The Ratchet effect in an ageing glass
Giacomo Gradenigo, Alessandro Sarracino, Dario Villamaina, Tomas Grigera, Andrea Puglisi
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Ratchet effect in an aging glass
Giacomo Gradenigo
CNR-ISC and Dipartimento di Fisica, Universit`a Sapienza - p.le A. Moro 2, 00185,Roma, Italy
Alessandro Sarracino
CNR-ISC and Dipartimento di Fisica, Universit`a Sapienza - p.le A. Moro 2, 00185,Roma, Italy
Dario Villamaina
CNR-ISC and Dipartimento di Fisica, Universit`a Sapienza - p.le A. Moro 2, 00185,Roma, Italy
Tom´as S. Grigera
Instituto de Investigaciones Fisicoqu´ımicas Te´oricas y Aplicadas (INIFTA), andDepartamento de F´ısica, Facultad de Ciencias Exactas, Universidad Nacional de LaPlata, and CCT La Plata, Consejo Nacional de Investigaciones Cient´ıficas yT´ecnicas, c.c. 16, suc. 4, 1900 La Plata, Argentina
Andrea Puglisi
CNR-ISC and Dipartimento di Fisica, Universit`a Sapienza - p.le A. Moro 2, 00185,Roma, ItalyE-mail: [email protected], [email protected]
PACS numbers: 05.60.-k,61.43.Fs,05.70.Ln
Abstract.
We study the dynamics of an asymmetric intruder in a glass-formermodel. At equilibrium, the intruder diffuses with average zero velocity. After anabrupt quench to T deeply under the mode-coupling temperature, a net average driftis observed, steady on a logarithmic time-scale. The phenomenon is well reproducedin an asymmetric version of the Sinai model. The subvelocity of the intruder growswith T eff /T , where T eff is defined by the response-correlation ratio, corresponding toa general behavior of thermal ratchets when in contact with two thermal reservoirs. atchet effect in an aging glass T eff . Here we show that the energy flowing from slowto fast modes can be rectified to produce a directed motion. The properties of theobserved current characterize the non-equilibrium behavior of the glass. In particulara striking monotonic relation is observed between the ratchet sub-velocity and T eff /T .The experience with other kinetic models of ratchets [10, 11] teaches that - when in thepresence of a temperature unbalance - the heat flux also governs the ratchet velocity.These observations suggest the conjecture that one can obtain the effective temperatureby replacing the measure of linear response and correlations with the simple measure ofan average current. In what follows we test the reliability of this procedure.The first of the numerical experiments proposed here involves the 3D soft-spheresmodel, which is a well known fragile glass-former [16, 17, 18, 19]. The thermodynamicproperties of soft-spheres are controlled by a single parameter Γ = ρT − / , whichcombines the temperature T and the density ρ of the system, with ρ = N/V σ and σ the radius of the effective one-component fluid. We study a binary mixture (50:50) withradii ratio, σ /σ = 1 .
2. Particles interact via the soft potential U ( r ) = [( σ i + σ j ) /r ] atchet effect in an aging glass N attempted MC moves). The model has adynamical crossover at a mode-coupling temperature T MC corresponding to the effectivecoupling Γ MC = 1 .
45 [16]. The ratchet is formed by the asymmetric interaction of asingle particle, the intruder, with all the other particles of the system. Denoting with x i , i > x coordinate of the i -th particle, and with x the abscissa of the intruder,we choose U ( r , r i ) = ( U ( | r − r i | ) if x i < x ,εU ( | r − r i | ) otherwise, (1)with ε = 0 .
02. The spatial symmetry along the x -axis is therefore broken, fulfilling oneof the requirement to get a ratchet device. Let us stress that here, in analogy withthe model proposed in [20], the asymmetry is inherent to a single object, the intruder,which is embedded in a symmetric environment, at variance with flashing and rocketratchets [5]. Moreover, by exploiting condition (1), we are able to build an intruder withan intrinsic asymmetry, even if it is a point-like particle.We equilibrate configurations embedding an asymmetric particle at a hightemperature T liq ≫ T MC . At T liq the system has simple liquid behaviour with fastexponential relaxation. The dynamics of the asymmetric intruder is then studied bothalong equilibrium trajectories and after quenches to different temperatures T < T MC .The average displacements of the intruder along different axes is compared at andoutside equilibrium, namely both h ∆ x ( t ) i and h ∆ y ( t ) i are studied, with ∆ x ( t ) = x ( t ) − x ( t ). In particular t is the time elapsed since the instantaneous quench, namelywe take t = 0. The averages denoted by h . . . i are realized considering 5000 initialconfigurations with a single intruder, equilibrated at T liq = 4 . T MC , each followedby an independent thermal history. Quenches are done to T = 0 . T MC , 0 . T MC ,0 . T MC , 0 . T MC .Let us consider first the effect of the asymmetric interaction on the arrangementof particles around the intruder. We study the pair distribution function of neighborsto the right and to the left of the intruder, g RL ( r | δx ≷
0) = P j | δx j ≷ δ ( r j − r ), with r j the distance between the intruder and the j -th particle, in equilibrium or at a fixedelapsed time after the quench (see Fig. 1, left panels). Right neighbors stay closer tothe intruder due to the reduced repulsion. This asymmetric clustering of neighborsis slightly enhanced during aging, but at this level of analysis there is no qualitativedifferences between the equilibrium and off-equilibrium regimes.Dynamical measurements, on the contrary, show important differences betweenthe two regimes. Let us consider the mean square displacement around the averageposition at time t of the intruder particle (Fig. 1 right): it is linear with time at hightemperatures, while it is logarithmic after a quench, h ∆ x ( t ) i − h ∆ x ( t ) i ∼ log t .This behavior, typical of activated dynamics in a rough potential [21, 17, 22], isalso observed for host particles. The behavior of the average displacement of theintruder, h ∆ x ( t ) i , is more striking. At equilibrium (Fig. 2, black circles) there is no atchet effect in an aging glass g R (r)g L (r)g(r) r
024 0.11100 1000 10000 t < ∆ x (t)> ~ t< ∆ x (t)> ~ log(t) e qu ili b r i u m a g i ng Figure 1.
Left panels: pair distribution function centered on the intruder, g RL ( r | δx ≷ g ( r ) for symmetric interactions.Right panels: h ∆ x ( t ) i at equilibrium (top) and after the quench (bottom).net displacement: this is because parity along x axes is broken whereas time reversalsymmetry is preserved. The same happens to h ∆ y ( t ) i (green diamonds) during aging:in this case only (macroscopic) time reversal symmetry is broken. But parity and time-reversal symmetry are both violated for h ∆ x ( t ) i after the quench (red squares), andin this case a net average drift is found (Fig. 2), linear on a logarithmic timescale, h ∆ x ( t ) i ∼ τ with τ = log / t . The logarithmic timescale again points to a non-equilibrium phenomenon ruled by activated events. Moreover, the simple scaling relation h ∆ x ( t ) i ∼ p h ∆ x ( t ) i − h ∆ x ( t ) i between the displacement along the x axis and them.s.d. around the average position is observed, in analogy with the Sinai model discussedbelow.Our first exploration of ratcheting effects in glassy models leaves open one importantquestion, namely the dependence of the drift on the parameters (e.g. temperature) ofthe system. In order to sketch a preliminary answer, we need to extract from the curve h ∆ x ( t ) i a synthetic observable. The finding of a logarithmic timescale h ∆ x ( t ) i ∼ τ with τ ∼ log / t suggests to define an average sub-velocity as [23]: v sub ( t, t w ) = h ∆ x ( t ) i − h ∆ x ( t w ) i δτ = h x ( t ) i − h x ( t w ) i δτ , (2)with δτ = log / t − log / t w . This average subvelocity depends in general on boththe running time t and the waiting time t w elapsed since the quench. More precisely,considering fig. 2, by fixing t w and t we choose the time lag where the slope of thecurve h ∆ x ( t ) i is measured. Clearly, the instantaneous sub-velocity only depends on t w atchet effect in an aging glass log (t) < ∆ x ( t ) > asym. equil.asym. agingsymm. aging ~ log (t) Figure 2.
Average intruder displacement: h ∆ x ( t ) i for equilibrium trajectories (blackcircles), and h ∆ x ( t ) i and h ∆ y ( t ) i after a quench far below T MC ( T = 0 . T MC ) (redsquares and green diamonds respectively). The displacement h ∆ x ( t ) i is measured inunits of the average inter-particle distance, as obtained from the position of the firstpeak in the pair distribution function g ( r ) (see also fig. 1). and corresponds to a slowly decaying velocity v ( t w ) = d h ∆ x ( t w ) i /dt w ∼ /t w (withlogarithmic corrections). For large enough waiting times we can define the “orderparameter” v sub , namely we find v sub ( t w ) ∼ const, and then we probe its dependenceupon the external parameters. Here we focus on the quench temperature T , drawinga connection between the drift of the glassy ratchet, which is a pure non-equilibriumeffect, and the more customary equilibrium-like descriptions of aging media in terms ofeffective temperatures. In the right panel of Fig. 3 we show the drift of the asymmetricintruder for four quench temperatures: T = 0 . T MC , 0 . T MC , 0 . T MC , 0 . T MC .We observe that when decreasing the quench temperature the drift grows in intensity,i.e. the sub-velocity increases. This somehow is a stronger evidence that the ratchetdrift cannot be described as an equilibrium-like effect, e.g. trying to connect averagekinetic or potential energy with a mobility: one would expect that at lower temperatureseverything is slowed down, whereas we find that the ratchet drift is enhanced.Typical examples of ratchets in (ideally) statistically stationary configurations areobtained by coupling the system with two or more reservoirs. It is the existence ofdifferent temperatures within the same system which allows the production of workwithout violations of the second principle of thermodynamics. But what is the secondtemperature in a glassy system? According to the well-established description of theaging regime of glasses [13], it is the effective temperature defined as the violationfactor of the fluctuation-dissipation theorem (FDT). The quench of a fragile glass, forinstance our the soft spheres model, below its mode-coupling temperature produces atchet effect in an aging glass X ( t, t w ) < T χ ( t, t w ) = X ( t, t w )[ C ( t, t ) − C ( t, t w )], with χ ( t, t w ) the integratedresponse and C ( t, t w ) the correlation, yields the definition of the effective temperature T eff ( t, t w ) = T /X ( t, t w ). The last is usually higher than the bath temperature T eff ( t, t w ) > T and is understood as the temperature of slow, still not equilibrated,modes. Clearly, the ratio T eff /T may be regarded as the parameter which tunes nonequilibrium effects and we study here how the glassy ratchet drift depends on it. Weobtain T eff from the parametric plot of C ( t, t w ), taken as the self-intermediate scatteringfunction, versus the integrated response T χ ( t, t w ), measured according to the field-freemethod of [24]. The parametric plot T χ ( t, t w ) vs C ( t, t w ) is shown in the left panel offig. 3 for different temperatures. C(t,t w ) T χ ( t , t w ) T=0.67 T MC T=0.53 T MC T=0.42 T MC T=0.31 T MC log (t) < ∆ x ( t ) > MC MC MC MC T eff / T v s ub Figure 3.
Left panel: parametric plot
T χ ( t, t w ) vs C ( t, t w ). Right panel: asymmetricintruder drift for different quench temperatures; linear fits yield sublinear velocities.Inset: v sub vs T eff /T . In both panels colors of data correspond to different effectivetempeartures: T eff = 1 . T MC (black), 1 . T MC (red), 1 . T MC (green) and1 . T MC (blue). At large times v sub exhibits finite-size effects, namely the drift saturates at a timethat can be increased by increasing the size of the simulation box. This is why wecompute v sub by fitting data only in a relative early time window, namely 10 < t w < .Accordingly, the measure of X ( t, t w ) is obtained from the parametric plot T χ ( t, t w ) vs C ( t, t w ) with t = 10 and t w ranging from 10 to 10 , namely the early aging regime isconsidered. The inset of the right panel of fig. 3 shows the behavior of v sub vs T eff /T ,revealing that the subvelocity increases when T eff /T is increased. Namely the intensityof the ratchet effect, traced in the measure of v sub , grows as the distance from equilibriumis increased.Let us remark here that the mechanism governing our ratchet differs from that of aflashing ratchet [3]. Irreversibility is achieved by choosing an initial condition which isout-of-equilibrium with respect to the bath temperature T liq = T . The extreme slownessof an aging glass prevents the system from thermalizing, so that energy continuouslyflows from fast to slow modes, supplying power to the Brownian ratchet. At variancewith a flashing ratchet, here there is no external time-dependent modulation of a atchet effect in an aging glass independent potential: as a enlightening example, in the following we discuss the Sinaimodel, where it is made clear how irreversibility comes solely from the choice of initialconditions. Sinai model is one of the simplest describing the diffusion of a single particlethrough a random correlated potential [25]. Its long-time dynamics is ruled by activatedevents and is characterized by a logarithmic time-scale: for this reasons it appears tobe a well fitted candidate to reproduce the previous experiment in a more controlledsetup. In the original Sinai model the random potential is built from a random-walk ofthe force on a 1 d lattice. The force F i at each lattice site i is an independent identicallydistributed random variable extracted from a zero mean symmetric distribution p ( F ).The potential on a lattice site n is given by U ( n ) = P ni =1 F i . Because R dF p ( F ) F = 0the average force experienced by a particle is zero. The potential excursion betweentwo sites grows like h| U ( i ) − U ( j ) |i ∼ | i − j | / . The above relation, together with theexpression of the typical time needed to jump a barrier, ∼ exp( β ∆ U ), yields the long-time scaling of the mean square displacement of: h ∆ l ( t ) i ∼ log ( t ), with l ( t ) denotingthe site occupied by the particle at time t and ∆ l ( t ) = l ( t ) − l (0). A logarithmic timescale for the growth of domain size is quite ubiquitous in the low temperature regime ofglassy systems, where activated processes dominate [26]. The time-reversal symmetrybreaking is inherent in the Sinai model: by averaging over initial positions extractedfrom a flat distribution one reproduces an initial infinite temperature T liq = ∞ , while thequench to a glassy phase is reproduced evolving the system at temperature T ≪ √ L where L is the linear size of the system. We propose a spatially asymmetric versionof the Sinai model, where the symmetric force distribution p ( F ) is replaced with anasymmetric one ˜ p ( F ), in analogy with the asymmetric potential of the glassy ratchet,in order to find a non-equilibrium drift. Namely the distribution ˜ p ( F ) is such that˜ p ( F ) = ˜ p ( − F ), but still R dF ˜ p ( F ) F = 0. In particular, the random force is obtainedaccording to the following procedure: at each site the sign of a random variable f i ischosen with probability 1 / e −| f i | /λ − or e −| f i | /λ + , with λ + > λ − . In the present simulation λ + = 1 and λ − = 0 . U ( n ) = P ni =1 F i , with F i = f i − ( λ + − λ − ) / h F i = 0.The results of the simulations of this model are shown in Fig. 4: they clearly showa behavior in striking similarity with those of Fig. 2 for the glass-former model: a driftis observed only when both time reversal and spatial symmetry are broken. Indeed,as demonstrated by the black and red solid curves, it is sufficient that one of the twosymmetries is restored to have a zero drift. In particular, to obtain an “equilibrium”dynamics in this model it is sufficient to distribute all initial positions of the simulationaccording to a distribution ∼ exp[ − βU ( i )]: in this case, even with the asymmetricdistribution of forces described above, the average drift is zero. The large time behaviorof the drift is compatible with a squared logarithm, h ∆ l ( t ) i ∼ log ( t ) ∼ p h ∆ l ( t ) i , in atchet effect in an aging glass log (t) < ∆ l ( t ) > asym. equil.asym. non-equil.symm. non-equil. L /T v s ub ~log (t) Figure 4.
Main: Average displacement in Sinai model. Black circles, diffusionin asymmetric potential at equilibrium; green diamonds, diffusion during aging insymmetric potential (quench from T = ∞ ); red squares, diffusion during aging inasymmetric potential. Inset: the sublinear drift velocity v sub grows with √ L/T . agreement with what observed previously for the glassy ratchet.The out-of-equilibrium dynamics of the Sinai model closely reproduces theseobservations: in the inset of Fig. 4 shows the sub-velocity as a function of √ L/T , with T ≪ L / the quench temperature and L the size of the linear chain, which fixes themost relevant energy scale of the model, i.e. the maximum depth of potential minima.The inset of Fig. 4 shows that v sub grows monotonously with √ L/T : when the lastquantity is increased particles condensate on the bottom of the deepest valleys. Themonotonic increase of v sub with √ L/T signals that also for the Sinai model the largerthe distance from equilibrium the larger the velocity of the drift.In conclusion, through numerical simulations in different models and differentchoices of the quench temperature, always chosen in the deep slowly relaxing regime,we have given evidences of the existence of a “glassy ratchet” phenomenon. The driftvelocity slowly decays in time and can be appreciably different from zero for at leastthree orders of magnitude in time. The overall intensity of the drift, measured in termsof a “sub-velocity”, is monotonically increasing with the distance from equilibrium, i.e.with the difference between the quench and effective temperatures. This observationsupports the idea of regarding the ratchet drift as a “non-equilibrium thermometer”: itcan be used as a device capable to say how far is a system from equilibrium. Nevertheless,for such a thermometer to be effective, a more accurate calibration procedure should becarried on. Namely, one should verify that v sub is a function of T eff and T only witha small number of parameters. In this case the calibration of our thermometer would atchet effect in an aging glass T eff to fix those parameters.As an experimental realization of our glassy ratchet, one should consider a particlewith anisotropic interaction with the surrounding ones with a small magnetic dipoleplaced on it, orthogonal to the asymmetry axis. Then, switching on a constant magneticfield the orientation of the two faces of the “Janus” [27] particle is preserved with respectto a fixed reference frame, so that the spatial symmetry of the interaction is broken.Recent theoretical and experimental advances in the study of functionalized or “patchy”particles [28, 29] promise an experimental verification of our hypothesis in the nearfuture. Acknowledgments
We thank A.Baldassarri and A.Vulpiani for many useful discussions. The work ofGG, AS, DV and AP is supported by the “Granular-Chaos” project, funded by ItalianMIUR under the grant number RBID08Z9JE. TSG was partially supported by ANPCyT(Argentina).
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