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.
(A.7), and aking in o accoun ha he op imal p o ile
j(x) e i ies he Eule -Lag ange equa ion, we ge
δG =−pj,2δj2−pj0,2δj0
2+pj,1δj1+pj0,1δj0
1.(A.10)
We ha e in oduced he no a ion pj,i =pj(xi), δji=
δj(xi), i= 1,2 and so on. Equa ion (A.10) implies ha
pj,2=−∂G
∂j2
, pj0,2=−∂G
∂j0
2
, pj,1=∂G
∂j1
, pj0,1=∂G
∂j0
1
.
(A.11)
Equa ion (2.33) o he main pape is he pa icula iza ion
o his esul o he case (i) x2=−x1= 1/2, (ii) solu-
ions o he Eule -Lag ange equa ion wi h well-de ined
pa i y, in which pj,2=pj,1,pj0,2=−pj0,1, and (iii) he
bounda y condi ions o Eq. (2.30).
[1] L. D. Landau and E. M. Li shi z, S a is ical Physics 3 d
edi ion, Cou se o Theo e ical Physics Vol. 5 (Pe gamon
P ess, Ox o d, 1980).
[2] H. B. Callen and T. A. Wel on, Phys. Re . 83, 34 (1951).
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g angian wi h only i s -o de de i a i es o he cu en j
a ose. Thus, in he canonical desc ip ion, he e was only
one canonical momen um pj, and he pa icula solu ion
o he canonical equa ions wi h pj= 0 also co esponded
o he a e age beha iou .
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[46] The pa ame e sappea ing in he Legend e ans o m o
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mul iplie in oduced in Sec. II A, s=−λ.