On the transient Fluctuation Dissipation Theorem after a quench at a critical point
Isaac Theurkauff, Aude Caussarieu, Artyom Petrosyan, Sergio Ciliberto
eepl draft
On the transient Fluctuation Dissipation Theoremafter a quench at a critical point
Isaac Theurkauff, Aude Caussarieu, Artyom Petrosyan, Sergio Ciliberto
Universit´e de Lyon Laboratoire de Physique, ´Ecole Normale Sup´erieure, C.N.R.S. UMR567246 All´ee d’Italie, 69364 Lyon, France
PACS – Abstract. - The Modified Fluctuation Dissipation Theorem (MFDT) proposed by G. Verley et al. (EPL 93, 10002, (2011)) for non equilibrium transient states is experimentally studied. We applyMFDT to the transient relaxation dynamics of the director of a liquid crystal after a quench closeto the critical point of the Fr´eedericksz transition (Ftr), which has several properties of a secondorder phase transition driven by an electric field. Although the standard Fluctuation DissipationTheorem (FDT) is not satisfied, because the system is strongly out of equilibrium, the MFDT isperfectly verified during the transient in a system which is only partially described by Landau-Ginzburg (LG) equation, to which our observation are compared. The results can be useful in thestudy of material aging.
After a sudden change of a thermodynamic parameter,such as temperature, volume and pressure, several systemsand materials may present an extremely slow relaxationtowards equilibrium. During this slow relaxation, usuallycalled aging, these systems remain out-of equilibrium fora very long time, their properties are slowly evolving andequilibrium relations are not necessarily satisfied duringaging. Typical and widely studied examples of this phe-nomenon are glasses and colloids where many questionson their relaxation dynamics still remain open [1,2]. Thusin order to understand the minimal ingredients for aging,slow relaxations have been studied theoretically in sec-ond order phase transitions when the system is rapidlyquenched from an initial value of the control parameterto the critical point [3–6]. Because of the critical slow-ing down and the divergency of the correlation length therelaxation dynamics of the critical model shares severalfeatures of the aging of more complex materials. One ofthe questions analyzed in this models is the validity ofthe Fluctuation Dissipation Theorem (FDT) during theout of equilibrium relaxation [8–11]. In equilibrium, FDTimposes a relationship between the response of the systemto a small external perturbation and the correlation of thespontaneous thermal fluctuations. When the system is outof equilibrium FDT does not necessarily hold and it has been generalized as k B T X ( t, t w ) χ ( t, t w ) = C ( t, t ) − C ( t, t w ) (1)where k B is the Boltzmann constant, T the bath temper-ature, C ( t, t w ) = < O ( t ) O ( t w ) > − < O ( t ) >< O ( t w ) > ( < . > stands for ensemble-average) the correlation func-tion of the observable O ( t ). The function χ ( t, t w ) is theresponse to a small step perturbation, of the conjugatedvariable h of O , applied at time t w < t : χ ( t, t w ) = < O ( t ) > h − < O ( t ) > h | h → (2)where < O ( t ) > h and < O ( t ) > o denote respectively themean perturbed and unperturbed time evolution. Thefunction X ( t, t w ) is equal 1 in equilibrium whereas in out-equilibrium it measures the amount of the FDT violationand it has been used in some cases to define an effectivetemperature T eff ( t, t w ) = X ( t, t w ) T .The above mentioned models of the quench at criticalpoints allows a precise analysis of the pertinence of thisdefinition of T eff [8]. In spite of the theoretical interest ofthese models only one experiment has been performed onthe slow relaxation dynamics after a quench at the criticalpoint [7]. The role of this letter is to experimentally ana-lyze the theoretically predictions in a real system affectedby finite size effects and unavoidable imperfections. Wep-1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] J un saac Theurkauff, Aude Caussarieu, Artyom Petrosyan, Sergio Cilibertoalso analyze another important aspects of the FDT in outof equilibrium systems. Indeed several generalizations ofFDT has been proposed [9–17] but almost all of them canbe applied to non-equilibrium steady states [18, 19] andare not useful for the transient time evolution which fol-lows the quench at the critical point. As far as we knowthere are only two formulations of FDT [20,21], which areuseful for these transient states and the second purposeof this letter is to discuss the application of the ModifiedFDT (MFDT) of ref. [20].Before describing the experimental set-up we summa-rize briefly the formulation of the MFDT of ref. [20] . Letus consider the relaxation dynamics of a system, whichhas been submitted at time t = 0 to a sudden change ofits control parameters. At time t after the quench, thisrelaxation is characterized by the variable x ( t ), by the ob-servable O ( x ( t )) and by the probability density function π ( x ( t ) , h ( t )). Here h ( t ) is an external control parameterwhich is used to perturb the dynamics. We define a pseu-dopotential Ψ( t, h ) = − ln[ π ( x ( t ) , h )] and an observable B ( t ) = − ∂ h ψ ( t, h ) | h → , with h (cid:54) = 0 constant for t > h = 0 for t <
0. The MFDT reads: χ ( t, t w ) = < B ( t ) O ( t ) > − < B ( t w ) O ( t ) > (3)Eq.3 defines the response function χ ( t, t w ) of O ( t ) to astep perturbation of h applied at t w , with 0 < t w < t .Notice that in this case h can be any parameter of thesystem and it does not need to be the conjugated variableof O ( t ). In this letter we will analyze how MFDT (eq.3)can be applied to the experimental data using the quenchat the critical point in a non ideal system. (cid:161) (cid:111) / (cid:111) (cid:161) < (cid:98)(cid:101) > *4 *10 (rad ) b) < (cid:101) > (rad ) a) Fig. 1: Phase diagram of the Freedericksz Transition in 5CB.a) Dependence on (cid:15) of θ o = < θ > (blue dots) and of thevariance ( dashed red line) of θ multiplied by 4 10 to be onthe same scale. The solution of the LG equation ( µ = 0) isthe straight black line. b) τ o /τ is plotted as a function of (cid:15) . The prediction of LG ( µ = 0) is the black straight line.The measured relaxation time deviates significantly from thepredicted one even for values of (cid:15) where the stationary solutionof LG, plotted in a) seems to reproduce the data. The experimental system where we study these proper-ties and the MFDT is the Freedericksz Transition (FrTr)in a nematic liquid crystal (LC) submitted to an externalelectric field (cid:126)E . Specificlly in our experiment we use the 5CB (p-pentyl-cyanobiphenyl, 5CB, produced by Merck).The experimental apparatus has been already described[22, 23] and we summarize here only the main features.The LC is confined between two glass plates, separated bya distance L =13.5 µ m. The surfaces in contact with LCmolecules are coated by ITO to apply an electrical field.Then, a polymer layer (rubbed PVA) is deposited to in-sure a strong anchoring of the 5CB molecules in a directionparallel to the plates. In the absence of any external field,the molecules in the cell align parallel to those anchored atthe surfaces. Applying a voltage difference U between theelectrodes, the liquid crystal is submitted to an electricalfield perpendicular to the plates. To avoid polarization,the applied voltage is modulated at a frequency f = 10kHz (cid:2) U = √ U cos(2 πf t ) (cid:3) . When U exceeds a thresh-old value U c , the planar states becomes unstable and themolecules rotate to align with the electrical field. To quan-tify the transition we measure the spatially averaged align-ment of the molecules determined by the angle θ betweenthe molecule director and the surface. Such measurementrelies upon the anisotropic properties of the nematic. Thisoptical anisotropy can be precisely measured using a verysensitive polarization interferometer [22] which gives a sig-nal ϕ ∝ θ . At U (cid:39) U c the dynamics is usually describedby a Landau-Ginzburg (LG) equation although as pointedout in ref. [23] this is a very crude approximation, whichhas several drawbacks. In a very first approximation thedynamics of the mean relaxation θ ( t ) = (cid:104) θ ( t ) (cid:105) is ruled bythe following Ginzburg-Landau equation : τ ˙ θ = (cid:15)θ − α (cid:0) θ − µ (cid:1) (4)where (cid:15) = ( U − U c ) /U c is the reduced control parameter,the τ o is a characteristic time of the LC and α a parameterwhich depends on the elastic and electric anisotropy ofthe LC. For 5CB α = 3 .
36 and τ o = 2 . s for the cellthickness L = 13 . µm . The residual angle µ (cid:39) . (cid:15) = 0 comes from cell assembling and preparation and hasbeen discussed in ref. [23]. Furthermore τ o is not strictlyconstant but it slightly depends on θ o . The dimensionalequation for the fluctuations δθ ( t ) = θ ( t ) − θ ( t ) is γAL ˙ δθ = K [( (cid:15) − αθ o ) δθ + δ(cid:15) θ o ] + η (5)with K = π k AL (6)where A is the laser cross section, k is one of the elas-tic constant of LC and η is a delta correlated thermalnoise such that < η ( t ) η ( t (cid:48) ) > = k B T ( γAL ) δ ( t − t (cid:48) ). Theterm with δ(cid:15) takes into account that during the measureof the response we have to perturb the value of (cid:15) by ap-plying a short pulse of duration τ p and amplitude δ(cid:15) . Thetwo eqs.4,5 describe in principle the dynamics of the meandeflection θ o and of the fluctuations δθ . However thereare several discrepancies with the experimental data whichare widely discussed in ref. [23]. We summarize here themost important, which are useful for the discussion. Thep-2 n the transient Fluctuation Dissipation Theorem after a quench at a critical point phase diagram of FrTr in 5CB and the relaxation timesare plotted in figs. 1 a),b) respectively. The solution ofLG θ o (cid:39) (cid:15)/α ( µ = 0 in eq.4) reproduces the station-ary experimental data for (cid:15) ≤ .
6. Instead the measuredrelaxation time (fig. 1 b) deviates significantly from thepredicted one even for values of (cid:15) where the stationarysolution of LG, (fig. 1 a) seems to reproduce the data.The characteristic time increases but it does not divergebecause of µ (cid:54) = 0 (see fig.1 and ref. [23]). In fig. 1 a) thevariance σ θ of δ is plotted too. From eq.5 this variance is σ θ = k B T / ( K ( (cid:15) − αθ o )) which does not diverge at (cid:15) = 0because µ (cid:54) = 0.Thus although eq. 4 and eq.5 are only a rough approx-imation of the FrTr dynamics, especially at (cid:15) > .
1, weuse them to fix the framework, and because they are veryclose to the theoretical mean field approach to the quenchat critical point discussed in ref. [6]. Thus it is interestingto check the analogies and differences with respect to thegeneral theory.The quench is performed by commuting (cid:15) from an initialvalue (cid:15) i to an (cid:15) f (cid:39) t = 0. As an example we showin fig.2 the time evolution of θ o for a quench from (cid:15) i =0 .
25 to (cid:15) f = 0 .
01. The system is relaxing from its initialequilibrium value towards the new one. We describe herethe time evolution of the statistical properties and we willdiscuss at the end the dependence on the initial and final (cid:15) values. The mean values of the statistical properties areobtained by repeating the quench at least 3000 times. < (cid:101) ( t ) > < (cid:98) (cid:101) ( t ) > / < (cid:98) (cid:101) > equ i a) b) Fig. 2: Quench close to the critical point from (cid:15) i = 0 .
25 to (cid:15) f = 0 .
01. a) Time evolution of the order parameter θ o beforeand after the quench preformed at t = 0. b) Time evolution ofthe variance as a function of time. The variance and the mean θ ( t ) has been obtained by performing 3000 quenches and thenmaking an ensemble average on the quenches at each time. The time evolutions of θ ( t ) and of the variance σ θ ( t )are shown in figs.2a) and b). We see that both quantitiesrelax from the initial to the final equilibrium values ,whichare θ e (cid:39) (cid:15)/α and σ θ ( t ) = k B T / ( K ( (cid:15) − αθ e )), where θ e isthe equilibrium value, which is not exactly 2 (cid:15)/α becauseof the presence of the imperfect bifurcation µ (cid:54) = 0 (seefig.1 and ref. [23]). We see that the fluctuation amplitudeincreases when approaching the critical point.In Fig.3 we plot C ( t, t w ) = < δθ ( t ) δθ ( t w ) > as a functionof t − t w at various t with t > tw >
0. We see that C ( t, t w )develops very long decays when t is increased. In order to (cid:239) t w (s) C ( t,t w ) / C ( t,t ) t=26 st=16 st=5 st=0.8 st=0.35 sequilibrium 0 5 10 1500.20.40.60.81 t (cid:239) t w (s) (cid:114) ( t,t w ) k B T / C ( t,t ) t=26t=16 st=5 st=0.8 st=0.35 s a) b) Fig. 3: The correlations functions (a) and the integrated re-sponses (b) (computed at various fixed times t and 0 < t w < t during the relaxation after the quench) are plotted as a func-tion of t − t w . study FDT we need to measure the response by perturb-ing the systems with a pulse of amplitude δ(cid:15) = 0 . τ p = 1 ms at time t w . As an example, in fig.4a) we plot thetime evolution perturbed at t w = 5 s and in fig.4b) the timeevolution of the difference < ∆ θ ( t ) > = < θ δ ( t ) > − θ o ( t )between the perturbed θ δ ( t ) and the unperturbed θ o ( t ).As it can be seen in eq.5 the amplitude of the pertur-bation is Kδ(cid:15) ( t w ) θ ( t w ) τ p . Thus the impulse responsefunction is R ( t, t w ) = < ∆ θ ( t ) > / ( δ(cid:15) ( t w ) θ ( t w ) τ p ) for t w < t . We repeat the experiments N p times by send-ing at each quench a pulse at a different time t w,i with[ t w, = 0 , ........., t w,Np = 20 s ]. Then the integrated re-sponse is χ ( t, t w,m ) = N t − (cid:88) i = m R ( t, t w,i +1 )( t w, ( i +1) − t w,i ) (7)such that t = t w,N t and χ ( t, t ) = 0 . The measured χ ( t, t w ) is plotted as a function of t − t w forvarious t in fig.3.b.To check the validity of the standard FDT, we plot, infig.5, χ ( t, t w ) k B T /C ( t, t ) as a function of C ( t, t w ) /C ( t, t )at various fixed t with t w varying in the interval 0 ≤ t w ≤ t . In this plot FDT is a straight line of slope -1. We seethat for t relatively short, compared to τ , the FDT is notsatisfied. In fig. 5 we also plot the prediction of ref. [6]for a quench done at (cid:15) = 0 in a Landau-Ginzburg (LG)equation. We see that for short time the behavior is quitedifferent from that of the LG equation confirming that thedynamics is not very well described by this equation. Thebehavior at long time is instead related to the fact thatthe quench is not performed exactly at (cid:15) f = 0.We now apply the MFDT to these data. In order to dothat one has to considers that δθ has a Gaussian distri-bution whose variance is plotted in fig.2. As observablein eq.3 we use O ( x ( t )) = x ( t ) = δθ . Following the for-mulation of the MFDT one has to consider the dynamicsof Ψ( t ) when a small perturbation h is applied at t = 0,therefore by the definitions of χ ( t, t w ) (eq.7) and of O ( t ),we get < δθ ( t ) > h = χ ( t, h because < δθ ( t ) > = 0.Thus at h (cid:54) = 0 (switched on at t = 0) the probabilityp-3saac Theurkauff, Aude Caussarieu, Artyom Petrosyan, Sergio Ciliberto < (cid:101) ( t ) > < (cid:101) ( t ) > h (cid:239) < (cid:101) ( t ) > a) b) t w t w Fig. 4: a) The time evolution has been perturbed at time t w by a short pulse of amplitude δ(cid:15) = 0 . θ to the delta perturbation. C(t,t w )/C(t,t) (cid:114) ( t,t w ) k B T / C ( t,t ) equilibriumLG eq. FDT t= 26 st= 16 st= 5 st= 0.8 st= 0.35 s Fig. 5: FDT plot. The function χ ( t, t w ) k B T /C ( t, t ) is plottedas a function of C ( t, t w ) /C ( t, t ) at various fixed times t and0 < t w < t after the quench. In equilibrium this plot is athe straight line of slope −
1. We see that at short times t the curves strongly deviate from the equilibrium position. Theequilibrium FDT is recovered only for very large t . The reddashed straight line is the prediction [6] for a quench done at (cid:15) = 0 in a Landau-Ginzburg equation. density function for of δθ around the mean is π ( δθ ( t ) , h ) = (cid:112) / (2 πσ ( t )) exp( − ( δθ − χ (0 , t ) h ) (2 σ θ ( t )) ) , (8)where we assume that if h is small enough then the depen-dence of σ θ ( t ) on h can be neglected. . Therefore fromthe expression of π ( δθ ( t ) , h ), the definition of Ψ( δθ ( t ) , h )and of B ( t ) one finds: B ( t ) = δθ ( t ) χ ( t, /σ θ ( t ). Thus eq.3 for this particular choice of variables becomes : χ ( t, t w ) χ ( t,
0) = − C ( t, t w ) χ ( t w , χ ( t, σ θ ( t w ) + 1 (9)All the quantities in eq.9 have been already measured.Thus in fig.6 we plot χ ( t, t w ) /χ ( t, C ( t, t w ) χ ( t w , / ( χ ( t, σ θ ( t w )) for variousfixed t and 0 < t w < t . We see that all the data points arealigned on a straight line of slope − This has been verified experimentally and for the dynamic de-scribed by eq.6 (cid:114) (t w ,0) C(t,t w ) / [ (cid:114) (t,0) C(t w ,t w ) ] (cid:114) ( t,t w ) / (cid:114) ( t, ) GFDTt=26 st=16 st=5 st=0.8 st=0.35 s
Fig. 6: In order to verify MFDT the left hand sideof eq.9, i.e. χ ( t, t w ) /χ ( t, C ( t, t w ) χ ( t w , /χ ( t, σ θ ( t w ). All the data collapse on thestraight line of slope − t . Notice that the re-sponse and the correlations are the same than those used infig.5, but now they have been normalized as prescribed by theMFDT, i.e. eq.9. (cid:114) ’(t w ,0) C(t,t w ) / [ (cid:114) ’(t,0) C(t w ,t w ) ] (cid:114) ’ ( t,t w ) / (cid:114) ’ ( t, ) GFDTt=26 st=16 st=5 st=0.8 st=0.35 s
Fig. 7: MFDT recomputed using in eq.9 the χ (cid:48) ( t, tw ) definedin the text. The left hand side of eq..9 is plotted as a functionof the right hand side. All the data collapse on the straight lineof slope − t in this case too. We clearly see that, in contrast to fig.5 where the stan-dard formulation of FDT is recovered only for very large t , MFDT is verified for all times. As pointed out there isno need in MFDT to use for h the conjugated variable of O ( t ). We can use simply h = δ(cid:15) . In such a case we definethe response function as R (cid:48) ( t, t w ) = < ∆ θ > / ( δ(cid:15) τ p ) andthe χ (cid:48) ( t, t w ) is obtained by inserting R (cid:48) ( t, t w ) in eq.7. TheMFDT computed using in eq.9 χ (cid:48) ( t, t w ) instead of χ ( t, t w )is checked in fig.7 where the left hand side of eq.9 is plot-ted as a function of the right hand side. We see that theMFDT is verified in this case too .All the data presented in this paper correspond to aquench from (cid:15) i (cid:39) .
25 to (cid:15) f (cid:39) .
0, however the mainstatistical features, here described, are independent on thep-4 n the transient Fluctuation Dissipation Theorem after a quench at a critical point starting and final points. The final point influences theduration of the out of equilibrium state, which depends onthe distance from the critical point. The small differencewith the results in ref. [7] is due to a slightly non linearresponse in that reference.As a conclusion in this letter we have applied to aquench a the critical point of Fr´eedericksz transition (Ftr),the Modified Fluctuation Dissipation Theorem for tran-sient, proposed in ref [20]. We find that although theequilibrium FDT is strongly violated the GFDT is verywell satisfied, independently of the chosen response. Itis interesting to point out that the result is interestingbecause although the system presents several differenceswith respect to the LG equation, it is affected by finitesize effects and the quench is not performed exactly at thecritical point the dynamics still presents features at thecritical quenching.We acknowledge useful discussion with G. Verley andD. Lacoste. This work has been supported by the ERCcontract OUTEFLUCOP.