ie he aiPK k=1 Akσk = 0. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 14 Feddy B he Ceae adii ad T  Tagaife 3 E iibi  aii a e hai  ad he ea (cid:28)ed vaiaia be a a age deviai e  3.1 age deviai hey i 2D  b e e he e iibi  ea (cid:28)ed vaiaia be The (cid:28) age deviai e  i w dieia  b e e have bee baied i he ex f he hey f he 2D E e e ai. i he ad Rbe [44℄ have  died he age deviai f Y g ea e ad have  ggeed ha he ey f he ie(cid:21)Rbe(cid:21)Seia hey i he aag e f a age deviai ae f  i. By ideig a i diib i f he vi iy ivaia i a faewk whee he ivaia ae ideed i a ai a eebe ahe ha i a i  ai a e B he ad aba [5℄ have give a deivai f a age deviai e  baed  (cid:28)ie dieia axiai f he vi iy (cid:28)ed. The begiig f he ieie ha a bee a ie f iee  dy f he aii a e hai  f he i vex de [4; 18; 27; 28; 38; 39℄ a e ia a f  i f he w dieia E e e ai. Ag he  dy age deviai e  f he e iibi  ea e whee a baied. The ai f hi e i i  ee a he ii  i f i  ai a ivaia ea e f he 2D E e e ai. Thi  i iaiy fw he iiia idea f he evi  wk [5; 44℄ b  i  h ii(cid:28)ed. eve f edaggi a ea he eadig f hi he ii eeai de  iy ay kwedge f age deviai hey ad avid ay e hi a di i. Thee ea e ae  ed ig (cid:28)ie dieia axiai f he vi iy (cid:28)ed wih N 2  be f degee f feed. N 2 i he he age deviai aaee ad he ey aea a he aag e f he age deviai ae f  i.  de  ae he ai e  e  de(cid:28)e p(r, σ) a he  a babiiy  beve vi iy va e e a  σ a i r: p(r, σ) = hδ(ω(r) − σ)i whee δ i he Dia dea f  i we ide aveagig h i ve he i  ai a aveage. The he age deviai ae f  i f p(r, σ) i S(E0) − S[p, E0] ea e ee e i 3.2. We a de(cid:28)e ω(r) =R dσ σp(r, σ) he  a vi iy whee S[p, E0] = S[p] ≡ZDXk pk log pk dr 9 if he ai N [p] = 1 ∀k, A [pk] = Ak ad E[ω] = E0 ae ai(cid:28)ed ad S[p, E0] = −∞ hewie ad whee S(E0) = sup {p | N [p]=1}{S[p] | E[¯ω] = E0, ∀k A[pk] = Ak} , 10 wih E0  Ak ad N  he eegy he vi iy diib i ad he babiiy aizai de(cid:28)ed i e i 3.3 ee ivey. The ieeai f hi e  i ha he  babe va e f he  a babiiy i he axiize f he vaiaia be 10 ad ha he babiiy  beve a dea e f hi  babe ae i exeiay ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 15 age wih aaee N 2 (cid:28)ed e ai f he eaf  i ψ a be deived f 9 a di ed i ad ae f  i 9. F hee he ai a ea efee e [6℄.  ex e i we de(cid:28)e e iey he i  ai a ea e f he 2D E e e ai e i 3.2 ad ve ha he ey S[p, E0] i a age deviai ae f  i f p e i 3.3. Thi j i(cid:28)e he ea (cid:28)ed vaiaia be 10. 3.2 i  ai a ea e  de  ey   a i  ai a ea e we di eize he v i iy (cid:28)ed  a if gid wih N 2 dig (cid:28)ie dieia a e ad ake he ii N → ∞. A if gid ha gid i de(cid:28)e a ea e  he e  be he i de  y wih a fa i vie hee f he 2D E e e ai [14; 59℄. he N , j N(cid:1) wih 0 ≤ i, j ≤ N − 1 ad dee . ωij ≡ ω(rij) he vi iy va e a i rij . The a  be f i i N 2 k=1 Akδ(σ − σk). F hi (cid:28)ie N axiai   e f i ae  (cid:28)g ai a e i We dee he ai e i by rij =(cid:0) i A di ed i he evi  e i we a e D(σ) = PK XN =(cid:8)ωN = (ωij)0≤i,j≤N−1 | ∀i, j ωij ∈ {σ1, . . . , σK} ad ∀k #{ωij | ωij = σk} = N 2Ak(cid:9) . ee #(A) i he adia f e A. We e ha XN deed  D(σ) h gh Ak ad σk ee 8. Uig he abve exei we de(cid:28)e he eegy he ΓN (E0, ∆E) a ΓN (E0, ∆E) =(cid:8)ωN ∈ XN | E0 ≤ EN(cid:2)ωN(cid:3) ≤ E0 + ∆E(cid:9) , whee EN = 1 2N 2 N−1Xi,j=0 v2 ij = − 1 2N 2 N−1Xi,j=0 ωij ψij, i he (cid:28)ie N axiai f he ye eegy wih vij = v(rij ) ad ψij = ψ(rij ) beig he di eized ve iy (cid:28)ed ad eaf  i (cid:28)ed ee ivey. ∆E i he widh f he eegy he. S h a (cid:28)ie widh i e eay f   di ee axiai a he adia f XN i (cid:28)ie. The he e f a eibe eegie  XN i a (cid:28)ie. e ∆N E be he yi a di(cid:27)ee e bewee w  eive a hievabe eegie. We he a e ha ∆N E ≪ ∆E ≪ E0 . The ii ea e de(cid:28)ed bew i exe ed  be ideede f ∆E i he ii N → ∞. The f daea a i f aii a e hai  ae ha ea h i ae i he (cid:28)g ai a e i e ibabe. By vi e f hi a i he babiiy  beve ay i ae i Ω−1 N (E0, ∆E) whee ΩN (E0, ∆E) i he  be f a eibe i ae i.e. he adia f he e ΓN (E0, ∆E). The ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 16 Feddy B he Ceae adii ad T  Tagaife (cid:28)ie N e i(cid:28) Bza ey i de(cid:28)ed a SN (E0, ∆E) = 1 N 2 log ΩN (E0, ∆E). 11 The i  ai a ea e i he de(cid:28)ed h gh he exe ai va e f ay bevabe A. F ay bevabe A[ω] f ia e a h f  ia f 1 The exe ai va e f AN f he i  ai a ea e ead he vi iy (cid:28)ed we de(cid:28)e i (cid:28)ie dieia axiai by AN(cid:2)ωN(cid:3). AN(cid:2)ωN(cid:3) . (cid:10)µN (E0, ∆E), AN(cid:2)ωN(cid:3)(cid:11)N ≡(cid:10)AN(cid:2)ωN(cid:3)(cid:11)N ≡ ΩN (E0, ∆E) XωN∈ΓN (E0,∆E) The i  ai a ea e µ f he 2D E e e ai i de(cid:28)ed a a ii f he (cid:28)ie N ea e: hµ(E0), A[ω]i ≡ lim N→∞(cid:10)µN (E0, ∆E), AN(cid:2)ωN(cid:3)(cid:11)N . The e i(cid:28) Bza ey i he de(cid:28)ed a S(E0) = lim N→∞ SN (E0, ∆E). 12 3.3 The ea (cid:28)ed vaiaia be a a age deviai e  C ig he Bza ey by die  eva ai f E. 12 i  ay a ia abe be. weve we ha  eed i a di(cid:27)ee way ad hw ha hi aeaive  ai yied he ae ey i he ii N → ∞. We give he ii ag e i de  ve ha he  ai f he Bza ey E. 12 i e ivae  he axiizai f he aied vaiaia be 10  aed a ea (cid:28)ed vaiaia be. Thi vaiaia be i he f dai f he RS aa h  he e iibi  aii a e hai  f he 2D E e e ai. The eeia eage i ha he ey  ed f he ea (cid:28)ed vaiaia be 10 ad f Bza ey de(cid:28)ii 12 ae he e a i he ii N → ∞. The abiiy   e he Bza ey h gh hi ye f vaiaia be i e f he ee f aii a e hai .   he ii deivai i baed  he ae ye f biai  ag e a he e ed by Bza f he ieeai f i H f  i i he hey f eaxai  e iibi  f a di e ga. Thi deivai de e he e hi aiie f age deviai hey. The ai i  a  ay bai he age deviai ieeai f he ey ad  vide a he ii deadig ig bai aheai  y. The de aheai a f f he eaihi bewee he Bza ey ad he ea (cid:28)ed vaiaia be ivve Sav hee. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 17 a ae ae e f i  i (cid:28)g ai haig iia a  i behavi.   ai i  ey ideify a ae ha f y de ibe he ai fea e f he age  ae f 2D  b e (cid:29)w ad he   e hei babiiy  ey. e  (cid:28) de(cid:28)e a ae h gh  a ae gaiig. We divide he N × N ai e i (N/n)× (N/n)  veaig bxe ea h aiig n2 i n i a eve  be ad N i a  ie f n. Thee bxe ae eeed  ie (i, j) = (In, Jn), whee iege I ad J veify 0 ≤ I, J ≤ N/n − 1. The idi e (I, J) abe he bxe. IJ be he fe e y  (cid:28)d he va e σk i F ay i ae ωN ∈ ΓN  e f k gid he bx (I, J) F k IJ (ωN ) = 1 n2 I+n/2Xi=I−n/2+1 J+n/2Xj=J−n/2+1 δ d (ωij − σk), whee δ (x) i e a  e wheeve x = 0 ad ze hewie. We e ha f IJ (ωN ) = 1. d k=1 F k a (I, J)PK A a ae pN =(cid:8)pk ωN ∈ XN  h ha F k f ii iy pN =(cid:8)pk IJ (ωN ) = pk IJ(cid:9)0≤I,J≤N/n−1;1≤k≤K  i he e f a i ae f IJ(cid:9)0≤I,J≤N/n−1;1≤k≤K efe  bh he e f va e ad IJ f a I, J  ad k by ab e f ai ad  he e f i ae havig he edig fe e ie. The ey f he a ae i de(cid:28)ed a he gaih f he  be f i ae i he a ae SN [pN ] = 1 N 2 log(cid:0)#(cid:8)ωN ∈ XN(cid:12)(cid:12) f a I, J, ad k, F k IJ (ωN ) = pk IJ(cid:9)(cid:1) . 13 Fwig a ag e by Bza i i a ai a exe ie i aii a e hai  ig biai  ad he Siig f a  ve ha i he ii N ≫ n ≫ 1  wih  akig i a   f he aea ai Ak  he ey f he a ae w d vege  SN [pN ] N≫n≫1∼ SN [pN ] = − n2 N 2 N/n−1XI,J=0 KXk=1 IJ log pk pk IJ f a give n2 I,J=0 pk IJ = Ak ad ∀I, J, N [pIJ ] = 1. A eay geeaizai f he abve if ∀I, J, N [pIJ ] = 1 ad SN [pN ] ∼ −∞ hewie whee N [pIJ ] ≡ Pk pk The aea ai ae eaiy exeed a ai ve pN : AN(cid:2)pk N(cid:3) ≡ N 2PN/n−1 if ∀k, AN(cid:2)pk N(cid:3) = Ak  ad SN [pN ] ∼ −∞ hewie.  he hey f age devia wih eegy EN(cid:2)ωN(cid:3) veifyig E0 ≤ EN(cid:2)ωN(cid:3) ≤ E0 + ∆E he iee i f ΓN (E0, ∆E) ad pN . F a give a ae pN   a i ae have he ae eegy. The ai a ew a ae (pN , E0) whi h i he e f i ae ωN SN [pN ] N≫n≫1∼ SN [pN ] IJ . i hi e   d have bee baied ig Sav hee. We w ide ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 18 Feddy B he Ceae adii ad T  Tagaife  he eegy h  a  be e a a a ie ai  he a ae pN . The e ha  ea he eegy ai i a e  be way. The eegy i EN(cid:2)ωN(cid:3) = − 1 2N 2 N−1Xi,j=0 ij ψN ωN ij . The eaf  i ψN ij i eaed  ωN h gh ψij = 1 N 2 N−1Xi′,j′=0 Gij,i′j′ ωN i′j′ , i he aa ia Gee f  i i he dai D.  he ii whee Gij,i′j′ N ≫ n ≫ 1 he vaiai f Gij,i′j′ f (i′, j′)  ig ve he a bx (I, J) ae vaihigy a. The Gij,i′j′ a be we axiaed by hei aveage va e ve he bxe GIJ,I′J′ . The ψij ≃ ψIJ ≡ 1 N 2 N/n−1XI′,J′=0 GIJ,I′J′ I+n/2Xi′=I−n/2+1 J+n/2Xj′=J−n/2+1 ωN i′j′ = n2 N 2 N−1XI′,J′=0 GIJ,I′J′ ωN IJ , whee he ae gaied vi iy i de(cid:28)ed a ωN IJ = 1 n2 I+n/2Xi′=I−n/2+1 J+n/2Xj′=J−n/2+1 ωN i′j′ . We e ha ve he a ae pN  he ae gaied vi iy deed  pN y: ωN IJ = pk IJ σk f ωN ∈ pN . KXk=1 Uig iia ag e i i eay   de ha i he ii N ≫ n ≫ 1 he eegy f ay i ae f he a ae pN i we axiaed by he eegy f he ae gaied vi iy EN(cid:2)ωN(cid:3) N≫n≫1∼ ENhωN IJi = − n2 2N 2 N/n−1XI,J=0 IJ ψN ωN IJ . The he Bza ey f he a ae i SN [pN , E0] N≫n≫1∼ SN [pN ] N(cid:3) = Ak ad ENhωN 14 IJi = E0  ad SN [pN , E0] ∼ −∞ if ∀k, N [pk hewie. N ] = 1 AN(cid:2)pk Cide PN,E0(pN )  be he babiiy deiy  beve he a ae pN i he (cid:28)ie N i  ai a eebe wih eegy E0 . By de(cid:28)ii f he ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 19 i  ai a eebe f he ey SN (E0) ee E. 11 ad he e edig aagah we have log PN,E0(pN ) N→∞∼ N 2 [SN [pN , E0] − SN (E0)] . 15 F he geea de(cid:28)ii f a age deviai e  give by E. 1 we eay We w ide he i   ii. The a ae pk ee ha f a 14 i a age deviai e  f he a ae pN i he i  ai a eebe. The age deviai aaee i N 2 ae f  i i −SN [pN , E0] + SN (E0). N ae w ee a he (cid:28)ie N axiai f pk  he  a babiiy  beve ω(r) = σk : pk(r) = hδ(ω(r)− σk)i. The a ae i he haa eized by p = {p1, . . . , pK}. Takig he ii N ≫ n ≫ 1 aw   de(cid:28)e he ey f he a ae (p, E0) a ad he age deviai S[p, E0] = S[p] ≡Xk ZD pklogpk dr 16 if ∀k N [pk] = 1 A [pk] = Ak ad E[ω] = E0  ad S[p, E0] = −∞ hewie.  he ae ii i i eay ee f de(cid:28)ii 13 ad e  16 ha hee i a  eai f i ae e  he  babe a ae. The exeia  eai e  hi  babe ae i a age deviai e  whee he ey aea a he ie f a age deviai ae f  i    a ieeva a. The exeia vege e wad hi  babe ae a j i(cid:28)e he axiai f he ey wih he ey f he  babe a ae. Th  i he ii N → ∞ we a exe he Bza ey E. 12 a S(E0) = sup {p | N [p]=1}{S[p] | E[¯ω] = E0, ∀k A[pk] = Ak} , 17 whee p = {p1, . . . , pK} ad ∀ r N [p](r) = PK k=1 pk(r) = 1 i he  a  aizai. F hee A[pk] i he aea f he dai edig  he vi iy va e ω = σk . The fa  ha he Bza ey S(E0) E. 12 a be  ed f he vaiaia be 17 i a wef  ivia e  f age deviai hey. 3.4 Ai ai f e iibi  aii a e hai   he w evi  e i we have de(cid:28)ed he i  ai a ea e f he w dieia E e ad  ai gehi e ai ad we have ve ha he gaih f he babiiy f a a ae p i give by he a ae ey 16. We a  de ha  f he i ae wi ed  he  babe a ae he e ha a  ay axiize he vaiaia be 17. Thi  babe a ae i aed he e iibi  a ae. Thi ea ha if we ake a ad i ae i wi eay  ey have he ae ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 20 Feddy B he Ceae adii ad T  Tagaife bevai Vyage Saii a E iibi  Fig. 5: ef: he beved ve iy (cid:28)ed i f Vyage a e af daa f Dwig ad ge [25℄ ; he egh f ea h ie i ia  he ve iy a ha i. e he g je   e f widh f de R he Rby defai adi . Righ: he ve iy (cid:28)ed f he aii a e iibi  de f he Gea Red S. The a  a va e f he je axi  ve iy je widh vex widh ad egh (cid:28) wih he beved e. The je i ieeed a he iefa e bewee w hae; ea h f he ed  a di(cid:27)ee ixig eve f he eia vi iy. The je hae bey a iia egh vaiaia be a ieiei a be baa ed by he e(cid:27)e  f he dee aye hea. ve iy a he e f he e iibi  a ae. A a e e e we  de ha e iibi  a ae ae a a adidae  de ef gaized age  ae  b e (cid:29)w ike f ia e he Gea Red S f  ie hw  (cid:28)g e 1. A  be f wk have ideed he ai f ef gaized  b e (cid:29)w wih e iibi  a ae. eeed eade wi (cid:28)d ai  wih exeie ad  ei a i ai de ibed i he eview [64℄ wheea de f gehyi a (cid:29)w f ia e he Gea Red S f  ie  ea e ae vi e g id bai je iia  he G f Sea  he  hi ae di ed i he eview [16℄. Re e ai ai  de he vei a   e f  ea a be f d i he ae [70; 71℄. A a exae (cid:28)g e 5 hw he ai f he beved ve iy (cid:28)ed f he Gea Red S f  ie wih he ve iy (cid:28)ed f a e iibi  a ae f he  ai gehi de. The heei a aayi f hi e i ibi  a ae [11℄ i baed  a aagy wih Va De Wa(cid:21)Cah(cid:21)iiad de f (cid:28) de aii ad he hae f he g je bey a iia egh vaiaia be a ieiei a be baa ed by he e(cid:27)e  f he dee aye hea ee [16℄ f e deai. Ahe exae f e iibi  edi i i he hae diaga f aii a e iibia f he w dieia E e e ai  a d by eidi dai  . Thi hae diaga (cid:28)g e 6 hw ha he aii a e iibia ae eihe die e y e ad e ai y e  aae (cid:29)w. Thi exae i f he di ed i he wk [10℄ ad he eview [16℄. Thi e iibi  hae diaga ha a bee ed i de  edi   e iibi  hae aii [10℄ a i di ed i e i 4.4. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 21        ✞ ✟ ✠ ✻ ■✧ ❛✹ ❉✕✖✗✘✙ ■■✧ ✯❣ ❯✚✕✛✕✜✙✢✣✕✗✚✤✘ ✥✘✗✦ ✻ ✠ ✟ ✞ ✝ ✆ ☎ ☎ ✆ ❣ ✲❣✯ ✸✁✺ ✝ ✸ ✷✁✺ ✷ ✶✁✺ ✶ ✵✁✺ ✵ ✵ ✏ ✎ ✍ ✌ ☞ ☛ ✡ ✡ ✼ ✰ ✪ ✮ ✭ ✬ ✫ ★ ★ ☛ ☞ ✌ ✍ ✎ ✏ ★✩★✪ ★✩✫ ★✩✫✪ ✱ ★✩✬ ★✩✬✪ ✶✵ ✷✵ ✸✵ ❊ ✵ ✺✵ s✑✒✓ t✔✒✓ Fig. 6: Bif  ai diaga f aii a e iibia f he w dieia E e e ai i a d by eidi dai a i he g a4 ae g i eaed  he dai ae  ai ad a4  he f h de e f he vi iy diib i eae ee [16℄. b baied  ei ay i he E − a4 ae E i he eegy i he ae f d by eidi geey wih ae  ai δ = 1.1. The ed ie ae eaf  i ad he ie ve i ae gd ageee bewee  ei a ad heei a e  i he w eegy ii. 4  e iibi  hae aii ah iega ad ia hey The ai f hi e i i  di   e iibi  hae aii i  b e (cid:29)w e e i(cid:28) ay f he dyai  f he w dieia avie(cid:21)Ske e ai wih ad f e  ai gehi dyai  wih ad f e  eaed dyai . We wa  di  ie exae f whi h i ai wih ae aii bewee w aa  exi biabiiy. We wi e ah iega ad age deviai i de   e he  babe ah f he aii ad he aii ae.  de  give a edaggi a eeai f ah iega ad age devia i hey f  hai dyai  we (cid:28) di  he exeey ai a ae f he ae be: he ve daed dyai  f a ai e i a d be we eia i e i 4.1. We geeaize hee e   a aba  e f dyai  aed agevi dyai  i e i 4.2. We ay hee e   w dieia E e ad ai Gehi agevi dyai  i e i 4.3 f whi h we ae abe  edi  biabiiy  e aii ae ad he  babe aii ah. Fiay we di  ah iega aa he ad a i iiize f he  hai avie Ske e ai i a  e iibi  ex i e i 4.4. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== ✂ ✄ ✳ ✴ 22 Feddy B he Ceae adii ad T  Tagaife 4.1 age deviai f he vedaed agevi dyai  We wih (cid:28)  give a edaggi a de ii f age deviai hey i  e iibi  ye  e e i(cid:28) ay f dyai  iig f  hai di(cid:27)eeia e ai. Theefe we begi by ayig age deviai hey  a ie a adei exae f a ve daed ai e i a d be we eia he ae be whee a age deviai e  exi. We wi hw ha we a  e he aii ae f he i f he ai e f e we  he he ad ha he e  i a Ahei  fa  i i ia  he exeia f he eegy baie heigh bewee he w we.  fa  hi i a age deviai e . Thi e i deve ai a idea. We e he ah iega fai f  hai  ee [52; 74℄. Siia e  ae di ed by aheai ia i he faewk f he Feidi Weze hey [30; 67℄. We ae  h ieeed by he ie evea yeie f he a i ad i e e e f he yey bewee eaxai ad (cid:29)  ai ah ad i e e e f he  ai f he  babe aii ia. The yeie ae di   h e fe ha he he aeia b  hee ae a vey ai a e ee ay i dae f age we d  kw exa y. 4.1.1 The vedaed agevi dyai  We ide a ige vedaed ai e i a 1D d be we eia V (x) ad  b je ed  ad f e d e  a a  ig  a hea bah. F ii iy we ideed he vedaed ii f whi h he dyai  f he ai e ii x i gveed by he  hai di(cid:27)eeia e ai ˙x = − dV dx +r 2 β η, 18 whee η i a ad whie ie wih a Ga ia diib i haa eized by E [η(t)η(t′)] = δ(t−t′) V (x) i a d be we eia ee Fig. 7 ad β = 1/kBT whee T i he eea e.  he deeiii i ai whe 1/β = 0 he ai e eaxe  e f he w abe eady ae f he eia V  i.e. i vege eihe  x = −1   x = 1.  he ee e f hea ie he ai e ay gai e gh eegy  j  he eia baie a x = 0 ad ee i he he eia we. f he f ig i weak i.e. 1 ≪ β∆V  he he j  bewee we wi be ae eve ad wi be aii ay ideede f e ahe. They wi he be de ibed by a i  e haa eized by a aii ae λ. We wi hw ha e a ay he hey f age deviai i de   e λ. eve he hey f age deviai wi ead  he  i ha  f he aii ah  eae e  he  babe aii ah. A wi be di ed e e iey bew hi  babe aii ah i hi i ai i aed a ia.  de  bai hee e  we wi e fa  ai baed  a ah iega f ai f he aii babiiie f he  hai  e ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 23 18. S h a ah iega f ai i efeed a age(cid:21)a h  fa i a age ad a h  (cid:28) ed i few yea afe he ah iega f ai f  a  e hai  by Feya. ) x ( V 1 0.8 0.6 0.4 0.2 0 -2 -1.5 -1 -0.5 DV 0 x 0.5 1 1.5 2 Fig. 7: Gah f he d be we eia V (x) = (x2 − 1)2/4. We beve w abe eady ae a x = ±1 ad a adde a x = 0 wih heigh ∆V = 1/4. 4.1.2 The aii babiiy a a ah iega T give a ie deadig f he age a h  fai we (cid:28)  ide a ve  η = {ηi}1≤i≤N f ideede Ga ia ad vaiabe wih ze ea E(ηi) = 0 ad vaia e E(ηiηj ) = δij . By de(cid:28)ii he babiiy ea e f η i he Ga ia ea e 19 dµ = exp − 1 2 η2 NXi=1 . dηi√2π i! NYi=1 (xi−1) +s 2∆t β The E e axiai f he agevi e ai 18 i wihi he  ve i xi = xi−1 − ∆t dV dx ηi 20 f 1 ≤ i ≤ N ad wih x0 = x(0) a give iiia ae. The babiiy ea e f a ai a ah x = {xi}1≤i≤N i give by iveig 20 ad ieig i i 19 dµ = exp − β 4 NXi=1(cid:18) xi − xi−1 ∆t + dV dx (xi−1)(cid:19)2 ∆t! J(η|x) dxi√2π . NYi=1 21  vei 20 he edig aix i we iag a wih e i he  hi exei J(η|x) i he a bia f he hage f vaiabe η → x.  he diaga  ha J(η|x) = 1. ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 24 Feddy B he Ceae adii ad T  Tagaife The ea e f a Ga ia  hai  e η(t) f ze ea E[η(t)] = 0 ad vaia e E[η(t)η(t′)] = δ(t − t′)  a ie ieva [0, T ] wih T = N ∆t i he fa geeaizai f he abve (cid:28)ie dieia ea e 19 dµ = exp − 1 2Z T 0 η2(t)dt!D[η]. 22 The di(cid:27)eeia eee D[η] i he abve exei i he fa ii f he dηi√2π f N → ∞, ∆t → 0 whee ηi = η(i∆t) = η(iT /N ). ee we aied i aheai  kw he di(cid:30) y  de(cid:28)e  h (cid:28)ie dieia  aiyQN i=1 a b je  b  we wi kee   di i a a fa eve ad ae ha hi fa ai ai a he aheai a  beie eaed  he ii N → ∞, ∆t → 0. The he babiiy ea e f a ai a a je y {x(t)}0≤t≤T i a he fa ii f 21 dµ = exp − β 0 (cid:18) ˙x + dV dx(cid:19)2 4Z T dt! J[η|x]D[x], 23 whee J[η|x] i he a bia f he hage f vaiabe η → x ad i a e a  e we efe  [74℄ f a e geea eae ig ha [74℄ a  ay e he Savi h vei. The aii babiiy f a iiia ae x0 a ie 0  a (cid:28)a ae xT a ie T i he   ve a ibe ah {x(t)}0≤t≤T  h ha x(0) = x0 ad x(T ) = xT f he babiiy f a ige ah 23. S h a   a be fay wie a he ah iega P (xT , T ; x0, 0) =Z x(T )=xT x(0)=x0 exp(cid:18)− β 2 A[x](cid:19) D[x], wih he a i f  ia A[x] = 1 0 (cid:18) ˙x + dV dx(cid:19)2 2Z T dt. 24 25 F 24 i i ea ha he  babe a je ie wih e ibed iiia ad (cid:28)a ae ae iiize f he a i wih e ibed iiia ad (cid:28)a i. The ia a i i deed A(x0, xT , T ) = min{A[x]| x(0) = x0, x(T ) = xT} . 4.1.3 F  ai ah Whe he iiia i x0 = xa beg  a aa  f he deeiii dy ai  f he ae be if x0 = xa = ±1 i a abe (cid:28)xed i i i exe ed ha he a i A(xa, X, T ) de eae wih ie. The a i iia aig f e aa  ad havig a i(cid:28)ie d ai wi h  ay a i ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 1 S hai avie(cid:21)Ske E ai ad Gehi T b e e 25 a e. eve he i(cid:28)ie ie a i iiize ae eeia be a e he aii babiiy P (X, T ; xa, 0) vege  he aiay diib i f he  hai  e whe he ie T ge  i(cid:28)iy. The a i iiize aig f e aa  ad wih a i(cid:28)ie d ai ae aed (cid:29)  ai ah hey ve A(xa, X,∞) = minnA[x]| lim T→∞ x(−T ) = xa, x(0) = Xo . 4.1.4 Reaxai ah We ide a ae X ha beg  he bai f aa i f a aa  xa f he deeiii dyai . The eaxai ah aig a x deed {xr(t)}0≤t≤T i de(cid:28)ed by ˙xr = − dV dx (xr) wih iiia dii xr(0) = X . A he ah vege  xa  we have xr(+∞) = xa . Uig he exei f he a i 25 we ee ha A[xr] = 0 a he eaxai ah i a deeiii  i ad we a i e ha A[x] ≥ 0 f ay ah {x(t)}0≤t≤T . A a e e e eaxai ah ae gba iiize f he a i A[x]. Thi i be a e fwig he deeiii dyai  xr i de  ea h he aa  xa aig f X de e ie ay  hai e bai  ha he  i ze ad he babiiy i axia. 4.1.5 Tie evea yey ad he eai bewee (cid:29)  ai ad eaxai ah  de  haa eize (cid:29)  ai ah ad ia we wi ake (cid:28) f he ie evea yey f he ve daed agevi dyai . We ide a ah {x(t)}0≤t≤T ad he eveed ah R[x] = {x(T − t)}0≤t≤T . The a i f he eveed ah ead 1 1 dt = R[x] + dV dx dt (R[x])(cid:19)2 0 (cid:18) d 2Z T 0 (cid:18)− ˙x(t′) + dV 2Z T A[R[x]] = wih he hage f vaiabe t′ = T − t. The wiig =(cid:18) ˙x + dV dx(cid:19)2 =(cid:18) ˙x + dV dx(cid:19)2 (cid:18) ˙x − dV dx(cid:19)2 − 4 ˙x dV dx (x(t′))(cid:19)2 dt′, dx − 4 d dt V (x), we ge A[R[x]] = A[x] − 2 (V (x(T )) − V (x(0))) . 26  ggig hi eai i he ah iega exei f he aii babi iy 24 we bai ##7#52#aSUZPUk1BVC1WaXJ0dWFsbw== 26 Feddy B he Ceae adii ad T  Tagaife P (R[xT ], T ; R[x0], 0) = P (xT , T ; x0, 0) exp(cid:18) V (x(T )) − V (x(0)) kBT (cid:19) . We e gize he Gibb aiay diib i f he ve daed agevi e a i PS(x) = 1 Z e−V (x)/kB T eai   ha he abve exei give he deaied baa e P (xT , T ; x0, 0)PS(x0) = P (x0, T ; xT , 0)PS(xT ). We have h  ve ha deaied baa e i a e e e f he ie evea yey a exe ed  geea g d. We w ide he (cid:29)  ai ah f e aa  xa  ay i X f i bai f aa i. Uig eai 26 ad he fa  ha he a i i away iive we have A[x] ≥ 2 (V (x(T )) − V (x(0))) , 27 iiia ae i a aa  ad he (cid:28)a ae i ahe i i he a iaed wih e aiy if ad y if x i a iiize f he eveed a i A[R[x]]. f he bai f aa i he eveed a i A[R[x]] i a ay iiized by he eaxai ah R[x
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5th Warsaw School of Statistical Physics
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