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, Université 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 simulations. 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 reverse 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

A fluctuation relation for weakly ergodic aging systems

Similar A fluctuation relation for weakly ergodic aging systems