Typical and rare fluctuations in nonlinear driven diffusive systems with dissipation

Typical and a e luc ua ions in nonlinea d i en di usi e sys ems wi h dissipa ion Pablo I. Hu ado∗ Ins i u o Ca los I de F´ısica Te´o ica y Compu acional, and Depa amen o de Elec omagne ismo y F´ısica de la Ma e ia, Uni e sidad de G anada, G anada 18071, Spain A. Lasan a†and A. P ados‡ F´ısica Te´o ica, Uni e sidad de Se illa, Apdo. de Co eos 1065, Se illa 41080, Spain (Da ed: Feb ua y 27, 2013) We conside luc ua ions o he dissipa ed ene gy in nonlinea d i en di usi e sys ems subjec o bulk dissipa ion and bounda y d i ing. Wi h his aim, we ex end he ecen ly-in oduced mac o- scopic luc ua ion heo y o nonlinea d i en dissipa i e media, s a ing om he luc ua ing hyd o- dynamic equa ions desc ibing he sys em mesoscopic e olu ion. In e es ingly, he ac ion associa ed o a pa h in mesoscopic phase-space, om which la ge-de ia ion unc ions o mac oscopic obse - ables can be de i ed, has he same simple o m as in non-dissipa i e sys ems. This is a consequence o he quasi-elas ici y o mic oscopic dynamics, equi ed in o de o ha e a non i ial compe i ion be ween di usion and dissipa ion a he mesoscale. Eule -Lag ange equa ions o he op imal densi y and cu en ields ha sus ain an a bi a y dissipa ion luc ua ion a e also de i ed. A pe u ba i e solu ion he eo shows ha he p obabili y dis ibu ion o small luc ua ions is always gaussian, as expec ed om he cen al limi heo em. On he o he hand, s ong sepa a ion om he gaussian beha io is obse ed o la ge luc ua ions, wi h a dis ibu ion which shows no nega i e b anch, hus iola ing he Galla o i-Cohen luc ua ion heo em as expec ed om he i e e sibili y o he dynamics. The dissipa ion la ge-de ia ion unc ion exhibi s simple and gene al scaling o ms o weakly and s ongly dissipa i e sys ems, wi h la ge luc ua ions a o ed in he o me case bu hea ily sup essed in he la e . We apply ou esul s o a gene al class o di usi e la ice models o which dissipa ion, nonlinea di usion and d i ing a e he key ing edien s. The heo e ical p e- dic ions a e compa ed o ex ensi e nume ical simula ions o he mic oscopic models, and excellen ag eemen is ound. In e es ingly, he la ge-de ia ion unc ion is in some cases non-con ex beyond some dissipa ion. These esul s show ha a sui able gene aliza ion o mac oscopic luc ua ion heo y is capable o desc ibing in de ail he luc ua ing beha io o nonlinea d i en dissipa i e media. I. INTRODUCTION Fluc ua ions a e inhe en o many physical phenom- ena, e lec ing he hec ic mic oscopic dynamics a mac o- scopic scales. In spi e o hei appa en andom o i- gin, essen ial physical in o ma ion is encoded he ein [1]. A classical example is he luc ua ion-dissipa ion heo- em, which ela es he linea esponse o a sys em o an ex e nal pe u ba ion o he luc ua ion p ope ies o he sys em in he mal equilib ium [2, 3]. Mo e e- cen ly, he in es iga ion o gene al p ope ies o luc u- a ions in nonequilib ium s eady s a es is opening new pa hs o unde s anding physics a om equilib ium [4– 14]. The s udy o luc ua ion s a is ics o mac oscopic obse ables p o ides an al e na i e pa h o ob ain he - modynamic po en ials, a complemen a y app oach o he usual ensemble desc ip ion. This obse a ion, alid bo h in equilib ium [1] and nonequilib ium [7, 9], is mos ele- an in he la e case because no gene al bo om-up ap- p oach, connec ing mic oscopic dynamics o mac oscopic nonequilib ium p ope ies, has been ound ye . In his way, he la ge de ia ion unc ion (LDF) con olling he ∗ph[email p o ec ed] †alasan[email p o ec ed] ‡[email p o ec ed] s a is ics o hese luc ua ions may play in nonequilib- ium s a is ical mechanics a ole simila o he equilib- ium ee ene gy [4, 10]. A cen al poin o his eme g- ing pa adigm is he iden i ica ion o he ele an mac o- scopic obse ables cha ac e izing he ou o equilib ium beha io o he sys em a hand. The sys em dynamics o en conse es locally some magni ude (a densi y o pa - icles, ene gy, momen um, cha ge, e c.), and he essen ial nonequilib ium obse able is hus he cu en o lux sus- ained by he sys em when subjec o bounda y-induced g adien s o ex e nal ields. The e o e, he unde s anding o cu en s a is ics in e ms o he mic oscopic dynam- ics ep esen s one o he main p oblems o nonequilib- ium s a is ical mechanics, igge ing an in ense esea ch e o in ecen yea s. In his con ex , key esul s a e he Galla o i-Cohen luc ua ion heo em [5], which e- la es he p obabili y o obse ing a gi en cu en luc u- a ion ~ Jwi h he p obabili y o he e e sed e en −~ J, o he ecen ly-in oduced Isome ic Fluc ua ion Rela ion [14], which ela es he p obabili y o any pai o isome - ic cu en luc ua ions ( ~ J, ~ J0), wi h |~ J|=|~ J0|. These in iguing “symme ies” appea as a consequence o he in a iance unde ime e e sal o he unde lying mic o- scopic dynamics. These and o he ecen esul s [5–14] a e howe e es ic ed o nonequilib ium “conse a i e” sys ems cha ac e ized by cu en s. On he o he hand, many nonequilib ium sys ems a e inhe en ly dissipa i e, ha is, hey need a con inuous in- a Xi :1302.6544 1 [cond-ma .s a -mech] 26 Feb 2013 2 pu o ene gy in o de o each a s eady s a e. In his class o sys ems, he ele an mac oscopic obse able is no only he cu en : he dissipa ed ene gy is also expec ed o play a main ole. These sys ems include g anula media [16], dissipa i e biophysical sys ems [18], u bulen luids [19], ac i e ma e [20], chemical eac ions [21], popula- ion dynamics [22], e c. In gene al, his class comp ises all so o eac ion-di usion sys ems whe e dissipa ion, di usion and d i ing a e he main physical mechanisms. Fluc ua ions in dissipa i e media ha e been much less in- es iga ed, mos p obably because hei physics is mo e complica ed as a esul o he i e e sibili y o hei mi- c oscopic dynamics. In p inciple, mos o he esul s o nonequilib ium s eady s a es we ha e e e ed o in he p e ious pa ag aph a e no applicable o dissipa i e sys- ems, since hey s em om he e e sibili y o he un- de lying mic oscopic dynamics in he conse a i e case. The e o e, a ques ion na u ally a ises as o whe he i is possible o ex end some o hese ideas o dissipa i e me- dia. One o he main goals o he p esen pape is o gi e a (pa ial) answe o his ques ion. In his wo k, we analyze bo h ypical and a e luc- ua ions in nonlinea d i en di usi e sys ems wi h dis- sipa ion. This is done by combining a sui able gene al- iza ion o mac oscopic luc ua ion heo y (MFT) [7] o he ealm o dissipa i e media, and ex ensi e nume ical simula ions o a pa icula albei b oad class o mic o- scopic models. Ou s a ing poin is a gene al luc u- a ing balance equa ion o he (ene gy) densi y, wi h a d i e m p opo ional o he spa ial de i a i e o he cu en and a sink e m. This mesoscopic desc ip ion is expec ed o be alid o many d i en dissipa i e media o e a ce ain “hyd odynamic” ime scale, much la ge han he one cha ac e is ic o he mic oscopic dynam- ics. O e he as (mic oscopic) ime scale, he sys em o ge s he ini ial condi ions and elaxes o a local equi- lib ium s a e in which all he p ope ies o he sys em become unc ionals o a ew “hyd odynamic” ields, he e he densi y, he cu en and he dissipa ion. A e wa ds, o e he much slowe hyd odynamic ime scale, he sys- em e en ually app oaches he s eady s a e ollowing he mesoscopic balance equa ion. We ocus on he luc ua- ions o he sys em in his nonequilib ium s eady s a e, in which he dissipa ion and he injec ion o ene gy bal- ance each o he . By using his luc ua ing hyd odynamic pic u e oge he wi h a pa h in eg al o mula ion, we de- i e a gene al o m o he ac ion associa ed o a his o y o he densi y, cu en and dissipa ion ields ( ha is, a pa h in mesoscopic phase space). Rema kably, his ac ion akes he same o m as in conse a i e nonequilib ium sys ems [7–14], simpli ying he analysis in he dissipa i e case. This is bo h an impo an and a su p ising esul , which s ems om he quasi-elas ic cha ac e o he un- de lying mic oscopic dynamics in he la ge sys em size limi . This quasi-elas ici y is necessa y in o de o ha e a balanced compe i ion be ween di usion and dissipa ion a he mesoscopic le el. F om he de i ed ac ion unc- ional, and using he ecen ly-in oduced addi i i y con- jec u e [8, 11–13], a gene al o m o he LDF o he dis- sipa ed ene gy is de i ed, wi h a “Lag angian” including second o de de i a i es. The e om, we de i e he Eule - Lag ange equa ion (a ou h-o de di e en ial equa ion) o he op imal ields esponsible o an a bi a y luc- ua ion. This Lag angian a ia ional p oblem can be mapped on o an equi alen Hamil onian p oblem ( ou coupled i s -o de di e en ial equa ions) which u ns ou o simpli y he analysis. We use his Hamil onian pic- u e o analyze in de ail h ee di e en limi s, namely small luc ua ions a ound he a e age o a bi a y dissi- pa ion coe icien , and he whole spec um o luc ua ions ( ypical and a e) o weakly- and s ongly-dissipa i e sys ems. The s a is ics o ypical ( ha is, small) luc- ua ions is gaussian as expec ed om he cen al limi heo em. Howe e , s ong sepa a ion om gaussian be- ha io is obse ed o a e luc ua ions, wi h a dis i- bu ion which shows no nega i e b anch, hus iola ing he Galla o i-Cohen luc ua ion heo em as o he wise expec ed om he i e e sible cha ac e o mic oscopic dynamics. We s udy in gene al he weakly-dissipa i e sys em limi using a singula pe u ba ion expansion. This yields a simple scaling o m o he dissipa ion LDF, showing ha la ge dissipa ion luc ua ions a e a o ed in his weakly-dissipa i e limi , wi h a LDF which ex ends o e a b oad egime and decays slowly in he a posi i e ail. On he o he hand, a di e en pe u ba i e analy- sis in he s ongly-dissipa i e sys em limi can be ca ied based on he o ma ion o bounda y ene gy laye s in his limi o all luc ua ions, which e ec i ely decouples he sys em in wo almos -independen pa s. This analysis shows ha la ge dissipa ion luc ua ions a e hea ily sup- p essed in his limi , as opposed o he weakly-dissipa i e egime esul . We apply his heo e ical scheme o a gene al class o d-dimensional dissipa i e la ice models wi h s ochas- ic mic oscopic dynamics, o which he hyd odynamic luc ua ing pic u e used abo e as s a ing poin can be demons a ed in he la ge sys em size limi [39] (we will ocus he e in one dimension o simplici y). In hese mod- els he e is one pa icle a each la ice si e, cha ac e ized by i s ene gy. Dynamics is s ochas ic and p oceeds ia collisions be ween nea es neighbo s, a a a e which de- pends on he ene gy o he colliding pai . In a collision, a ce ain ac ion o he pai ene gy is dissipa ed, and he emaining ene gy is andomly dis ibu ed wi hin he pai . This mechanism gi es ise o a nonlinea compe i- ion be ween di usion and dissipa ion in he mac oscopic limi , p o ided ha he mic oscopic dissipa ion coe i- cien scales adequa ely wi h he sys em size. This class o models ep esen s a a coa se-g ained le el he physics o many eac ion-di usion sys ems o echnological as well as heo e ical in e es . In pa icula , when he collid- ing pai is chosen comple ely a andom, independen ly o he alue o i s ene gy, he Kipnis-Ma chio o-P esu i (KMP) model [23] o hea conduc ion is eco e ed in he conse a i e case. The KMP model plays a main ole in nonequilib ium s a is ical physics as a ouchs one o es 3 heo e ical ad ances [5, 6, 8, 11–15, 23]. Ou gene al class o models con ains he essen ial ing edien s cha - ac e izing mos dissipa i e media, namely: (i) di usi e dynamics, (ii) bulk dissipa ion, and (iii) bounda y injec- ion. The chances a e ha ou esul s emain alid o mo e complex dissipa i e media desc ibed a he meso- scopic le el by a simila e olu ion equa ion. He e we epo analy ical and simula ion esul s o he s a is ics o he dissipa ed ene gy in his gene al class o models using bo h s anda d simula ions and an ad anced Mon e Ca lo me hod [27]. The la e allows he sampling o he ails o he dis ibu ion, and implies simula ing a la ge numbe o clones o he sys em. The plan o he pape is as ollows. Sec ion II desc ibes a sui able gene aliza ion o mac oscopic luc ua ion he- o y o nonlinea d i en dissipa i e media. The la ge- de ia ion s a is ics o he dissipa ed ene gy, a cen al obse able in his ype o sys ems, is in es iga ed he e wi hin bo h he Lag angian and Hamil onian equi alen amewo ks. Sec ion III is de o ed o he de ailed s udy o di e en asymp o ic beha io s wi hin he Hamil onian o mula ion, which u ns ou o simpli y he analysis. In sec ion IV we de ine a gene al class o mic oscopic la ice models whose s ochas ic dynamics is dissipa i e [25, 39]. The heo e ical amewo k de eloped in he p e ious sec- ions is applied o his amily o models in sec ion V, and he LDF o he dissipa ed ene gy is explici ly wo ked ou . The analy ical p edic ions a e compa ed o ex en- si e nume ical simula ions o he mic oscopic models, and a e y good ag eemen is ound. A summa y o he main esul s o he pape , oge he wi h a physical discussion he eo , is gi en in sec. VI. Finally, he appendix deals wi h some echnical de ails ha , o he sake o cla i y, we ha e p e e ed o omi in he main ex . II. MACROSCOPIC FLUCTUATION THEORY FOR DRIVEN DISSIPATIVE SYSTEMS In his wo k, we will analyze a gene al class o sys- ems whose dynamics a he mesoscale is desc ibed by he ollowing luc ua ing e olu ion equa ion ∂ ρ(x, ) = −∂xj(x, ) + d(x, ).(2.1) We ocus he e in one dimension o simplici y, bu ou analysis can be ca ied ou in an equi alen manne in d- dimensions. In eq. (2.1), ρ(x, ), j(x, ) and d(x, ) a e he densi y, cu en and dissipa ion ields, espec i ely, and and x∈[−1/2,1/2] a e he mac oscopic ime and space a iables, ob ained a e a di usi e scaling limi such ha x= ˜x/L and =˜ /L2, wi h ˜xand ˜ he mic oscopic space and ime a iables and L he sys em leng h. These coa se-g ained spa ial and empo al scales eme ge om a sui able con inuum limi o he unde lying mic oscopic dynamics [39]. The cu en ield is a luc ua ing quan i y, and can be w i en as j(x, ) = −D(ρ)∂xρ(x, ) + ξ(x, ).(2.2) The i s e m is Fou ie ’s law, whe e D(ρ) is he di u- si i y (which migh be a nonlinea unc ion o he local densi y), and ξ(x, ) is he cu en noise ha is gaussian and whi e, hξ(x, )i= 0,hξ(x, )ξ(x0, 0)i=σ(ρ) Lδ(x−x0)δ( − 0), (2.3) wi h σ(ρ) being he so-called mobili y. This gaussian luc ua ing ield is expec ed o eme ge o mos si ua- ions in he app op ia e mesoscopic limi as a esul o a cen al limi heo em: al hough mic oscopic in e ac ions o a gi en model can be highly complica ed, he ensuing luc ua ions o he slow hyd odynamic ields esul om he sum o an eno mous amoun o andom e en s a he mic oscale which gi e ise o gaussian s a is ics, wi h an ampli ude o he o de o L−1/2, in he mesoscopic egime in which eq. (2.1) eme ges. On he o he hand, he dissipa ion ield d(x, ) is d(x, ) = −νR(ρ(x, )),(2.4) whe e νis he mac oscopic dissipa ion coe icien , and R(ρ) is a ce ain unc ion o he densi y ρ. Fo he calcu- la ions which ollow h oughou his sec ion, i is use ul o in oduce a new a iable y, such ha y=R(ρ),(2.5a) d(x, ) = −νy(x, ).(2.5b) The dissipa ion ield is p esen a he mesoscopic le el because he mic oscopic s ochas ic dynamics o he mod- els o in e es dissipa es some ene gy, ha is, we ha e he equi alen o a mic oscopic es i u ion coe icien α, so ha he amoun o dissipa ed ene gy is p opo ional o 1 −α. The mac oscopic dissipa ion coe icien νis hus p opo ional o 1 −α. No e howe e ha he e is no noise e m in eq. (2.4), so he local luc ua ions o he dissipa ion ield a e ensla ed o hose o he densi y ρ(x, ). The physical eason o his beha io is ha he mic oscopic dynamics mus be quasi-elas ic in o de o ensu e ha dissipa ion and di usion ake place o e he same ime scale in he he modynamic limi . Typically, 1−αmus scale as L−2 ha is he o de o magni ude o he di usi e e m in a sys em o leng h L[25, 39]. The bounda y condi ions o eq (2.1) depend on he physical si ua ion o in e es . Fo ins ance, we may con- side ha he sys em is kep in con ac wi h wo he - mal ese oi s a x=±1/2, a he same empe a u e T, so ρ(±1/2, ) = T. In ha case, he sys em e en ually eaches a s eady s a e in he long ime limi , o which he injec ion o ene gy h ough he bounda ies and he dissipa ion balance each o he . The s a iona y a e age (mac oscopic) solu ion o (2.1) e i ies j0 a (x) + νR(ρa (x)) = 0, ja (x) = −D(ρa (x))ρ0 a (x), (2.6) whe e he p ime indica es spa ial de i a i e. The i s equa ion in (2.6) ollows om (2.1), and he second one 4 is Fou ie ’s law o he a e ages. Equi alen ly, a closed second-o de equa ion o ρmay be w i en, d dx [D(ρa )ρ0 a ] = νR(ρa ),(2.7) wi h he bounda y condi ions ρa (±1/2) = T. Equa ions (2.6) and (2.7) can be also w i en o he a iable y in oduced in eq. (2.5a), j0 a (x) + νya (x)=0, ja (x) = −ˆ D(ya (x))y0 a (x), (2.8) wi h ˆ D(y) = dy dρ−1 D(ρ),(2.9) since ja (x, ) = −ˆ D(ya )∂xy(x, ).(2.10) Thus ˆ Dis an “e ec i e” di usi i y: i is he ac o mul i- plying he spa ial g adien when w i ing Fou ie ’s equa- ion in e ms o he new a iable y. Equa ion (2.8) can also be summa ized in a second o de di e en ial equa- ion o ya , hˆ D(ya )y0 a i0=νya , ya (±1/2) = R(T).(2.11) In e es ingly, i can be shown (see below) ha ˆ Dis con- s an , independen o y, whene e y=R(ρ) depends al- geb aically in ρ, a case we will s udy in de ail in sec ion V. This obse a ion conside ably simpli ies he subsequen analysis. The p obabili y o obse ing a his o y {ρ(x, ), j(x, )}τ 0 o du a ion τ o he densi y and cu en ields, s a ing om a gi en ini ial s a e, can be w i en now as a pa h in eg al o e all he possible ealiza ions o he cu en noise {ξ(x, )}τ 0, weigh ed by i s gaussian measu e, and es ic ed o hose ealiza ions compa ible wi h eq. (2.1) a e e y poin o space and ime [12]. This p obabili y hence obeys a la ge de ia ion p inciple o he o m [4, 7– 13, 25] P({ρ, j}τ 0)∼exp (+LIτ[ρ, j]) ,(2.12) wi h a a e unc ional [7, 9] Iτ[ρ, j] = −Zτ 0 d Z1/2 −1/2 dx [j+D(ρ)∂xρ]2 2σ(ρ)(2.13) wi h ρ(x, ) and j(x, ) coupled ia he balance equa- ion (2.1), and he dissipa ion d(x, ) gi en in e ms o ρ(x, ) by (2.4). Equa ion (2.13) exp esses he gaussian na u e o he local cu en luc ua ions a ound i s a e - age (Fou ie ’s law) beha io . The unc ional in (2.13) is he same as in he conse a i e case ( ha is, wi h no bulk dissipa ion), due o he quasi-elas ici y o he mic o- scopic dynamics, which makes he cu en noise be he FIG. 1. (Colo online) Con e gence o he space& ime- in eg a ed dissipa ion o i s ensemble alue o many di e en ealiza ions, and ske ch o he p obabili y concen a ion as ime inc eases, associa ed wi h he la ge de ia ion p inciple, eq. (2.15). only ele an one in he hyd odynamic desc ip ion, see discussion in sec ion IV [39]. We ocus now on he luc- ua ions o he dissipa ed ene gy, in eg a ed o e space and ime d=−1 τZτ 0 d Z1/2 −1/2 dx d(x, ) (2.14) =ν τZτ 0 d Z1/2 −1/2 dx R(ρ(x, )) >0, whe e we ha e in oduced a minus sign o he sake o con enience, in o de o make dposi i e. As discussed abo e, his is a undamen al obse able o unde s and he s a is ical physics o d i en dissipa i e media. The p obabili y o such a luc ua ion Pτ(d) scales in he long- ime limi as Pτ(d)∼exp [+τL G(d)] , G(d) = 1 τmax ρ,j Iτ[ρ, j]. (2.15) This de ines a new la ge de ia ion p inciple o d, see Fig. 1, such ha G(d) is ob ained om Iτ[ρ, j] ia a saddle- poin calcula ion o long imes ( ha is, i ollows om he con ac ion o he o iginal a e unc ion Iτ[10]). The op imal ields ρ0(x, ;d), j0(x, ;d) which a e he solu ion o he a ia ional p oblem (2.15) mus be consis en wi h he p esc ibed alue o he dissipa ed ene gy din (2.14), and a e also ela ed by he balance equa ion (2.1), sup- plemen ed wi h (2.4) and he app op ia e bounda y con- di ions. These op imal ields can be in e p e ed as he ones adop ed by he sys em o sus ain a long- ime luc- ua ion o he space& ime-in eg a ed dissipa ion d. Fo he sake o simplici y, we ha e no explici ly in oduced in ou no a ion he pa ame ic dependence o he LDF G(d) and he associa ed op imal p o iles on he bound- a y empe a u e T, hough his should be bo ne in mind o la e e e ence. 5 A. The cons ained a ia ional p oblem We now assume ha hese op imal p o iles do no de- pend on ime. In conse a i e sys ems, his conjec u e has been shown [7] o be equi alen o he addi i i y p in- ciple ecen ly in oduced o s udy cu en luc ua ions in di usi e media [8]. The alidi y o his addi i i y sce- na io has been ecen ly con i med in ex ensi e nume ical simula ions o a b oad in e al o luc ua ions [11, 14], hough i may e en ually b eak down o ex eme luc u- a ions ia a dynamic phase ansi ion [30, 31]. As we will see below, he applicabili y o his gene aliza ion o he addi i i y conjec u e o dissipa i e sys ems is well sup- po ed by nume ical e idence. Unde his simpli ying hy- po hesis, he luc ua ing balance equa ion (2.1) educes o j0(x) + νy(x)=0, y(x) = −j0(x)/ν, (2.16) making use o he a iable yde ined in eq. (2.5a). Mo e- o e , we can in eg a e o e ime in he de ini ion (2.14) o he in eg a ed dissipa ion d, d=νZ1/2 −1/2 dx y(x),(2.17a) o , equi alen ly, d=−Z1/2 −1/2 dx j0(x) = j(−1/2) −j(1/2) >0.(2.17b) In his way, by using he addi i i y hypo hesis we can elimina e ρ(x) and w i e G(d) in e ms o only one a i- able as G(d) = −min j(x)S[j],wi h S[j] = Z1/2 −1/2 dx L(j, j0j00), (2.18a) L(j, j0, j00) = [j−ˆ D(−j0/ν)j00 ν]2 2ˆσ(−j0/ν),(2.18b) whe e ˆ Dis he e ec i e di usi i y de ined in eq. (2.9), and ˆσis he mobili y, de ined in eq. (2.3), bo h w i en in e ms o y=−j0/ν. The unc ion L(j, j0, j00) is a gene alized Lag angian wi h dependence on i s and also second o de de i a i es, see he appendix. We ha e o ind he op imal cu en p o ile j0(x;d), ha is, he solu ion o he a ia ional p oblem (2.18), wi h he cons ain ha he in eg a ed dissipa ion dhas a de ini e alue, as gi en by (2.17). The e o e we mus use he Lag ange mul iplie p ocedu e [32, 33], ha is, look o an ex emum o Sλ[j] = S[j]−λZ1/2 −1/2 dx (j0+d) (2.19) =Z1/2 −1/2 dx Lλ(j, j0, j00) (2.20) whe e Lλ(j, j0, j00) = L(j, j0, j00)−λ(j0+d),(2.21) wi h λbeing he Lag ange mul iplie . The ex emum o Sλ ollows om wo condi ions: (i) δSλ= 0, and (ii) ∂Sλ/∂λ = 0. The i s condi ion implies d2 dx2∂Lλ ∂j00 −d dx ∂Lλ ∂j0+∂Lλ ∂j = 0,(2.22) which is he Eule -Lag ange equa ion o a Lag angian Lλcon aining second o de de i a i es (see he ap- pendix). Condi ion (ii) leads o he cons ain on he in eg a ed dissipa ion, gi en by Eq. (2.17). The bound- a y condi ions o he Eule -Lag ange equa ion a e j0(±1/2) = −νR(T), pλj(±1/2) = 0.(2.23) We ha e in oduced he gene alized momen um pjcon- juga e o j, o he new Lag angian Lλ, as pλj =∂Lλ ∂j0−d dx ∂Lλ ∂j00 .(2.24) These bounda y condi ions a ise om (1) he alues o he densi y a he bounda ies, which a e p esc ibed, ρ(±1/2) = T, and (2) he condi ion δSλ= 0, which p o- ides he addi ional needed condi ions when he e a e no enough alues o he a iables ixed a he bound- a ies (see he appendix, and also [33]). B. Mapping he cons ain o bounda y condi ions Taking in o accoun he ela ion be ween Lλand L, Eq. (2.21), he gene alized momen um pλj e i ies pλj =pj−λ, (2.25) whe e pjis he gene alized momen um o he Lag angian L, ha is, pj=∂L ∂j0−d dx ∂L ∂j00 .(2.26) Mo eo e , he Eule -Lag ange equa ion (2.22) implies ha d2 dx2∂L ∂j00 −d dx ∂L ∂j0+∂L ∂j = 0,(2.27) ha is, we also ob ain he Eule -Lag ange equa ion co - esponding o he o iginal Lag angian L. The de ini ion o pjin eq. (2.26) gua an ees ha p0 j=∂L/∂j, as in he case o he usual a ia ional p oblem wi h a Lag angian including only i s -o de de i a i es. Now, he bounda y condi ions can be w i en as j0(±1/2) = −νR(T), pj(±1/2) = λ, (2.28) 6 which ollow om Eqs. (2.23) and (2.25). The abo e esul imply ha ou cons ained a ia ional p oblem can be mapped on o a uncons ained a ia ional p ob- lem wi h he o iginal Lag angian L, i s associa ed Eule - Lag ange equa ion (2.27) and he bounda y condi ions (2.28). The unknown alue λ o he gene alized momen- um pja he bounda ies mus be de e mined by impos- ing he p esc ibed alue o he in eg a ed dissipa ion, as gi en by Eq. (2.17), ha is, λ=λ(d). In pa icula , λ= 0 is equi alen o imposing no es ic ions on he in eg a ed dissipa ion, so ha we should eco e (as we will see la e ) he a e age p o iles and dissipa ion in his case. In his sense, a non-ze o alue o λ=pj(±1/2) is a measu e o he depa u e om he a e age hyd odynamic beha iou . On physical g ounds, we expec he co esponding op- imal densi y p o ile o be an e en unc ion o x, because o he symme y o ou sys em a ound he cen e x= 0. In ac , he Eule -Lag ange equa ion (2.27) admi s so- lu ions wi h well-de ined pa i y. Since he Lag angian has he symme y p ope y L(−j, j0,−j00) = L(j, j0, j00), Eq. (2.27) has solu ions wi h jbeing an odd unc ion o x, which implies ha y(and he e o e ρ) is an e en unc ion o x. F om now on, we will es ic ou sel es o hese symme ic solu ions o he a ia ional p oblem. Thus, Eq. (2.17) educes o d= 2j(−1/2; d) = −2j(1/2; d),(2.29) so he bounda y condi ions o he Eule -Lag ange equa- ion boil down o j0(±1/2; d) = −νR(T), j(−1/2; d) = −j(1/2; d) = d/2, (2.30) i.e. much simple han Eq. (2.28). This ollows om pj being an e en unc ion o x o he solu ions wi h well- de ined pa i y we a e conside ing. In his way, symme y conside a ions lead o he simple bounda y condi ions (2.30), which in u n allow us o ge id o he Lag ange mul iplie λ. In summa y, we ha e mapped ou o iginal a ia ional p oblem wi h he subsidia y condi ion ha he dissipa ion has a gi en alue o an uncons ained a i- a ional p oblem, wi h he o iginal Lag angian L(j, j0, j00) and p esc ibed alues o jand j0a he bounda ies. Once he op imal cu en p o ile is ob ained, he op- imal densi y p o ile can be calcula ed om he balance equa ion (2.16). O cou se, he densi y p o ile so ob- ained obeys he bounda y condi ions ρ(±1/2; d) = T. I mus be s essed ha he Eule -Lag ange equa ion (2.27) wi h bounda y condi ions (2.30) gi es he co ec solu ion o he cons ained a ia ional p oblem when he op imal p o iles ha e a well-de ined pa i y. Ne e heless, one canno ule ou he exis ence o symme y-b eaking solu ions wi hou well-de ined pa i y, since in gene al a a ia ional p oblem may ha e mul iple solu ions [32]. In ha case, one mus sol e he mo e complex a ia ional p oblem comp ising he Eule -Lag ange equa ion (2.27) wi h he bounda y condi ions (2.28), whe e he Lag ange mul iplie λ=λ(d) is de e mined by imposing he con- s ain (2.17). We no e howe e ha nume ical e idence s ongly suppo s he alidi y o symme ic solu ions (see below). The LDF G(d) depends on dand T h ough he bound- a y condi ions ( ecall ha i s T-dependence has been omi ed in no a ion o simplici y). The de i a ion o he Eule -Lag ange equa ion (2.27) (see he appendix) shows ha , o he solu ions wi h well-de ined pa i y, δG(d) = pj(1/2)δd + 2pj0(1/2)νdR(T) dT δT, (2.31) whe e pjis he gene alized momen um conjuga e o j, de- ined in Eq. (2.26), and pj0is he gene alized momen um conjuga e o j0, pj0=∂L ∂j00 ,(2.32) which is an odd unc ion o x o he solu ions wi h well- de ined pa i y. In his way Eq. (2.31) o e s a geome ic in e p e a ion o he alues o he gene alized momen a a he bounda ies, as hey a e di ec ly ela ed o he pa - ial de i a i es o he LDF, ∂G ∂d =pj(1/2),∂G ∂T = 2νdR(T) dT pj0(1/2).(2.33) C. A Hamil onian o mula ion o he p oblem We will no w i e he de ailed o m o he gene al ou h-o de di e en ial equa ion (2.27) o he op imal p o ile j(x;d), since i is no pa icula ly illumina ing. Ins ead we now w i e a se o ou equi alen i s o de di e en ial equa ions a ising in he equi alen “Hamil o- nian” desc ip ion. In he ollowing, we ske ch he p oce- du e o in oduce he Hamil onian o a Lag angian wi h highe -o de de i a i es [32, 33], adap ed o he p esen case. As he Eule -Lag ange equa ion is a ou h-o de di e en ial equa ion, we should ha e wo canonical coo - dina es and hei wo co esponding canonical momen a. The i s canonical coo dina e is he cu en j, and we choose he second one o be y, which is p opo ional o j0, as gi en by Eq. (2.16). This choice is sugges ed by he s uc u e o he Lag angian in eq. (2.18). I is wo h ecalling ha he densi y p o ile can be di ec ly ob ained om yby making use o i s de ini ion, Eq. (2.5a). Nex , we in oduce he canonical momen a pyand pjconjuga e o yand j, espec i ely. The momen um pjhas been de- ined in Eq. (2.26), and pyis gi en by py≡ −ν∂L ∂j00 ,(2.34) which ollows om he de ini ion o pj0, Eq. (2.32), and Eq. (2.16) o y. The Hamil onian is hen in oduced in he usual way, H ≡ y0py+j0pj−L ≡ y0py−νypj−L.(2.35) 7 A e some algeb a, we ge H=1 2Q(y)p2 y−ˆ D−1(y)jpy−νypj,(2.36a) Q(y)≡ˆσ(y) ˆ D2(y),(2.36b) whe e we ha e de ined he auxilia y unc ion Q(y), wi h Q(y)>0 o all y. We ha e also made use o eq. (2.16), o he exp ession o pywhich ollows om i s de ini ion (2.34), py=ˆ D(y)j+ˆ D(y)y0 ˆσ(y),(2.37) and o he Lag angian L=ˆσ(y)p2 y 2ˆ D2(y)=1 2Q(y)p2 y,(2.38) w i en in e ms o he canonical a iables, wi h he aid o eq. (2.37). As usual, His a unc ion o (y, j, py, pj), which sa is y he ollowing se o ou ”canonical” i s - o de di e en ial equa ions, y0=∂H ∂py =Q(y)py−ˆ D−1(y)j, (2.39a) j0=∂H ∂pj =−νy, (2.39b) p0 y=−∂H ∂y =−dQ(y) dy p2 y 2+dˆ D−1(y) dy jpy+νpj,(2.39c) p0 j=−∂H ∂j =ˆ D−1(y)py(2.39d) ha a e equi alen o he ou h-o de Eule -Lag ange equa ion (2.27). No e ha , as is usual in physics and in o de no o clu e ou o mulae, we ha e d opped he subindex 0 o he op imal p o iles, which a e now solu- ions o he abo e canonical equa ions; he same no a ion is used o he canonical a iables in he Hamil onian and o he solu ions o Hamil on’s equa ions. On he o he hand, as he Hamil onian does no depend explici ly on x, i is a i s in eg al o he sys em (2.39): H= cons . o e any o i s solu ions. This p ope y may be used o simpli y he in eg a ion o he sys em. In gene al, o a gi en alue o he dissipa ion d, we ha e o sol e he sys em o equa ions (2.39) wi h he bounda y condi ions y(±1/2) = R(T), j(−1/2) = −j(1/2) = d/2,(2.40) which ollow om Eqs. (2.30) and (2.16). Again, we ha e o look o solu ions o Eq. (2.39) wi h well-de ined pa i y, ha is, yand ja e e en and odd unc ions o x, espec i ely (and, he e o e, pyis odd and pje en). The solu ion o hese canonical equa ions is hen inse ed in o he exp ession o he LDF G(d), which can be w i en in e ms o he canonical a iables as G(d) = −Z1/2 −1/2 dx L=−1 2Z1/2 −1/2 dx Q(y)p2 y,(2.41) by combining eqs. (2.18) and (2.38). In his way, we ob ain he LDF o an a bi a y alue o he in eg a ed dissipa ion dwi hin he Hamil onian o mula ion o he a ia ional p oblem. Equa ion (2.41) shows clea ly ha he mos p obable (a e age) p o iles co espond o a so- lu ion wi h py= 0 o all x, o which G(d) anishes. By subs i u ing py= 0 in Eqs. (2.39c)-(2.39d), we also ha e ha pj= 0 o all x. Mo eo e , Eqs. (2.39a) and (2.39b) simpli y o eq. (2.8), ha is, he a e age p o iles a e eob ained. The e o e, he e is always a solu ion o he canonical equa ions (2.39) wi h iden ically anishing canonical momen a, which co esponds o he a e age solu ion o he hyd odynamic equa ion (2.8) [40]. These a e age hyd odynamic p o iles {ρa , ja }lead o he a - e age alue o he in eg a ed dissipa ion da =νZ1/2 −1/2 dx R(ρa ) = νZ1/2 −1/2 dx ya (x) = 2ja (−1/2).(2.42) This discussion is consis en wi h he one below Eq. (2.28), which was done wi hin he amewo k o he equi alen Lag angian desc ip ion. Fluc ua ions in ol e non-ze o alues o he canonical momen a, whose magni- ude is hen a measu e o he depa u e om he a e age beha iou (d−da )/da . III. ANALYSIS OF THE LDF IN SOME LIMITING CASES In he ollowing subsec ions, we u he analyze he o m o he LDF in ce ain limi s o in e es , o which some gene al esul s can be ob ained. Fi s , we ocus on he beha io o G(d) o small luc ua ions a ound he a e age, whe e a quad a ic shape o he LDF is expec ed (co esponding o gaussian luc ua ions). We hen ana- lyze he limi o weakly-dissipa i e sys ems, ν1, o which an adequa e pe u ba i e expansion allows us o ob ain a non- i ial and in e es ing scaling o m o he LDF. Finally, we conside he opposi e limi o s ongly- dissipa i e sys ems, ν1, o which a di e en scaling o he LDF is ound. A. Small luc ua ions a ound he a e age As he a e age beha io co esponds o he pa icula solu ion o he canonical equa ions co esponding o an- ishing momen a pj= 0, pρ= 0, small luc ua ions can be 8 hus analyzed by assuming ha he canonical momen a a e small. Le us de ine he dimensionles pa ame e =d−da da (3.1) o measu e he sepa a ion om he a e age in eg a ed dissipa ion da . As we ha e jus discussed, he canonical momen a anish o = 0. We w i e y=ya +∆y, j =ja +∆j, py=∆py, pj=∆pj, (3.2) and linea ize Eqs. (2.39) a ound he a e age solu ion, ha is, we only e ain e ms linea in . Then, ∆y0=Q(ya )∆py−ˆ D−1(ya )∆j−ja dˆ D−1(ya ) dya ∆y, (3.3a) ∆j0=−ν∆y, (3.3b) ∆p0 y=ja dˆ D−1(ya ) dya ∆py+ν∆pj,(3.3c) ∆p0 j=ˆ D−1(ya )∆py.(3.3d) The bounda y condi ions o hese equa ions a e ∆y(±1/2) = 0, ∆j(−1/2) = −∆j(1/2) = da /2. The solu ion o his sys em o equa ions mus be inse ed in he la ge de ia ion unc ion (2.18). Using he exp ession (2.41) o he LDF, i ollows ha G(d)∼ −2 2Z1/2 −1/2 dx Q(ya )∆p2 y,(3.4) o small luc ua ions o he dissipa ion a ound he a - e age. Taking in o accoun (3.1) and he la ge de ia ion p inciple (2.15), eq. (3.4) means ha he p obabili y o such small luc ua ions o he in eg a ed dissipa ion dis app oxima ely gaussian, Pτ(d)||1 ∝exp −Lτ (d−da )2 2d2 a Λ2 ν(3.5) wi h Λ2 νgi en by Λ2 ν= Z1/2 −1/2 dx Q(ya )∆p2 y!−1 .(3.6) In his way, he gaussian es ima ion o he s anda d de- ia ion o he dissipa ion, by compa ing (3.5) o (3.6), is gi en by χ≡da Λν/√τL. In o de o make a mo e de- ailed s udy o he LDF, conc e e unc ional dependences o he di usi i y D, he mobili y σand he dissipa ion R on he densi y ρmus be conside ed. This is done in he ollowing sec ions o he pape , whe e we will conside a b oad amily o models o which he anspo coe - icien s can be explici ly ob ained. On he o he hand, i is impo an o no ice ha gaussian s a is ics is only expec ed o small luc ua ions a ound he a e age dissi- pa ion. In gene al, he solu ion o he a ia ional p oblem gi en by he in eg a ion o eq.(2.39), when inse ed in o (2.41), will gi e ise o non-gaussian s a is ics ( ha is, a non-quad a ic dependence o he LDF) o an a bi a y luc ua ion o he dissipa ed ene gy d. B. Weakly-dissipa i e sys ems, ν1 We p oceed now by analysing he canonical equa ions (2.39) in he limi ν1. Unsu p isingly, a egula pe - u ba ion expansion in powe s o νb eaks down, since i is no possible o impose he necessa y bounda y con- di ions o he cu en . This singula i y o he elas ic limi was o be expec ed on a physical basis, as i is no possible o ob ain he beha io o weakly dissipa i e sys- ems (ν1) as a co ec ion a ound he conse a i e case ν= 0, o which ρ(x) = Tand j(x) = 0. The e o e, a singula pe u ba ion analysis should be done, looking o a sui able escaling o he a iables o ν1. Equa- ion (2.8) o he a e ages implies ha ya =R(T) + O(ν), ja =−νR(T)x+O(ν2), ν 1. (3.7) The a e age cu en anishes linea ly in νin he limi ν→0+, as expec ed. Mo eo e , he a e age dissipa ion, ob ained by combining eqs. (2.42) and (3.7), is gi en by da =νR(T) + O(ν2).(3.8) The e o e, i is sensible o p opose he ollowing escaling o a iables j(x) = νψ(x), pj(x) = Πψ(x) ν,(3.9) which is consis en wi h he canonical equa ions (2.39), since ψ0=1 νj0=1 ν ∂H ∂pj =∂H ∂Πψ ,(3.10a) Π0 ψ=νp0 j=−ν∂H ∂j =−∂H ∂ψ ,(3.10b) wi h he same Hamil onian H. In o he wo ds, eq. (3.9) de ines a “canonical ans o ma ion” om he pai o canonical conjuga e a iables {j, pj} o {ψ, Πψ}, a ans- o ma ion ha heals he singula beha io in he ν→0+ limi . The Hamil onian can be now w i en as H=1 2Q(y)p2 y−yΠψ−νˆ D(y)−1ψpy(3.11) in he escaled a iables. No ice ha he ans o ma ion in oduced is essen ial o ob ain he co ec “dominan balance” [42] o he lowes o de . In pa icula , be o e he escaling, he e m p opo ional o ypjwas o he o de o νand he e m p opo ional o jpywas o he o de o uni y; a e he escaling he o de s o magni ude 9 a e in e changed, he e m p opo ional o yΠψis o he o de o uni y while he e m p opo ional o ψpyis o he o de o ν. We now s a om he ze o- h o de escaled Hamil onian by pu ing ν= 0 in eq. (3.11), H0=1 2Q(y)p2 y−yΠψ.(3.12) om which we we a i e a y0=∂H0 ∂py =Q(y)py,(3.13a) p0 y=−∂H0 ∂y =−1 2 dQ(y) dy p2 y+ Πψ,(3.13b) ψ0=∂H0 ∂Πψ =−y , (3.13c) Π0 ψ=−∂H0 ∂ψ = 0 .(3.13d) In o de no o clu e ou o mulas, we do no in oduce a di e en no a ion o he canonical a iables, al hough he app oxima e canonical equa ions (wi h H0) a e di - e en om he exac ones (wi h H). We ha e only o emembe ha ou esul s a e alid o he lowes o de in ν. The canonical equa ions (3.13) ha e o be sol ed wi h he bounda y condi ions y(±1/2) = R(T), ψ(−1/2) = −ψ(1/2) = ∆/2, (3.14) whe e ∆ = d ν=R(T)d da (3.15) is assumed o be o he o de o uni y, ha is, d=O(ν) o d/da =O(1). Thus, ou escaling allows us o ob ain a solu ion o he op imal p o iles o he densi y ρ, by in e ing he ela ion y=R(ρ), and he cu en j= νψ, o in eg a ed dissipa ions d e y di e en om i s a e age alue da . I is wo h no icing ha ψis a cyclic a iable and i s conjuga e momen um is hus cons an , Πψ≡Πψ0= cons ., see Eq. (3.14); his ac allows us o ob ain a closed i s o de di e en ial equa ion o y(x) in he ν1 limi . Mo eo e , by ecalling Eq. (2.33), we ha e ha Πψ0=∂G ∂∆,(3.16) which gi es he physical in e p e a ion o his i s in- eg al o he app oxima e canonical equa ions: i is he pa ial de i a i e o he LDF wi h espec o he escaled dissipa ion. The Hamil onian H0is also cons an , since i does no depend explici ly on x, and combining (3.12) and (3.13a), y02= 2Q(y)(H0+yΠψ0), y(±1/2) = R(T).(3.17) Once his is sol ed, he escaled cu en ψcan be ob- ained om (3.13c) ψ0=−y, ψ(−1/2) = −ψ(1/2) = R(T)d 2da ,(3.18) so ha he wo cons an s H0and Πψ0will be gi en in e ms o he empe a u e Tand d/da . The e a e no mo e cons an s o be adjus ed in he solu ion o eqs. (3.17) and (3.18) due o he pa i y p ope ies o (y, ψ): yis and e en unc ion o xand ψis an odd unc ion o xin he in e al [−1/2,1/2]. O cou se, he a e age p o iles ya (x) and ja (x) a e eob ained om he canonical equa ions by pu ing Πψ= 0 and py= 0 he ein. Equa ion (3.12) implies ha H0= 0 o e he a e age p o iles. The simple o m o he di e en ial equa ion (3.17) al- lows us o in e some o he p ope ies o he op imal p o ile y(x) associa ed o a gi en dissipa ion luc ua- ion in he limi o weakly-dissipa i e sys ems. Fi s , no ice ha in gene al he solu ion o eq. (3.17) will be non-mono onic, exhibi ing ex ema in he in e al x∈[−1 2,1 2]. Mo eo e , aking in o accoun ha he unc- ion Q(y) is posi i e de ined, i ollows ha he p o ile a he ex ema will ake an unique alue y0≡ − H0 Πψ0 (3.19) No e ha Πψ06= 0 o d6=da and, mo eo e , i mus ha e a di e en sign ha H0, i.e. sgn(Πψ0)6= sgn(H0), since y(x)>0∀x. The e o e, he op imal p o ile y(x) can only ha e a single ex emum (minimum o maxi- mum) [41], ha is loca ed a x= 0 because o symme y easons. By ew i ing eq. (3.17) as y02= 2Q(y)H01−y y0(3.20) we conclude ha he cons an H0and y0−y(x) mus ha e he same sign ∀x∈[−1 2,1 2]. Thus, o H0>0 he p o ile y(x) has a single maximum, y(x)> y(±1/2) = R(T)∀x, and hus d>da . On he o he hand, H0<0 implies a single minimum, y(x)< R(T)∀xand d < da . All hese p ope ies a e con i med below o pa icula examples, bo h analy ically and nume ically. In e es ingly, he leading beha io o he LDF can be also easily ob ained in e ms o he i s in eg als H0and Πψ0. In ac G(d)∼ −1 2Z1/2 −1/2 dx Q(y)p2 y=−1 2Z1/2 −1/2 dx y02 Q(y), (3.21) and making use o eq. (3.17), G(d)∼ − H0+ Πψ0Z1/2 −1/2 dx y(x)! =−(H0+ Πψ0∆) = −H01−∆ y0.(3.22) Thus, he emaining ask consis s in w i ing he con- s an s H0and y0(o equi alen ly H0and Πψ0) in e ms o he in eg a ed dissipa ion dand he empe a u e a he bounda ies T. Once his is done, he LDF ollows om he simple exp ession gi en by eq. (3.22). Fu he - mo e, we may ob ain bounds o he p o ile ex emum y0 16 ∆j0=−ν∆y, (5.19b) ∆p0 y=ν∆pj,(5.19c) ∆p0 j=1 ˆ D∆py,(5.19d) which pa icula izes eq. (3.3) o ou amily o mod- els. The bounda y condi ions a e ∆y(±1/2) = 0, ∆j(−1/2) = −∆j(1/2) = da /2. The solu ion o his sys em mus be inse ed in o eq. (3.6), which gi es he a iance o he gaussian dis ibu ion, χ2≡d2 a Λ2 ν/Lτ. The canonical momen um ∆pyis di ec ly ob ained by in eg a ing eqs. (5.19c) and (5.19d), ∆py=Ksinh x ν ˆ D,(5.20) since ∆pymus be an odd unc ion o xas a consequence o ybeing e en. The cons an Kis o be de e mined wi h he aid o he bounda y condi ions, bu his can only be done a e sol ing eqs. (5.19a) and (5.19b), ha ing p e iously inse ed (5.20) in o hem. Subs i u ion o eq. (5.20) in o eq. (3.6) gi es Λ2 ν=K2 Z1/2 −1/2 dx Q(ya ) sinh2x ν ˆ D!−1 .(5.21) In o de o e alua e he in eg al, eq. (5.14) o Q(y) mus be used, Q(y)∝yγ, wi h he pa ame e γbeing a unc ion o β, 1 < γ ≤2. We now analyze he simples choice β= 0, ha is, γ= 2, ha co esponds o he dissipa i e e sion o he KMP model in oduced in [25]. In his case, he calcula ion is s aigh o wa d and yields Λ2 ν=sinh(2√2ν)−2√2ν 4√2νsinh2(√2ν).(5.22) In e es ingly, Λ2 ν∼1/3 independen o νin he limi o weakly-dissipa i e sys ems ν1. This can be unde - s ood as a eminiscence o he scaling o G(d) de i ed in sec ion III. In ac , eq. (3.34) ells us ha , o γ= 2, G(d) is jus a unc ion o d/da . In he gaussian ap- p oxima ion, his implies he con e gence o Λ2 ν o a con- s an alue in he quasi-elas ic limi as ν→0+. On he o he hand, Λ2 ν∼(2√2ν)−1 o ν1, which is con- sis en wi h he supp ession o dissipa ion luc ua ions p e iously ound in he s ongly inelas ic egime, as ex- p essed by he gene al scaling o he LDF gi en by Eq. (3.48). The same quali a i e obse a ions apply o o he alues o β, hough he calcula ion is mo e con olu ed. We ha e es ed he abo e p edic ions in s anda d Mon e Ca lo simula ions o he dissipa i e KMP model desc ibed in his sec ion, o he pa icula case β= 0. Figu e 3 shows he p obabili y densi y unc ion (pd ) o he dissipa ed ene gy, in eg a ed o e he whole sys em and o e a long ime τ o many di e en alues o he 10-3 10-2 10-1 100101102103 ν 10-3 10-2 10-1 100 101 102 da , Λν 2 ~ν1/2 ~ν-1/2 Λν 2~1/3 da ~ν FIG. 4. (Colo online) Measu ed a e age dissipa ion and i s a iance as a unc ion o ν o β= 0. The solid line co e- sponds o he heo e ical p edic ion o da , eq. (5.18), while he dashed line is he gaussian es ima ion o he dissipa ion a iance pa ame e , Λ2 ν, see eq. (5.22). The ag eemen is excellen in all cases. No ice in pa icula he scaling wi h νo bo h obse ables in he weakly- and s ongly-dissipa i e sys em limi s. mac oscopic dissipa ion coe icien ν∈[10−3,103]. In o - de o minimize ini e-size e ec s in he measu emen s, we pe o med simula ions o sys ems wi h inc easing size as νg ows, L∝`−1 ν, in such a way ha he numbe o la ice si es pe uni ypical leng h is cons an and la ge enough so we a e wi hin he hyd odynamic egime. Fu - he mo e, he in eg a ion ime τ=O(1) o he con in- uous, di usi e, imescale o e which he hyd odynamic p edic ions should hold [45]. S anda d Mon e Ca lo sim- ula ions do no allow us o sample he ails o he dis i- bu ion, bu hey a e use ul o s udy he ypical luc ua- ions a ound he a e age we a e in e es ed in he e (e.g., a egime o 5 s anda d de ia ions a ound he a e age). Figu e 3 shows ha , when plo ed agains he educed a iable z≡(d−da )/χ, he dis ibu ion Pτ(z) ollows app oxima ely a no mal dis ibu ion o ypical luc u- a ions. Mo eo e , all cu es o di e en νcollapse in his egime. Howe e , e en a his s anda d simula ion le el, i becomes appa en ha he ails o he dis ibu- ion (co esponding o mode a e dissipa ion luc ua ions) de ia e om gaussian beha io , see inse in Fig. 3, show- ing asymme ic ails and b eaking he collapse o gaus- sian beha io obse ed o small luc ua ions. As we will show below, he analysis o he dissipa ion LDF shows ha he la ge luc ua ions s a is ics is a om gaussian. In o de o u he check ou heo y, we ha e also com- pa ed he measu ed a e age dissipa ion and i s a iance wi h he analy ical esul s abo e, as a unc ion o he mac oscopic dissipa ion coe icien ν, a ying in a ange which co e s 6 o de s o magni ude. Again, we see in Fig. 4 ha he ag eemen is excellen in all cases. In pa - 17 icula , he a e age dissipa ion g ows as ν( esp. ν1/2) in he weakly ( esp. s ongly) dissipa i e sys em limi , while he a iance emains cons an o ν1 bu de- cays as ν−1/2 o ν1. Rema kably, he gaussian ap- p oxima ion o he a iance u ns ou o be an excel- len es ima o o he empi ical dissipa ion a iance. Fo Lτ 1, la ge luc ua ions o he dissipa ion a e e y a e and mos o he p obabili y concen a es in a egion o wid h p opo ional o (Lτ)−1/2a ound he a e age alue, a egime desc ibed by he gaussian app oxima ion. B. Comple e luc ua ion spec um o he in eg a ed dissipa ion We now in es iga e he whole spec um o luc ua- ions (bo h ypical and a e) o he in eg a ed dissi- pa ion. Thus, we need o e alua e he LDF G(d) o a bi a y alues o d, in gene al no close o i s a e - age alue da , bo h analy ically and nume ically. Ex- plo ing in s anda d simula ions he ails o he dissipa- ion dis ibu ion associa ed o he non i ial s uc u e o G(d) is an daun ing ask, since LDFs in ol e by de ini- ion exponen ially-unlikely a e e en s, see eq. (2.12). This has been co obo a ed in Fig. 3, whe e he dissi- pa ion dis ibu ion has been measu ed di ec ly bu we a e unable o ga he enough s a is ics in he ails o he pd o ob ain clea -cu esul s in he non-gaussian egime. A ecen se ies o wo ks ha e add essed his is- sue, de eloping an e icien me hod o measu e di ec ly LDFs in many pa icle sys ems [27–29]. The me hod is based on a modi ica ion o he dynamics so ha he a e e en s esponsible o he la ge de ia ion a e no longe a e [27], and i has been de eloped o disc e e- [27] and con inuous- ime Ma ko dynamics [28]. Fo a e- cen e iew, which also discusses Hamil onian sys ems, see e . [29]. The me hod yields he Legend e-Fenchel ans o m o he dissipa ion LDF, which is usually de- ined as µ(s) = maxd[G(d) + sd] [10, 46]. In pa icula , i UC0Cis he ansi ion a e om con igu a ion C o C0o he associa ed s ochas ic p ocess, he modi ied dy- namics is de ined as ˜ UC0C(s) = UC0Cexp(s dC0C), whe e dC0Cis he ene gy dissipa ed in he elemen a y ansi ion C→C0. I can be hen shown [11, 27–29] ha he na - u al loga i hm o he la ges eigen alue o ma ix ˜ U(s) gi es µ(s), which in u n can be Legend e- ans o med back o ob ain a Mon e Ca lo es ima e o G(d). The me hod o e s. [27–29] hus p o ides a way o measu e µ(s) by e ol ing a la ge numbe Mo copies o clones o he sys em using he modi ied dynamics ˜ U(s). This me hod is exac in he limi M→ ∞, bu in p ac ice we a e able o simula e a la ge bu ini e popula ion o clones, ypically M∈[103,104]. This in oduces addi- ional ini e-size e ec s ela ed o he popula ion o clones which mus be conside ed wi h ca e, see [13] o u he discussion along his line. The nume ical esul s o he LDF in he ollowing sec ions ha e been ob ained using hese ad anced Mon e Ca lo echniques. 1. Weakly-dissipa i e sys ems, ν1 We now ocus ou a en ion on he analysis o LDF o he in eg a ed dissipa ion o weakly dissipa i e sys ems, in which ν1. In he gene al amewo k de eloped in sec ion II, we ound a scaling p ope y o G(d), as gi en by eq. (3.34), da νγ−2 G(d) = −"1−d/da Y0(e H)#e H,(5.23) whe e γ= (2 + β)/(1 + β), Y0(e H) is de e mined by Eq. (3.29), and he cons an e Hdepends only on he a io d/da , as gi en by eq. (3.31). Fo he sake o conc e eness, le us conside now he simples case β= 0, co esponding o he dissipa i e KMP model in oduced in [25]. Equa ion (3.26) o he escaled densi y p o ile now eads Y0(x)2= 8 e HY21−Y Y0, Y (±1/2) = 1,(5.24) which can be explici ly in eg a ed, wi h he solu ion Y(x, e H) = Y0sech2(xp2e H), Y0= cosh2se H 2,(5.25) whe e we ha e al eady used ha Y(x) mus be an e en unc ion o x. The escaled cu en p o ile Ψ(x) in o- duced in (3.27) is Ψ(x, e H) = −cosh2qe H 2 p2e H anh(xp2e H).(5.26) The op imal p o iles o he densi y and he cu en can be now eadily w i en by combining he p e ious wo equa ions wi h Eqs. (3.35)-(3.36), yielding ρ(x) = TY (x) = Tcosh2se H 2sech2(xp2e H),(5.27a) j(x) = da Ψ(x) = −da cosh2qe H 2 p2e H anh(xp2e H) (5.27b) in e ms o e H=e H(d). We ha e aken in o accoun ha y≡ρ o β= 0. No e ha he cu es ρ(x)/T =Y(x, e H) o di e en alues o νplo ed as a unc ion o xonly depend on he ela i e dissipa ion d/da . Now, eq. (3.31) implies ha d da = 2Ψ(−1/2) = sinh p2e H p2e H ,(da ∼νT ),(5.28) which gi es he cons an e Himplici ly in e ms o d/da . Finally, pa icula izing eq. (5.23) o he case γ= 2 we 18 0123456 d/da -8 -6 -4 -2 0 G(d) ν=0.01 ν=0.1 0123456 d/da -4 -3 -2 -1 0 G(d) β=1 β=0.5 β=0 β=1 β=0.5 β=0 FIG. 5. (Colo online) Scaling o he dissipa ion LDF in he quasi-elas ic limi (ν1) o N= 50, T= 1 and a ying β o wo di e en alues o ν, namely ν= 0.01 ( illed symbols) and ν= 0.1 (open symbols). The solid lines a e he MFT p edic ions in each case. Cu es ha e been shi ed e ically o con enience, G(da ) = 0, ∀ν, β. Fo he case β= 0, he simula ion cu es a e plo ed o d<dI, wi h dIbeing he in lec ion poin a which G(d) changes con exi y in he limi ν1 (see he ex and also Fig. 6). Inse : Compa ison o he heo e ical G(d) o di e en β, whe e i is clea ha inc easing β a o s la ge dissipa ion luc ua ions. a e analyzing ( ha is, β= 0), we ob ain G(d) = p2e H anh se H 2−e H.(5.29) No e ha eq. (5.28) o e H(d) equi es some ca e ul anal- ysis. F om he gene al discussion in Sec ion III B, we ha e ha e H>0 o d > da , and hus p2e His a eal numbe , while e H<0 o d<da and p2e His imagina y. The la e case poses no p oblem o G(d), which is al- ways eal- alued. In ac , i we w i e pe H=iq|e H| we a i e a G(d) = −q2|e H| an q| e H| 2+|e H| o e H<0. In he limi as d→0, we ha e ha e H → −π2/2, and hus G(d)→ −∞ as expec ed on a physical basis. Equa ion (5.29) gi es a simple scaling o m o G(d), independen o ν, o he linea (β= 0) dissipa i e KMP model in he low-dissipa ion limi ν1 [25]. As an ici- pa ed by (5.23), he cu e o G(d) s he ela i e dissipa- ion d/da is independen o νin his quasi-elas ic egime. This scaling is ully con i med in Fig. 5, in which we plo G(d) o di e en , small alues o ν∈[10−2,10−1] mea- su ed in simula ions o he dissipa i e KMP model using he ad anced Mon e Ca lo echnique desc ibed a he be- ginning o his subsec ion. In pa icula , he ag eemen be ween heo y and simula ions is excellen in he b oad luc ua ion egime ha we could measu e (see below). The dissipa ion LDF is highly skewed wi h a as dec ease o luc ua ions d<da and no nega i e b anch, so luc u- -100 -80 -60 -40 -20 0 s da -25 -20 -15 -10 -5 0 5 µ(s) ν=0.01 ν=0.1 0 0.5 1 s da 0 0.5 1 µ(s) 0 2 4 6 8 10 d/da -1 -0.5 0 G(n)(d) G G' G'' FIG. 6. (Colo online) Scaling plo o he Legend e ans o m o he dissipa ion LDF, µ(s) = maxd[G(d) + sd], in he quasi- elas ic limi ν1 o N= 50, T= 1, β= 0 and wo di e en alues o ν, namely ν= 0.01 (ci cles) and ν= 0.1 ( iangles). The solid line is he MFT p edic ion, see eq. (5.34). No ice ha µ(s) is de ined up o a h eshold alue sI= 0.878458/da , beyond which he Legend e-Fenchel ans o m di e ges. The op- igh inse shows a zoom a ound he h eshold sI. This is ela ed o he exis ence o an in lec ion poin in G(d) o dI= 2.27672da , i.e. a poin a which G00(dI) = 0, see middle- le inse , beyond which he dissipa ion LDF is non-con ex, see discussion in main ex . a ion heo em- ype ela ions linking he p obabili ies o a gi en in eg a ed dissipa ion dand he in e se e en −ddo no hold [5, 6]. This was o cou se expec ed om he lack o mic o e e sibili y, a basic ene o he luc ua ion he- o em o apply [36]. The limi e H  1 co esponds o la ge dissipa ion luc ua ions, whe e G(d)≈ −1 2[ln(d/da )]2, ha is, a e y slow decay which shows ha such la ge luc ua ions a e a mo e p obable han expec ed wi hin gaussian s a is ics (∼ −3 2(d/da )2). In ac , such slow decay implies he p esence o an in lec ion poin in G(d): he e is a alue dIsuch ha G00(dI) = 0. The con ex- i y o G(d) changes a d=dI,G00(d)<0 o d<dI while G00(d)>0 o d > dI. Speci ically, Eqs. (5.28) and (5.29) imply ha dI/da = 2.27672 (see middle- le inse in Fig. 6). The comple e measu emen o non-con ex LDFs in compu e simula ions is a challenge which emains unsol ed. The eason is ha he ad anced Mon e Ca lo me hod desc ibed abo e o di ec ly mea- su e la ge-de ia ion unc ions in simula ions is based on he Legend e-Fenchel ans o m o he LDF o in e es , which is no well-beha ed in egimes whe e he LDF is non-con ex [10]. To be e unde s and his issue, ecall ha he Legend e-Fenchel ans o m o he dissipa ion LDF can be w i en as µ(s) = max d[G(d) + sd] = G[d∗(s)] + s d∗(s),(5.30) 19 whe e d∗(s) is solu ion o he equa ion ∂G(d) ∂d =−s . (5.31) No e ha , ma hema ically, µ(s) is he Legend e-Fenchel ans o m o −G(d), because he Legend e-Fenchel ans- o m is de ined o con ex unc ions [10]. The pa ial de i a i e o Gwi h espec o dis ela ed o he i s in- eg al o Hamil on equa ions Πψ0, see Eq. (3.16), which in u n can be ob ained om eqs. (3.37) and (5.25), yielding Πψ0=ν∂G ∂d =−e H Tsech2se H 2.(5.32) Equi alen ly, s=−∂G ∂d =e H νT sech2se H 2.(5.33) In his way, making use o Eqs. (5.28), (5.29) and (5.33), he Legend e ans o m o he dissipa ion LDF can be w i en as µ(s) = 2p2e H anh se H 2−e H,(5.34) in e ms o e H, which is ob ained implici ly as a unc ion o s om Eq. (5.33). No e ha he scaling o G(d) wi h d/da , see Eq. (5.23), implies a simila collapse o µ(s) when plo ed as a unc ion o s da . Eq. (5.31) has a single solu ion d∗(s) o s < 0 and hence poses no p oblem. On he o he hand, due o he exis ence o an in lec ion poin , G0(d) exhibi s a minimum a dI, inc easingly smoo hly a e wa d o each asymp o icaly ze o in he limi d→ ∞, see middle-le inse in Fig. 6. The e o e, o s > 0 he e exis wo solu ions d∗ 1(s)≤ dI≤d∗ 2(s) o eq. (5.31), bu only he i s one maximizes eq. (5.30). This means ha we canno ob ain G(d) by in e se Legend e- ans o ming µ(s) o dissipa ions abo e he in lec ion poin dI= 2.27672 da . In ac , µ(s) is de ined up o a c i ical sI, such ha sI= 0.87845/da ( he slope o −G(d) a he in lec ion poin ), beyond which µ(s) di e ges. This can be seen by no icing he main p ope ies o µ(s), namely ∂µ ∂s =d , ∂2µ ∂s2=−∂2G ∂d2−1 .(5.35) The e o e, µhas a singula i y a he alue o he slope sI co esponding o he in lec ion poin dI, whe e ∂2µ/∂s2 di e ges. The ansi ion o non-con ex beha io hus implies ha we can only measu e he s a is ics o a e dissipa ion luc ua ions up o dIusing he cloning algo- i hm [27–29]. Fig. 6 shows a compa ison be ween he measu ed µ(s) o wo di e en alues o ν1 and he heo e ical expec a ion, up o he c i ical sI. The ag ee- men is excellen in all cases, and he collapse o µ(s) -0.4 -0.2 0 0.2 0.4 x 0 1 2 3 ρ0(x;d) d/da =0.30 d/da =0.37 d/da =0.46 d/da =0.54 d/da =0.78 d/da =1.01 d/da =1.20 d/da =1.46 d/da =1.73 d/da =1.97 d/da =2.22 FIG. 7. (Colo online) Top: Op imal ene gy p o iles o a y- ing d/da and β= 0, measu ed o ν= 10−3(symbols) and ν= 10−2(dashed lines), and MFT p edic ions (solid lines). Ag eemen is e y good in all cases. Bo om: MFT p edic ion o he op imal densi y p o iles o a ying d/da . when plo ed agains s da is con i med. The challenge emains o de ise compu a ional echniques capable o explo ing a e-e en s a is ics e en in egimes whe e he associa ed LDF is non-con ex. We may sol e in a simila way he MFT o he in- eg a ed dissipa ion o a bi a y alues o he exponen β, hough ma hema ical exp essions a e a mo e con o- lu ed ha in he illus a i e case β= 0 desc ibed abo e. Fig. 5 also shows he dissipa ion LDF o o he exponen s β > 0, as well as he esul s o nume ical expe imen s in hese cases. Quali a i ely, he esul s a e equi alen o hose discussed abo e, wi h a ν-independen scaling o m o he LDF in he ν1 limi which goes apidly o ze o as d→0 and has a ela i ely a ail o dda . This ail changes con exi y (based on a nume ical analysis) a a la ge dissipa ion dIwhich inc eases wi h (a) ν o ixed β(b) β o ixed ν. Fo β=0, we ha e dI/da ≃2.8 o ν= 1, while dI/da >6 o ν= 10. On he o he hand, o β= 1, G00(d)<0 in he conside ed egion and no in lec ion poin he ein. Fo he in e media e alue, β= 0.5, he posi i e alues o G00(d) a e so small ha 20 FIG. 8. (Colo online) Collapse o he op imal ene gy p o iles measu ed o N= 50 and T= 1 as a unc ion o he ela i e dissipa ion d/da o di e en alues o ν1, namely ν= 10−2(do ed lines) and ν= 10−1(dashed lines), o β= 0 ( op, g een), β= 0.5 (middle, ed) and β= 1 (bo om, blue). The la ge β, he less p onounced he cen al o e shoo is o d > da . Solid lines co espond o MFT p edic ions. we ha e chosen no o elimina e he poin s behind he nume ical in lec ion poin , dI/da ≃3.2 o ν1 and dI/da ≃5.1 o ν= 1, al hough hey oughly coincide wi h he alues a which he heo e ical and he simula- ion cu es begin o sepa a e. Fu he mo e, compa ison wi h nume ical esul s is excellen in all cases. In e es - ingly, see inse in Fig. 5, inc easing β esul s in a b oade dissipa ion LDF, meaning ha la ge dissipa ion luc ua- ions a e enhanced as βg ows away om he linea case β= 0. We ha e also measu ed he ypical ene gy p o ile as- socia ed o a gi en dissipa ion luc ua ion o he case β= 0, see op panel in Fig. 7, inding also e y good ag eemen wi h he mac oscopic luc ua ing heo y de- eloped in his pape . Rema kably, op imal p o iles o a ying ν1 also collapse o cons an d/da (all he simula ions ha e been done wi h he same alue o he ene gy densi y a he bounda ies T= 1), as p edic ed by eq. (5.27a). Fu he mo e, p o iles exhibi he x↔ −x symme y conjec u ed in sec ion II B in all cases, wi h a single ex emum which can be minimum o maximum de- pending on he alue o he ela i e dissipa ion d/da , a p ope y which was deduced om he gene al o malism in sec ion III B. In e es ingly, p o iles associa ed o dissi- pa ion luc ua ions abo e he a e age exhibi an ene gy o e shoo in he bulk. This obse a ion sugges s ha he mechanism esponsible o la ge dissipa ion luc ua- ions consis s in a con inued o e -injec ion o ene gy om he bounda y ba h, which is anspo ed o and s o ed in he bulk be o e being dissipa ed. The same quali- a i e obse a ions and good ag eemen be ween heo y and simula ions is obse ed o o he alues o he ex- ponen β > 0, see Fig. 8. No ice in pa icula he nice collapse o op imal p o iles o di e en alues o ν1 0123456 d/da -12 -10 -8 -6 -4 -2 0 G(d) ν=10-2 ν=10-1 ν=1 ν=10 β=1.0 β=0.5 β=0.0 FIG. 9. (Colo online) Dissipa ion LDF o N= 50, T= 1 and a ying β= 0,0.5,1.0 and ν∈[10−2,10]. Cu es o β= 0.5 and 0 ha e been shi ed e ically o con enience ( ecall ha G(da ) = 0 ∀ν, β), so ha β= 1, 0.5 and 0 co espond o op (blue), medium ( ed) and bo om (g een). The MFT p edic ions a e plo ed wi h lines: solid o β= 1, dashed o β= 0.5, and do ed o β= 0. As in Fig. 5, o a ixed ν inc easing β esul s in la ge dissipa ion luc ua ions. bu equal ela i e dissipa ion. An in e es ing obse a ion is ha op imal densi y p o iles a e less p onounced he la ge de nonlinea i y exponen βis, see Fig. 8. This gi es a plausible explana ion o he widening o G(d) as βinc eases: o he same alue o d/da and inc easing β, he associa ed op imal p o ile is close o he hyd ody- namic solu ion he la ge βis, and hence his luc ua ion cos dec eases, ha ing a la ge associa ed p obabili y. 2. A bi a y dissipa ion coe icien ν Fo a bi a y alues o ν&1 no gene al scaling unc- ion can be de i ed in p inciple o G(d). Fo each pa - icula case, he whole a ia ional p oblem, eqs. (5.11)- (5.16), mus be sol ed, which is o en analy ically in- ac able. In o de o u he ad ance, we eso now o a nume ical e alua ion o he op imal p o iles, which a e used in u n o compu e he dissipa ion LDF. Fig. 9 shows he heo e ical p edic ions o G(d) o inc easing, non-pe u ba i e alues o ν, oge he wi h nume ical e- sul s om simula ions, o di e en alues o β. As o he weakly-dissipa i e sys em limi p e iously discussed, he ag eemen be ween heo y and measu emen s in Fig. 9 is qui e good. We a ibu e he obse ed di e ences be ween heo y and simula ion o ini e size e ec s in he la e , which a e mo e appa en o la ge νas compa ed o he weakly-dissipa i e sys em limi ν1, compa e wi h Fig. 5, see also [39]. Such s ong ini e-size e ec s a e expec ed since he na u al leng h scale associa ed o a gi en νis `ν=qˆ D/ν. As ollows om Eq. (5.7) and he associa ed discussion, `νdec eases as νg ows 21 FIG. 10. (Colo online) Top: Op imal ene gy p o iles as a unc ion o he ela i e dissipa ion measu ed o ν= 10, N= 50 and T= 1 o he pa icula case β= 0. Thick (g een) lines co espond o measu emen s while hin (pink) lines a e MFT p edic ions. Bo om: Measu ed op imal ene gy p o iles o ν= 10, N= 50 and T= 1, and a ying alues o he non- linea i y exponen β. Fo a gi en ela i e dissipa ion, ene gy localiza ion a ound he mal ba hs dec eases as βinc eases. so la ge sys em sizes a e needed o obse e con e gence o he mac oscopic limi . In addi ion, ini e-size e ec s ela ed o he numbe o clones Mused o he sampling become an issue in his limi [13, 31]. In any case, he sha pening o G(d) as νinc eases o any βshows ha la ge dissipa ion luc ua ions a e s ongly supp essed in his egime, as was a gued o ν1 on qui e gene al g ounds in Sec. III C). In his s ongly-dissipa i e sys em limi ν1 he scale `ν→0, and he sys em decouples e ec i ely in o wo indepen- den bounda y shells. Thus, he ene gy is concen a ed a ound he bounda y ba hs, a pic u e which ag ees again wi h he analysis o Sec. III C. This beha iou is e i- denced by he op imal ene gy p o iles o a gi en dmea- su ed o ν= 10, see Fig. 10, in con as o he beha io obse ed o ν1, see Figs. 7-8. The ag eemen o he obse ed p o iles wi h MFT p edic ions is a he good, aking in o accoun he non-negligible ini e-size e ec s a ec ing hese measu emen s. Bo om panel in Fig. 10 shows he measu ed ene gy p o iles as a unc ion o he ela i e dissipa ion and o di e en alues o he nonlin- ea i y exponen β. F om his igu e, i is clea ha o a gi en ela i e dissipa ion, ene gy localiza ion a ound he mal ba hs dec eases as βinc eases. This sugges s again ha , as in he ν1 limi , he p obabili y o a ixed ela i e dissipa ion luc ua ion d/da , inc eases as βg ows, gi ing ise o a b oadening o G(d) wi h β. VI. SUMMARY AND CONCLUSIONS In his pape we ha e de eloped a gene al heo e i- cal amewo k o calcula ing he p obabili y o la ge de- ia ions o he dissipa ed ene gy in a gene al class o nonlinea d i en di usi e sys ems wi h dissipa ion. Ou s a ing poin is a mesoscopic luc ua ing hyd odynamic heo y o he ene gy densi y in e ms i a ew slow hy- d odynamic ields, ha is, a luc ua ing eac ion-di usion equa ion wi h a d i e m compa ible wi h Fou ie ’s law and a sink e m which can be w i en in e ms o he local ene gy densi y. The alidi y o his hyd odynamic desc ip ion can be demons a ed o a la ge amily o s ochas ic mic oscopic models [39], bu i is expec ed o desc ibe he coa se-g ained physics o many eal sys ems sha ing he same main ing edien s, namely: (i) nolinea di usi e dynamics, (ii) bulk dissipa ion, and (iii) bound- a y d i ing. F om his luc ua ing hyd odynamic desc ip- ion, and using a s anda d pa h in eg al o mula ion o he p oblem, we can w i e he p obabili y o a pa h in mesoscopic phase space, ha is, he space spanned by he slow hyd odynamic ields. In e es ingly, he ac ion associa ed o his pa h, om which la ge-de ia ion unc- ions o mac oscopic obse ables can be de i ed, has he same simple o m as in non-dissipa i e sys ems. This is a consequence o he quasi-elas ici y o mic oscopic dy- namics, equi ed in o de o ha e a non i ial compe i ion be ween di usion and dissipa ion a he mesoscale [39]. We use he de i ed ac ion unc ional o in es iga e he la ge de ia ion unc ion o he dissipa ed ene gy. The ene gy dissipa ed in a non-conse ing di usi e sys em is, oge he wi h he ene gy cu en , he ele an mac o- scopic obse able cha ac e izing nonequilib ium beha - io . A simple and powe ul addi i i y conjec u e simpli- ies he esul ing a ia ional p oblem o he dissipa ion LDF, om which we a i e a Eule -Lag ange equa ions o he op imal densi y and cu en ields ha sus ain an a bi a y dissipa ion luc ua ion. A Hamil onian e- o mula ion o his a ia ional p oblem g ea ly simpli ies he calcula ions, allowing us o analyze he gene al he- o y in ce ain in e es ing limi s. A pe u ba i e solu ion he eo shows ha he p obabili y dis ibu ion o small ( ha is, ypical) luc ua ions o he dissipa ed ene gy is always gaussian, as expec ed om he cen al limi he- o em. Mo eo e , a gene al exp ession o he a iance o he dis ibu ion in he gaussian app oxima ion has been de i ed which compa es nicely wi h nume ical esul s. On he o he hand, s ong sepa a ion om he gaussian beha io is expec ed o la ge dissipa ion luc ua ions, wi h a dis ibu ion which shows no nega i e b anch, hus iola ing he Galla o i-Cohen luc ua ion heo em as ex- pec ed om he i e e sibili y o he dynamics. Fu he - mo e, he dissipa ion LDF exhibi s simple and gene al scaling o ms in he weakly- and s ongly-dissipa i e sys- em limi s, which can be analyzed in gene al wi hou knowing he explici solu ion o he canonical equa ions. We apply ou esul s o a gene al class o di usi e la ice models o which dissipa ion, nonlinea di usion 22 and d i ing a e he key ing edien s. The heo e ical p e- dic ions, which can be explici ely wo ked ou in ce ain cases, a e compa ed o ex ensi e nume ical simula ions o he mic oscopic models (which co e bo h ypical luc u- a ions and a e e en s), and excellen ag eemen is ound in all cases. In pa icula , he simple scaling o he dissi- pa ion la ge-de ia ion unc ion in he weakly-dissipa i e sys em limi is ully con i med o di e en alues o he nonlinea i y exponen β, exhibi ing non-con ex beha io o la ge enough luc ua ions. In e es ingly, in his limi ν1 ene gy p o iles associa ed o la ge dissipa ion luc- ua ions exhibi an o e shoo in he bulk esul ing om an excess ene gy injec ion om bounda y ba hs. On he o he hand, in he s ongly-dissipa i e sys em limi ν1 he ypical leng hscale goes o ze o and he sys em decouples in o wo almos -independen bounda y shells, gi ing ise o a di e en scaling o m o he LDF and a s ong supp ession o he dissipa ion luc ua ions in his egime. Recen ly, a simila hyd odynamic heo y has been de- eloped o s udy la ge luc ua ions in a pa icula class o d i en dissipa i e media [37], bu i s p edic ions do no compa e well wi h nume ical esul s o he dissipa- i e la ice models he e s udied. The eason is ha Re . [37] s udies sys ems wi h wo compe ing dynamics, one conse a i e and ano he nonconse a i e, hus esul ing in independen luc ua ions o he densi y and dissipa- ion ields. In ou heo y, as is he case in many d i en dissipa i e sys ems, dissipa ion is linked o he collision p ocess, and hence dissipa ion luc ua ions a e ensla ed o densi y p o ile de ia ions. In ac , bo h heo ies coin- cide in he limi whe e he op imal dissipa ion p o ile is gi en in e ms o he op imal densi y ield. In summa y, ou esul s show ha a sui able gene al- iza ion o mac oscopic luc ua ion heo y [7] is capable o desc ibing in de ail he luc ua ing beha io o gen- e al nonlinea d i en dissipa i e media. In his scheme, he dissipa ion LDF ollows om a a ia ional p oblem whose solu ion also gi es he op imal p o iles ha he sys em has o sus ain o achie e he conside ed luc ua- ion. The p oposed amewo k is e y gene al, as MFT is based only on (a) he knowledge o he conse a ion laws go e ning a sys em, which allow o w i e down he balance equa ions o he luc ua ing ields, and (b) a ew anspo coe icien s appea ing in hese luc ua ing bal- ance equa ions. This opens he doo o u he gene al esul s in he nonequilib ium s a is ical physics o dissi- pa i e media. In pa icula , i would be in e es ing o ex- plo e he exis ence o phase ansi ions and spon aneous symme y b eaking a he luc ua ing le el he ein, in a way simila o he phenomenon epo ed in conse a i e sys ems [30, 31]. Mo eo e , as he ele an magni udes cha ac e izing nonequilib ium beha io in dissipa i e sys- ems a e bo h he dissipa ed ene gy and he cu en , i would be wo h analyzing he join luc ua ions o hese wo obse ables wi hin he MFT app oach. ACKNOWLEDGMENTS We acknowledge inancial suppo om Spanish Min- is e io de Ciencia e Inno aci´on p ojec s FIS2011-24460 and FIS2009-08451, EU-FEDER unds, and Jun a de An- daluc´ıa p ojec s P07-FQM02725 and P09-FQM4682. Appendix: Va ia ional p oblem wi h a Lag angian including second-o de de i a i es Le us analyze a a ia ional p oblem in which he “ac- ion” is de ined as he in eg al o a “Lag angian” wi h second o de de i a i es, ha is S[j] = Zx2 x1 dx L(j, j0, j00).(A.1) The ac ion S[j] is a unc ional o he p o ile j(x) in he ixed in e al x1≤x≤x2. The a ia ional p oblem a ises when one looks o he “op imal” p o ile j(x) o which he unc ional S[j] is a ex emum. Fo he sake o conc e eness, le us conside a p oblem simila o he one analyzed in his pape : we a e in e es ed in calcula ing Gde ined as G=−min j(x)S[j].(A.2) Then, we conside he a ia ion δSo he unc ional when a gi en p o ile j(x) is sligh ly changed o j(x) + δj(x), δS=Zx2 x1 dx ∂L ∂j δj +∂L ∂j0δj0+∂L ∂j00 δj00.(A.3) Now, we ake in o accoun ha δj0=d dxδj, δj00 =d2 dx2δj (A.4) in o de o in eg a e by pa s (i) once he e m p opo - ional o δj0(ii) wice he e m p opo ional o δj00. We a i e hus a δS=∂L ∂j0−d dx ∂L ∂j00 δj +∂L ∂j00 δj0x2 x1 +Zx2 x1 dx ∂L ∂j −d dx ∂L ∂j0+d2 dx2∂L ∂j00 ,(A.5) 23 whe e [ ]x2 x1= (x2)− (x1). By analogy wi h he case o he usual Lag angian wi h only i s -o de de i a i es, we in oduce he gene alized momen a as pj=∂L ∂j0−d dx ∂L ∂j00 , pj0=∂L ∂j00 .(A.6) In his way, he bounda y e m has he usual o m and Eq. (A.5) can be ew i en as δS= [pjδj +pj0δj0]x2 x1+Zx2 x1 dx ∂L ∂j −d dx ∂L ∂j0+d2 dx2∂L ∂j00 δj. (A.7) The ex emum condi ion is δS = 0. I he alues o j and j0a e p esc ibed a he bounda ies, bo h δj and δj0 anish a x1,2and he bounda y e m anishes. Then, as δj is a bi a y o x1< x < x2, he “op imal” p o- ile solu ion o he a ia ional p oblem e i ies he Eule - Lag ange equa ion d2 dx2∂L ∂j00 −d dx ∂L ∂j0+∂L ∂j = 0,(A.8) which is a ou h-o de di e en ial equa ion. In e es - ingly, Eq. (A.8) can be w i en as dpj/dx =∂L/∂j ha is o mally iden ical o he usual Eule -Lag ange equa ion o Lag angians wi h only i s -o de de i a i es. The bounda y condi ions o he Eule -Lag ange equa ion a e he p esc ibed alues o jand j0a he bounda ies; ou condi ions o he ou h-o de di e en ial equa ion. Howe e , in physical p oblems he e a e some imes less p esc ibed quan i ies a he bounda ies han necessa y. In ha case, as poin ed ou by Lanczos [33], he ex- emum condi ion δS= 0 p o ides he “missing” bound- a y condi ions. Fo ins ance, i we only ha e ixed alues o j0a he bounda ies (as in he LDF p oblem we ha e deal wi h in he main ex ), δj0(x1) = δj0(x2) = 0 bu δj(x1) and δj(x2) a e ee pa ame e s. Equa ion (A.7) s ill implies he Eule -Lag ange equa ion bu also ha pj(x1) = pj(x2)=0.(A.9) The gene alized momen um conjuga e o he a iable ha is no ixed a he bounda y mus anish: he so- lu ion o he a ia ional p oblem e i ies hen he Eule - Lag ange equa ion (A.8) wi h he p esc ibed alues o j0 a he bounda ies and he “ex a” condi ions p o ided by Eq. (A.9). In his way, we ob ain he ou condi- ions needed o de e mine comple ely he solu ion o he Eule -Lag ange equa ion. The unc ion Gde ined in Eq. (A.2) depends on he alues o jand j0a he bounda ies. Making use o Eq. 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