A fluctuation relation for weakly ergodic aging systems

0.22 MB·English

by  A. Crisanti

A fluctuation relation for weakly ergodic aging systems
A. Crisanti1, M. Picco2, and F. Ritort3
1Dipartimento di Fisica, Universita` di Roma “La Sapienza”, and
Istituto dei Sistemi Complessi ISC-CNR, P.le Aldo Moro 2, 00185 Roma, Italy
2CNRS, LPTHE, Universit´e Pierre et Marie Curie,
UMR 7589, 4 place Jussieu, 75252 Paris cedex 05, France
3Departament de F´ısica Fonamental, Facultat de F´ısica,
Universitat de Barcelona, Diagonal 647, E-08028, Barcelona and
3 CIBER-BBN Center for Bioengineering, Biomaterials and Nanomedicine, Instituto de Salud Carlos III, Madrid
1
(Dated: $ V 13.3.0 2012/08/13 10:37 AC$)
0
2 A fluctuation relation for aging systems is introduced, and verified by extensive numerical simula-
tions. It is based on the hypothesis of partial equilibration over phase space regions in a scenario
n
a of entropy-driven relaxation. The relation provides a simple alternative method, amenable of ex-
J perimental implementation, to measure replica symmetry breaking parameters in aging systems.
The connection with the effective temperatures obtained from the fluctuation-dissipation theorem
3
is discussed.
2

PACSnumbers: 05.40.-a,05.70.Ln
]
h
c
e Non-equilibriumsystemsarecharacterizedbyaneten- [15]. The new relation is further supported by extensive
m
ergy transfer (in the form of work, heat, or mass) to the numerical simulations. The existence of an AFR was al-
- environment. Aging systems pertain to the category of ready suggested in Refs. [16, 17] and recently hinted at
t
a weakly ergodic non-equilibrium systems [1, 2] exhibiting ina quenchexperimentofagelatindropletthatexhibits
t slowrelaxationaldynamicsandstronghistorydependent a sol-gel transition [18]. In that reference heat distribu-
s
. effects where the fluctuation-dissipation theorem (FDT) tions were measured and shown to satisfy a fluctuation
t
a is violated [3–6]. This has led to the introduction of the relation for a system in contact with two baths at dif-
m
concept of non-equilibrium or effective temperature [7]. ferent temperatures. Yet it remains unclear whether the
- Despiteoftheinsightgainedfromexactlysolvablemodels same relation applies to other aging systems. In con-
d
n andotherconceptualattempts (e.g. [8,9])westilllacka trast, the new relation we are proposing should be gen-
o clear understanding of the generalpicture describing ag- erally valid in aging systems and is amenable to future
c ingsystems. Incontrasttostationarysystems,agingsys- experimental verification in noise measurements of glass
[
temsaredescribedbytwotimescales: thewaitingtimetw formers, critical systems and small systems (e.g. single

1 elapsed since the system was set in the non-equilibrium molecules).
v state and the time t > tw at which measurements are The aging fluctuation relation (AFR). Consider a sys-
1
taken. Acharacterizationofthefullspectrumoffluctua- tem quenched at time t = 0 from an high-temperature
6
4 tions appearskey for a satisfactoryunderstanding of the equilibrium state down to temperature T where it ages
5 aging state. exhibiting slow relaxation and activated dynamics phe-
. Overthepastyearsseveralresultsaboutenergyfluctu- nomena. Such aging state is characterized by low
1
0 ations in non-equilibrium states have been obtained un- entropy-productionrateandlossoftime-translationalin-
3 der the heading of fluctuation theorems (FTs) [10, 11]. variance [19]. During the aging process the system con-
1 FTs take slightly different mathematical forms depend- tinuously exchanges energy with the bath, but some re-
:
v ingonthespecificnon-equilibriumcontext[12,13]. How- laxationaleventsresultinlargerthantypicalamountsof
i everallthemsharethesamecommonfeature: theyrelate heatQreleasedto the bath, leadingto a netpositiveen-
X
probabilitiesofabsorbingandreleasingagivenamountof tropyproductionh∆Si=hQi/T >0,wherethebrackets
r
a energyundernon-equilibriumconditions;they areuseful h(···)i denote the average over dynamical histories.
in small systems and short times where energy fluctua- Toanalyzethespectrumofheatorentropyproduction
tions can be directly measured allowing us, for example, fluctuations in the aging state we consider the Bochkov-
to extract free energies of kinetic molecular states from Kuzovlevworkfluctuationrelation,originallyintroduced
irreversible pulling experiments [14]. as a generalization of the FDT [20]. After a time t ,
w
In this work we present a theoretical derivation of a elapsedsincethesystemwasquenched,aconstantexter-
fluctuation relation in aging systems. The aging fluctu- nal perturbation of strength h coupled to an observable
ation relation (AFR) is based on the hypothesis of par- A is applied to the system. This will cause a change
tial equilibration in a scenario of entropy-drivenslow re- in A during its subsequent evolution. The entropy pro-
laxation. It can be written in terms of a phase-space duction during the time interval [t ,t] (t ≥ t ) caused
w w
contraction factor x that bears resemblance to the or- by the perturbation is equal to ∆S = Q /T =
tw,t tw,t
der parameter x(q) defined in the context of spin glasses h[A(t)−A(t )]/T =h∆A /T. Thisquantityhasbeen
w tw,t
2
termed exclusive work in [21] and satisfies a fluctuation the case for mean-field models where metastable states
relation if the system is equilibrated at t (which is not have infinite lifetime.
w
the case here). ∆S is a fluctuating quantity changing To justify Eq. (1) we consider a system whose time
tw,t
uponrepetitionofthesameexperiment. Moreoverinthe evolution is ruled by the Langevin equation
aging regime, where relaxation dynamics is ruled by the
dϕ δ
complex topological structure of the phase space, made =− H(ϕ)+h+ξ (2)
dt δϕ
ofmanyalmostdegeneratemetastablestates,∆S dis-
tw,t
plays a strong intermittent behavior. This means that
whereϕis anN-dimensionalfield, ξ(t)aGaussianwhite
for fixed t and ∆S the requirement ∆S = ∆S de-
w tw,t noise (thermal noise) of zero average and correlation
fines a very broadintervalof times t coveringmany, well hξ(t)ξ(t′)i = 2Tδ(t − t′), −δH(ϕ)/δϕ the force aris-
separated, timescales. In such a context a data analysis
ing from the conservative energy H(ϕ), and h the con-
forfixedt andtmaybeambiguoussinceitmaymixup
w stant external field coupled to the macroscopic observ-
processeswithdifferenttimescales,asnoticedsomeyears able ψ = ϕi, where i denotes a site index. Fluc-
t i t
agointheframeworkofturbulence[22,23]. Toovercome
tuation relaPtions derive from the behavior of the prob-
thisproblemwedefineP (∆S)asthe probabilityofob-
serving the value ∆S t=w ∆S after t . ability P of a trajectory {ϕs}s∈[tw,t] and its reverse
In Ref. [16] a similtawr,tapproach, baswed on the concept {ϕs}s∈[tw,t] ≡ {ϕt+tw−s}s∈[tw,t]. These can be easily
computedusingthepathintegralformalism,seee.g. [24]:
of Inherent Structure, was used. The idea was to look e
for∆Stw,t associatedtothefirstjumpoutofanInherent e−∆Stw,t+∆SeqP[{ϕs}s∈[tw,t]] =P[{ϕs}s∈[tw,t]] (3)
Structure. While this definition could in principle be
used in a numerical simulation, it is far less useful in a where ∆Stw,t = βh(ψt −ψtw) and ∆Seeq = β[H(ϕt)−
H(ϕ )] = −ln[Peq(ϕ )/Peq(ϕ )]. The quantities in
real experiment. In contrast, the one proposed here is tw t tw
the exponent depend on the trajectory end-points only,
well suited for real experiments.
Despite the factthatthe perturbingfieldh favorspos- then summing over all trajectories from ϕtw at tw to ϕt
at time t, and including normalizedprobability distribu-
itive values of ∆S , trajectories with negative values
tw,t
tions P (ϕ ) and P (ϕ ) = P (ϕ ) for the initial and
can be observed as well. We argue that, for long enough 0 tw 1 tw 1 t
final states, we get
t , the probability of observing positive and negative
w e
values of ∆S satisfies the following fluctuation relation
e−∆StotP(ϕ ,t|ϕ ,t )P (ϕ )=P(ϕ ,t|ϕ ,t )P (ϕ )
(AFR) t tw w 0 tw tw t w 1 t
(4)
log Ptw(∆S) =x ∆S, (1) where P(ϕt,t|ϕtw,tw) and P(ϕtw,t|ϕt,tw) are the con-
(cid:20)Ptw(−∆S)(cid:21) tw kB ditional probabilities of the forward ϕtw → ϕt and re-
verse ϕ → ϕ trajectories, respectively, and ∆S =
where kB is the Boltzmann constant (equal to 1 in the ∆S −t ∆Seqt+w∆S ,with∆S =−ln[P (ϕ )/P (ϕtot )].
following). If at t the system is in equilibrium, then tw,t b b 1 t 0 tw
w From this relation the following identity follows
x = 1 and Eq. (1) reduces to the Bochkov-Kuzovlev
tw
work fluctuation relation [20, 21]. P (−∆S)= δ ∆S −∆S e−∆Stot (5)
tw,t tw,t tw,t
In aging systems, such as glasses, the FDT is not vio- (cid:10) (cid:0) (cid:1) (cid:11)
latedfor times t−tw <∼tw after switching onthe pertur- where the average is over the forward process ϕtw → ϕt
bation. This is because the degrees of freedom whose with initial probability distribution P0(ϕtw).
characteristic relaxation times are sufficiently smaller After the quench the system partially equilibrates in-
than t have equilibrated. In the present context this side independent phase-space regions (that we will call
w
means that small enough energy transfers between the cages),fromwhichitwillescapeonlyafteratimet−tw ∼
system and the bath occur in equilibrium, and hence tw. Therefore,whent−tw ≪tw thesystemisin(partial)
xtw ≃ 1. This remains true as long as |∆S| is smaller equilibrium with the thermal bath, P0(ϕ) = P1(ϕ) ∝
than a cross-over value ∆S∗, which sets the scale of the Peq(ϕ) so that ∆Seq = ∆Sb, ∆Stot = ∆Stw,t, and from
typicalminimumenergytransferinprocessesthatinvolve (5) one gets eq. (1) with xtw =1.
non-equilibrated degrees of freedom. For |∆S| > ∆S∗ To study the opposite limit t −tw ≫ tw, where the
non-equilibriumenergyexchangeprocessesresponsibleof systemcanaccessdifferentcages,weobservethat∆Stw,t
slow relaxation come into play, and the equality xtw =1 depends only on the macroscopic variables ψt and ψtw,
is violated. Not all degrees of freedom can contribute to thentheaverageonther.h.sof(5)canbedonebypartial
the relaxation process, some of them being frozen at t , classification,thatisbyaveragingfirstoverallpathswith
w
and hence xtw < 1. As the system ages more degrees of given initial and final states, and then over ψt and ψtw:
freedomequilibrate att , and hence ∆S∗ increaseswith
w
tw. In the limit tw ≫ teq, where teq is the equilibration Ptw,t(−∆S) = Z dψtwZ dψtδ ∆Stw,t−∆S
ctiomnve,eraglelsdteog1refeosroafllfr∆eeSd.oWmehnaovteeetqhuaitlitbhriastemdayanndotxbtwe × e−∆Stot (cid:0) . (cid:1) (6)
ψtwtw;ψtt
(cid:10) (cid:11)
3
where h(···)i denote dynamical averages re- namical MCT transition at T = 0.536. Below T the
ψtwtw;ψtt d d
stricted to those trajectories starting with ψ at t systemisdynamicallyconfinedintooneofthemany(ex-
tw w
and ending with ψ at t. Assuming the system is par- ponentially large in number) metastable states and can-
t
tially equilibrated over cages, the probability P(ϕ|ψ) not reach full equilibrium. The equilibrium transition
of a state ϕ in a cage with fixed ψ is proportional to occurs at the lower static (or Kauzmann) temperature
Peq(ϕ)timestheprobabilityofhavingψ inacage. Thus T =0.25. Thelow-temperaturebehaviorofthemodelis
c
P(ϕ|ψ) ∝ Peq(ϕ)×Ω (ψ)/Ω (ψ), where Ω (ψ) is describedbyaone-stepreplicasymmetry(1RSB)break-
cage tw cage
the number of states with ψ inside the cage, divided by ing order parameter. Typical signature of this is a two-
the total number Ω (ψ) of accessible states, not neces- slopes FD plot.
tw
sarily in the same cage, with ψ. Under the hypothesis
of partial equilibrium S (ψ)=lnΩ (ψ) is the ther-
cage cage
omnalsyesqtuemilibargiuemsinecnetrmopoyreinwethweaictamgeo.reΩdtewg(rψe)esdoefpefrnedes- 105 ∆S)1100--21 (a) 10
fduolml threelramxo.dSytnwa(mψ)ic=enlntrΩoptwy(aψn)discotnhvenergsmesatlolerittohnalnytfhoer ∆-S) 104 P(tw1100--43 (c)
tawnd≫finteaql.stUastiensg, athnids tahnesartezlaftoironth∂eSPcaDgeF(ψo)f/t∂hψe i=nitβiahl ) / P(tw103 10-5-20 0 2∆0S 40 60 0.0610 2 4 6 8
S
together with the analogous ∂Stw(ψ)/∂ψ = xtwβh, cor- ∆( 102
rected through the coefficient xtw < 1 to account for Ptw χ 0.03 (b)
the frozen degrees of freedom, we have ∆Seq −∆Sb = 101 ttw==614024
βh(ψt−ψtw)−xtwβh(ψt−ψtw)=(1−xtw)∆Stw,t. Insert- tw=16384 00 0.05 0.1 0.15
ingthisforminto(6)theAFR(1)follows. Thecoefficient w 1 - C
0
10
x measures the phase space contraction due to frozen 0 5 10 15 20 25 30
detwgrees of freedom at t . Clearly x →1 as t ≫t . ∆S
w tw w eq
Numericaltests. We havetested the AFR (1) through
FIG.1: Numerical test of the AFR in the ROM.Main
Monte Carlo simulations of several model systems, but
plotisatest ofEq.(1) forthemodelwith N =1000, T =0.2
wereportresultsforonlythreeofthem. Thesystem,ini-
andh=0.1atthreevaluesoftw. Thedashedlinecorresponds
tiallypreparedinanhightemperatureequilibriumstate,
to xtw = 1 while full lines to xtw = 0.271,0.287,0.299 for
is instantaneously quenched to a temperature T below tw = 64,1024,16384 respectively. (a) Ptw(∆S) and (b) FD
the freezing transition temperature. After tw a pertur- plot. The dashed and continuous lines shown in the FD plot
bation of small intensity, to ensure a good statistics for haveslopes equalto those shown in themain plot. (c) Zoom
trajectories with ∆S < 0, is applied and the fluctua- oftheregioncorrespondingtoxtw =1(intra-cagerelaxation).
tions ∆A of the conjugated variable A are recorded
tw,t
at fixed time intervals t−t . The maximum recording Figure 1 shows results for the ROM. Two regimes can
w
time t was taken much larger than tw to ensure good be distinguished, xtw = 1 for |∆S| < ∆S∗ and xtw < 1
statistics. The procedure was repeated several times, for |∆S|>∆S∗ (∆S∗ ≃2 for tw =1024). The values of
Ptw(∆S)calculatedfromdatabinning andEq.(1)tested xtw agree quite well with x=T/Teff, extracted from the
to extract the value of x . To compare it with the FD plot [inset (b)].
tw
parameter x = T/T (t ) obtained from the FDT, the As a more realistic system we have studied a 80 : 20
eff w
fluctuation-dissipation (FD) plots in time-domain were binary mixture oftype A and B particles interacting via
alsomeasuredusing the standardprocedures[3]. Both x a Lennard-Jones pair potential (BMLJ):
parameters (the one derived from the AFR, Eq.(1), and
σ 12 σ 6
the one derived from FDT) asymptotically coincide un- V (r)=4ǫ αβ − αβ (8)
αβ αβ
der general assumptions, see [25]. (cid:20)(cid:16) r (cid:17) (cid:16) r (cid:17) (cid:21)
The first model is the Random Orthogonal model where α,β = A,B, r is the distance between the two
(ROM) defined by the Hamiltonian [3] particles and the parameters σ ,ǫ stand for the ef-
αβ αβ
fective diameters and well depths between species α,β.
H=− J σ σ , (7)
ij i j The parametersfor length and energymeasured in units
1≤iX<j≤N of σ and ǫ are ǫ = 0.5, ǫ = 1.5, σ = 0.88
AA AA BB AB BB
where σ = ±1 are Ising spins and J = J quenched andσ =0.80,andaretakento preventcrystallization
i ij ji AB
Gaussian variables of zero mean and variance 1/N sat- [26]. Withthischoiceasystemofreduceddensityρ=1.2
isfying J J = 16δ , with J = 0. The system exhibits a glass transition well described by the MCT at
k ik kj ij ii
is pertuPrbed by a uniform magnetic field of strength h thecriticaltemperatureTMCT ≃0.435. Thestudyofthe
conjugated to the total magnetization M(t)= σ (t). AFR was done by adding at time t an external poten-
i i w
This model describes structural glasses in tPhe mode tial of the form V0 jǫjcos(k·r), where V0 <T and ǫj
coupling theory (MCT) approximation and has a dy- are i.i.d. (quenchedP) random variables equal to ±1 with
4
equal probability, and recording the conjugated observ-
able A (t) = ǫ exp[ik·r (t)] [27] . Results for the t = 95
k j j j 104 w
AFR are showPn, and compared with standard FD plots, tw = 1020
in Fig 2. Also in this case the agreement between the )
S
x extracted from AFR and that from FD plot is rather ∆ 3 10-2 (a)
ghboiebnoiddt.inthgIenitsbedreeunsedtiitnnogglfiytnhicatuetr-svcieazsneienbffeetchsteesenAthFianRt fFopDrlostphlodorottsn.twotSpuercxoh-- S) / P(-tw11002 0.3 ∆S)P(tw1100--43
duce an equilibration of the system, and a depletion of ∆( (b) 10-5
the statistics of rare events for longer tw. The plot then Ptw 1 χ 0.2 -15 0 15∆3S0 45 60
bends upwardfor shortt anddownwardfor longert . 10 0.1
w w
0
0 0.2 0.4 0.6 0.8
S) 104 11110000----2431 (a) ∆S)(Ptw tttwww=== 111000342 FIG.130:00Numeri4cal test8of t1h-Ce∆1SA2FR in16the 3D2±0J EA
∆ 3
P(-tw10 10-5-15 0 ∆1S5 30 45 mpaoradmele.terMs aLin=pl1o6t, iTs a=t0es.7t aonfdEqh.(=1)0f.o1ratthetwmoovdaelluewsitohf
∆S) / 102 t(wre.d,Tblhuee)dtaoshxetdw l=ine0.3co8r3r,e0s.p4o5n0dfosrttowx=tw95=,1102w0hrielespfeuclltivlienleys.
P(tw 0.4 (lian)esPt(wre(d∆,bSl)uea)ndin(bt)heFDFDplpolto.tThhaevedaslsohpeedsaenqduaclontotintuhoouses
101 χ 0.2 shown in the main plot.
(b)
0
0 0 0.5 1-C 1
10
0 3 6 9 12 15
∆S
equilibrium-entropy driven scenario in slowly relaxing
FIG. 2: Numerical test of the AFR in the BMLJ
systems from noise measurements. A theoretical deriva-
model. Main plot is a test of Eq.(1) for the model with
tionofthe AFRwasgivenanditsvalidityverifiedbyex-
parameters N = 500,V0 = 0.1,|k| = 7.25,T = 0.3 at
tended numerical experiments. The connection between
three values of tw. The dashed line corresponds to xtw = 1
wtwhi=le1f0u2ll,1li0n3e,s10(r4erde,sgpreecetniv,eblylu.e()a)toPxtwtw(∆=S)0.a4n0d,0(.b5)4,F0D.60plfootr. twhaesvsahlouwesn.ofRxetmwaerxktarbalcyteednofruogmh,tahnedAiFnRcoanntdraFstDtoplFotDs
Thedashedandcontinuouslines(red,green,blue)intheFD plots, extracting the value of x does not require mea-
tw
plot haveslopes equalto those shown in themain plot. suring aging correlation functions. By only measuring
the statistics of ∆S to an externally applied perturba-
Bothabovesystemsaredescribedbyatwo-stepsrelax- tion, P (∆S), we can test the validity of Eq.(1) to ex-
tw
ation, or 1RSB, scenario. As last example we have con- tract the value of x . We emphasize that in order to
tw
sidered the 3-dimensional ±J Edwards-Anderson model test Eq.(1) statistical events with ∆S < 0 must be ob-
(±J-EA) defined by the Hamiltonian (7), but with Jij served. Since the average value of ∆S continuously in-
randomly chosen equal to ±1 if the sites j and i are creases with t, only rare events with ∆S < 0 give full
nearest-neighbors on a cubic 3-dimensional lattice, and meaning to the AFR. A similar situation is encountered
zero otherwise. Numerical investigation indicates that in the Gallavotti-Cohen theorem for steady state sys-
below T ≃ 1.14 there is a spin-glass phase described tems [30]. Eq.(1) is ready to be employed in mesoscopic
by a continuous-step relaxation, or Full-RSB, scenario. systems (e.g. magneto-conductance fluctuations in spin
Typical signature of this is a continuous-slope FD plot. glasses and electron glasses [31, 32]) and single molecule
Numerical tests of Eq.(1) for the ±J-EA at two differ- experiments. The latter include molecular systems ex-
ent tw are shown in Fig.3, together with the FD plot. hibitingslowfoldingduetodisorderandfrustration(e.g.
The AFR is well verified also in this model. However,as RNA) or slow binding kinetics (e.g. peptides or proteins
shown in inset (b) and in contrast to the previous 1RSB binding DNA). Ultimately, small systems may provide a
models,the phasecontractioncoefficientxtw agreeswith direct access to experimentally measure the always elu-
T/Teff only at the early stage, where the FD plot de- sive spin-glass order parameter.
part from equilibrium. This is in agreement with the
aforementioned argument, according to which x gives AC thanks the LPTHE where part of this work was
tw
information on the phase-space partition at t . done. FR acknowledges support from HFSP Grant No.
w
Discussion. Summarizing, the AFR (1) shows a RGP55-2008, ICREA Academia 2008 and Spanish Re-
promising route to experimentally test the partial search Council Grant. No. FIS2010-19342.
5
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