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Recent theories of glasses as out of equilibrium systems

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Recent theories of glasses as out of equilibrium systems ?Jorge Kurchan P.M.M.H.Ecole Sup′e rieure de Physique et Chimie Industrielles,10,rue Vauquelin,75231Paris CEDEX 05,France (February 1,2008)Abstract We discuss a theoretical approach to structural glasses as they happen in real situations.Older ideas based on ‘con?gurational entropy’,on ‘?ctive temperatures’and on Edwards’‘compactivity’are sharpened and uni?ed in an out of equilibrium context.The picture is such that it may be supported or disproved by an experimental test,which we describe.In this note I brie?y review a rather general picture of structural glasses that has emerged in the last dozen years or so,and that to my knowledge has not been summarised elsewhere from quite this point of view.To the extent that this scenario applies,one can make a theory of glasses as they happen ,any reference to ideal in?nitely slow annealings being super?uous.Two sets of older ideas —one based on con?gurational entropy put forward by Kauz-mann,Adam,Gibbs and Di Marzio 1and the other on aging,‘?ctive temperatures’2,and Edwards’‘compactivity’3—are uni?ed and developed in two ways:?They either admit a quantitative de?nition,or,in certain aspects,their inherent am-biguity is uncovered.?Models and approximations are identi?ed within which one can calculate everything analytically,and the scenario holds strictly.This opens the way for systematic im-provements of these schemes,a line that is currently pursued.Solvable cases have indicated that certain features of the older ideas had to be reconsidered,of which we shall see some instances below.1.Aging

Consider a supercooled liquid.In order to probe its dynamics we shall use an autocor-relation function,say,C (t 1?t 2))=<ρ?k (t 1)ρk (t 2)>.In Fig.1we show how this looks

FIG.1.A correlation.

like when plotted in terms of log(t)=log(t1?t2)for two di?erent temperatures T a>T b. There is a fast relaxation which corresponds to rapid motion of the particles‘inside a cage’formed by its neighbours,much as atoms move around their lattice position in a crystal. On much longer timescales,the actual‘cages’will rearrange:these are the‘structural’or ‘alpha’relaxations.While the cage motion is weakly dependent on the temperature,the structural motion slows dramatically as we cool the system.

In Fig.1we also show two arbitrary ways of de?ning an alpha-relaxation time tα(T).If, as we decrease T,we?nd that t′α(T)and tα(T)stay of the same order,then we have only one slow timescale,i.e.we have in all a two-step process.For the moment it seems that this is the situation for structural glasses.Another possibility,encountered in mean-?eld spin glasses,is that on tuning the appropriate parameter any two timescales go to in?nity at di?erent paces,the ratio tα/t′αbecoming larger and larger:this is a multi-timescale situation. This serves to underline the fact that when one talks about several timescales,the concept becomes clear-cut only in an appropriate limit in which they separate completely.This will turn out to be relevant in what follows.

An alternative way to describe the relaxation is by considering the integrated response χ(t1?t2)conjugated to C:in this caseχ(t1?t2)is the compressibility of the mode k with respect to a?eld acting continuously from t2to t1.In equilibrium we have that Tχ(t1?t2)=C(t1?t1)?C(t1?t2),by virtue of the?uctuation-dissipation theorem (FDT),see Fig.2.

In cases in which we have a two-step relaxation,we can describe the slow dynamics of the system by a plot as in Fig.3.Such plots are in general done with the viscosity instead of tαin the y-axis.

At this point the tradition is to embark in a detailed discussion of the form of the curve Fig. 3.Is there a true divergence at some T o>0or not?If so,what is the nature of the equilibrium thermodynamic state below T o?Here I take the point of view that these questions,although of obvious interest,are not the most urgent as far as understanding glasses in real life,and one can make progress without addressing them.Let us here content ourselves with mentioning that there are some glasses(like plastics)that may have a T o>0, while others are expected to have a?nite tαat all temperatures above zero.Note that glasses from this last group,of which‘window’glasses are likely to be an example,are

FIG.2.The associated

response.

described from the equilibrium thermodynamic point of view as being at all temperatures just supercooled liquids.

Consider now a system having at T a a timescale tαof ten minutes,and at T b of an hour. What will happen if,from an equilibrium situation at T a we lower the bath’s temperature to T b in one minute?In the instant t w=0just after the quench,while the rattling of particles inside their cages will quickly adapt to T b,the cages themselves will not have the time to adapt to their new situation,as this needs structural rearrangements taking times t w much larger than an hour.

If we measure the correlations starting from di?erent t w,it is quite clear that we will observe a relaxation as in Fig. 4.Theα-relaxation time grows with t w and eventually saturates at the new value—and this can also be said about the viscosity.In other words, the two-time correlation function becomes waiting-time dependent:C=C(t+t w,t w).In particular,if there is a true divergence in the equilibrium value of tα(T)at some T o>T b, the t w-dependence will stay forever.This is the aging phenomenon:quantities depend on the waiting time for a long(eventually in?nite)time t w>>tα(T b).The situation described above will always be met in a real-life cooling procedure,as sooner or later the cooling time will be too short compared with the equilibrium tα(T).Our problem is to understand the t w<

FIG.4.Aging in

correlations.

on whether tα(T)is actually in?nite or not.

If instead of a correlation function we look at the associated response,then we also see aging.Figure5is a sketch of how this would look.If we are considering a uniform compression(k=0),the vertical arrow shows the beginning of the‘creep’deformation,as studied in the seminal experiments by Struik4.

We have mentioned that when the system is quenched it rapidly adapts its fast,rattling motion to the new temperature.Its structural motion instead remembers the situation in which it lost pace and fell out of equilibrium:thus the idea of using the temperature at which this happened as an extra history-encoding state variable—the‘?ctive temperature’2. This old idea,which had been somewhat abandoned by theorists5?nds a place and a clear de?nition within the framework of this and the following section.

Consider a plot of the y-axis of Fig.1versus the y axis of Fig.2,where time acts as a parameter.The?uctuation-dissipation theorem tells us that for temperatures T a and T b we obtain straight lines with gradients?1/T a and?1/T b,respectively.What happens if we attempt to put together in the same way the aging curves Figs.4and5,using both t and t w as parameters?A sketch of this is shown in Fig. 6.Soon after the quench,all curves superpose in the range(C>q,χ<χo)on a straight line with gradient?1/T b:just as we would have expected to be the case in a system thermalised to the new temperature T b.

FIG.6.E?ective temperature plot.

This is not unreasonable,as it corresponds to the times‘inside a cage’(cfr.Figs.4and5). The surprising result appears in the slow relaxation range(Cχq):all the curves superpose on another straight line with gradient(say)?1/T eff(curve(1)of Fig.6).Then, on a much slower timescale(manyα-relaxations),the straight line slowly drifts(curve(2) in Fig.6)eventually working its way to the dashed line of gradient?1/T b,at which time the system has?nally re-thermalised and aging has?nished.

The parametric response-versus correlation plots like Fig.6reveal at a glance the nature of the aging dynamics.They were introduced in the context of analytically solved models of aging6(see Section3),and were subsequently numerically implemented for realistic models7, for which an analytical solution is not available,with strikingly similar results8.

Let us argue that T eff obtained from Fig.6is a true temperature for the structural motion,in the sense that it has the good thermometry and heat exchange properties.It can be shown that9:

?Any thermometer coupled to an observable in the system and tuned to respond to the timescale of structural relaxation measures exactly the T eff associated with its ?uctuations and response.

?A frequency?lter tuned to the timescale of structural relaxation coupled to an observ-able in the system and to an auxiliary thermal bath of temperature T?will transfer energy from the highest to the lowest of(T eff,T?).

These two results are model-independent.In addition,within the framework of section 3it has been shown that:

?Every pair of observables in the system gives the same?uctuation-dissipation temper-atures T eff at a given timescale.

These three facts characterise the FDT-related e?ective temperature as the only bona ?de temperatures associated to the slow motion.Moreover,there is a lesson we can learn from exactly solvable models:T eff turns out to be of the order of,but not quite equal to the temperature at which the system falls out of equilibrium.We have here an instance of a fact

mentioned at the beginning of the article:the old?ctive temperature idea is promoted to a directly testable concept with thermodynamic(zero-th law)properties,but is also somewhat modi?ed.

2.Ensembles and complexity

Let us discuss now what are the suggestions of Fig.6from the point of view of phase-space distributions.We can picture a two-timescale dynamics as a quasi-equilibrium motion for the fast evolution‘inside the instantaneous cages’(corresponding to the interval of corre-lations C>q)combined with a slow structural motion of the con?guration of the cages.The moment the system falls out of equilibrium,the latter is not able to keep track ergodically. It is thus perfectly general that we should obtain a good equilibrium form in Fig.6for the motion in the interval C>q,and not for C

What is not general,and begs for an explanation,is that the structural,‘cage rearrange-ment’motion behaves,from the point of view of FDT,as thermalised to another temperature, itself possibly decreasing adiabatically.This immediately raises the question of whether there is a new,hidden form of‘ergodicity’for the structural motion,allowing us to construct a statistical mechanics ensemble for the slow motion at each energy level.

One such construction is better introduced in the context in which it was originally formulated by Edwards3.Consider a granular medium that is compactifying under gentle ‘tapping’10.After each tap,the particles settle into a blocked con?guration.

Now,Edwards proposed the extremely strong hypothesis that the blocked con?gurations reached dynamically at each density are the typical ones:at any given time,all macroscopic observables can be obtained by averaging their value over all blocked con?gurations of the appropiate density.This is an implicit de?nition of an ensemble for the dynamics.Once accepted,one immediately is led to de?ning an entropy(in the glass language a‘con?gura-tional entropy’),as the logarithm of the number of blocked con?gurations of given volume —and energy,in the case of soft particles.Di?erentiating the con?gurational entropy with respect to the volume we obtain a‘compactivity’,and with respect to energy an inverse con?gurational temperature.

This example of a granular medium is particularly simple because by looking at the system in repose some time after each tap,one concentrates on the slow rearrangements without having to deal with a superposition of simultaneous fast and slow motion.

Within the mean-?eld/mode-coupling scenario to be discussed in Sect.3,it is possible to calculate averages over all the energy minima(the relevant blocked con?gurations for a zero temperature dynamics),as well as the their number at each energy level.The remarkable result is that the con?gurational temperature thus calculated`a la Edwards and the FDT temperature T eff obtained after a deep quench are found to coincide exactly.Furthermore, the out of equilibrium macroscopic observables are correctly given by?at averages over the blocked regions11–14,19.

These models teach us something else:for quenches to non-zero temperature,it is not the energy minima that have to be computed in order to reproduce the dynamic result,but rather the number of metastable states.This immediately poses a problem:while in the

mean-?eld models a metastable state is unambiguously de?ned as a region in phase space from which the system never escapes,for realistic,?nite-dimensional models at T>0all metastable states have a?nite lifetime.In other words,there is no absolute separation between cage and structural motion and there is an essential ambiguity in the concept of metastable state.This ambiguity carries over to the concept of con?gurational entropy,and hence to the con?gurational temperature,as the number of states we count depends on the minimal lifetime we expect from a state to call it a‘state’14,15.

This is a rather rapidly developing and not fully settled?eld,several alternatives have been suggested but many questions remain open.One strategy that is easy to implement numerically is to identify as a state the Gibbs-Boltzmann distribution restricted to the set of con?gurations in the basin of attraction of each energy minimum—the‘inherent structure’construction16.Though this identi?cation is not without problems of principle17, the results in connection with the aging dynamics are encouraging18,19.Recently,a more direct comparison between compaction results and Edwards ensembles in a schematic three dimensional model has yielded very good agreement20.

Another strategy is to use a direct de?nition of‘states with a given lifetime’14,https://www.docsj.com/doc/497632650.html,ing a construction of Gaveau and Schulman21,one can compute averages over states having lifetimes of a any preassigned value15.Yet another possibility is to directly construct a phe-nomenological two-temperature thermodynamics,without invoking a statistical mechanical ensemble13.

Let us conclude by remarking that the role played by con?gurational entropy here is stronger than in the Kauzmann picture of an ideal glass transition,since it involves the out of equilibrium dynamics,and not only an in?nitely slow cooling.This has an advantage with respect to robustness,since the present scenario would be relevant in situations in which a thermodynamic glass transition either does not exist or is unobservable at experimentally accesible times.

At any rate,let us insist again,these ideas of generalised ensembles can only be taken se-riously to the extent that the existence of FDT temperatures,independent of the observable at each timescale,is veri?ed experimentally.

3.Methods and Models

Within a set of approximations and models the scenario discussed above holds rigorously. While these approximations are far from perfect,they have suggested tests that have been performed on realistic systems,with a considerable degree of success.

The paradigmatic and best studied model of supercooled liquids with two-step relaxations is the mode-coupling theory(MCT)22.These are approximate equations for the two-point dynamical correlation functions,which can be obtained through addition of a partial(though in?nite)set of terms in the perturbative expansion of the exact dynamics28.

An apparently di?erent point of view was developed by Kauzmann,and Adam,Gibbs and DiMarzio1,who?rst pictured the total entropy as composed of‘rapid’and‘con?gurational’degrees of freedom,and the‘ideal glass’transition as the temperature at which the system runs out of con?gurational entropy.

In a remarkably insightful set of papers,Kirkpatrick,Thirumalai and Wolynes23showed that both approaches can be seen as aspects of the same theory.To do this,they?rst noted that MCT is the exact solution of the two-time correlations of the high temperature phase of a family of fully connected models having spin glass-like Hamiltonians but with multi-spin interactions.(This is why we referred above to the‘mean-?eld/mode-coupling’models).They then showed that the partition function of these models ignores completely the dynamic transition at T MCT,but has an equilibrium transition at a lower temperature T K.They explained this fact by showing that between T MCT and T K the phase-space is composed of exponentially many,mutually inaccesible states that the Gibbs measure confuses into a single,large,deceivingly liquid-like state.The equilibrium transition at T K happens because at just that temperature the number of states contributing to the measure becomes of order one:precisely Kauzmann’s idea with the logarithm of the number of states playing the role of con?gurational entropy.The paradigm of this form of transition is Derrida’s Random Energy Model24.

A natural question that arises immediately is,what does a system satisfying exactly mode coupling equations for T>T MCT do if quenched below the dynamic glass transition T

As mentioned before,one can study the free energy minima26of the corresponding model:?at averages over metastable states of the appropiate energy give the correct out of equilib-rium values of macroscopic observables11–14.Furthermore,the con?gurational temperature obtained as the logarithmic derivative of the number of states with respect to the free en-ergy turns out to give exactly the dynamically de?ned1/T eff.Hence,within these models Edwards’‘compactivity’ideas are also strictly realised.

Now,it is a well known fact that the mode coupling transition becomes a crossover in real life22.This also has an explanation when the nature of the models is considered. The transition at T MCT relies on the existence of fully stable excited states,but these will eventually decay in any?nite-dimensional model through nucleation23.On the other hand, while the equilibrium transition at T K may or may not in a realistic model become a crossover as well,there is no a priori reason for this to be the case.

There are several strategies starting from these considerations.Given the limitations of the models considered,it is?rst mandatory to check directly with a simulation of a realistic model(say,a Lennard-Jones liquid)whether a two-temperature aging scenario remains valid in the presence of activated processes that may eventually lead the system to thermalisation. The evidence seems to be positive7.A much harder numerical check concerns the nature of the equilibrium phase at low temperatures27,using such small samples as one can thermalise.

From the analytic point of view,there are various di?erent approaches being pursued. Perhaps the most basic is to try to understand whether there is a generic reason that there should be only one FDT temperature for all observables per widely separated timescale —independently from the approximation scheme in which this scenario was?rst(perhaps accidentally)found.

On a more practical line,there are several schemes for improving the approximations for the dynamics by considering diagrammatic ressumations28,although dealing with activated processes is notoriously hard.One can also construct better approximations for the equilib-

rium calculations.This approach,advocated by Mezard and Parisi29,may be motivated by the following consideration:Clearly,an equilibrium calculation is relevant at such late times that energy,volume,and all macroscopic observables can be considered to have relaxed to their thermodynamic equilibrium value.If a glassy system has a transition below which tαdiverges strictly,even if all quantities have reached essentially their equilibrium value their T eff might still never reach the bath’s temperature:It turns out that in such cases we can calculate the limiting t→∞form of Fig.6using equilibrium techniques30,and the result is directly related to Parisi’s order parameter.(Obviously,within this approach,‘strong’glasses having T o=0are described as liquids).This rather surprising connection between equilibrium and the last stages of out of equilibrium relaxation involves certain general as-sumptions,and in the speci?c case of structural glasses it requires the exclusion from the equilibrium calculation of all con?gurations having a degree of crystallisation.

4.Experiments

As remarked in several places above,the main observable feature of the present scenario for glasses is the existence of well separated timescales and a single?uctuation-dissipation temperature per timescale,shared by all observables.For example,the relation between spontaneous di?usion and mobility in a given time interval should be the same for di?erent kinds of particles within the glass,or between rotational and translational degrees of freedom of the same particle.This relation de?nes a temperature that only coincides with the bath’s temperature for the fastest relaxations.

The?uctuation-dissipation experiments we have so far31are comparisons of the dielec-tric susceptibility and polarisation noise in low-temperature glycerol and in Laponite gel, respectively.They show clearly that the low frequency ratio between noise amplitude and susceptibility stays di?erent from the bath temperature after very long times.The analysis of the data is somewhat complicated by the fact that,for experimental reasons,these results are given in the frequency domain,and it is di?cult to unravel the fast and slow timescales.

Ideally,one should go on to measure the time correlation between several?uctuating quantities and compare these with the associated responses—and check that the propor-tionality factor is the same at equal timescales for all observables.If,and when,this is not the case,the scenario discussed in this paper does not apply.

The measurements are always complicated by the waiting-time dependence of correlations and responses.An approach that may be useful in practice is to study glasses that are gently driven,for example by shearing,and to use the surprising fact that in such conditions aging can be stopped without changing the response versus correlation curve12.

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8See also Ref.13for even simpler,instructive models.

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12Kurchan J.,Rheology and how to stop aging,in‘Jamming and Rheol-ogy:Constrained Dynamics on Microscopic and Macroscopic Scales’(1997), https://www.docsj.com/doc/497632650.html,/online/jamming2/,and Edwards,S.F.,Liu,A.and Nagel,S.R. Eds.,to be published,cond-mat9812347.

13See Th.M.Nieuwenhuizen,Thermodynamic picture of the glassy state gained from exactly solvable models,Phys.Rev.E61(2000)267,and references therein.

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Sasstry S.,Debenedetti P.G.and Stillinger F.H.,Signatures of distinct dynamical regimes in the energy landscape of a glass-forming liquid,Nature393,(1998)554-557,

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Sciortino F.,Kob W.,and Tartaglia P.,The phase space structure of supercooled liquids: Evidence of a dynamical critical temperature,Phys.Rev.Lett.83(1999)3214.

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26Rieger H,Number of solutions of the TAP equations for p–spin interaction spin glasses, Phys.Rev.B46,(1992)14665;Crisanti A.and Sommers H-J,On the TAP approach to the spherical p-spin SG model,J.Phys.I(France)5,(1995)805.

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Unit 9 How to Grow Old 课文翻译

Unit 9 How to Grow Old Bertrand A. Russell 1. In spite of the title, this article will really be on how not to grow old, which, at my time of life, is a much more important subject. My first advice would be, to choose your ancestors carefully. Although both my parents died young, I have done well in this respect as regards my other ancestors. My maternal grandfather, it is true, was cut off in the flower of his youth at the age of sixty-seven, but my other three grandparents all lived to be over eighty. Of remoter ancestors I can only discover one who did not live to a great age, and he died of a disease which is now rare, namely, having his head cut off. A great-grandmother of mine, who was a friend of Gibbon, lived to the age of ninety-two, and to her last day remained a terror to all her descendants. My maternal grandmother, after having nine children who survived, one who died in infancy, and many miscarriages, as soon as she became a widow devoted herself to women’s higher education. She was one of the founders of Girton College, and worked hard at opening the medical profession to women. She used to relate how she met in Italy an elderly gentleman who was looking very sad. She inquired the cause of his melancholy and he said that he had just parted fro m his two grandchildren. “Good gracious,” she exclaimed, “I have seventy-two grandchildren, and if I were sad each time I parted from one of them, I should have a dismal existence!” “Madre snaturale,” he replied. But speaking as one of the seventy-two, I prefer her recipe. After the age of eighty she found she had some difficulty in getting to sleep, so she habitually spent the hours from midnight to 3 a.m. in reading popular science. I do not believe that she ever had time to notice that she was growing old. This, I think, is the proper recipe for remaining young. If you have wide and keen interests and activities in which you can still be effective, you will have no reason to think about the merely statistical fact of the number of years you have already lived, still less of the probable brevity of your future. 2. As regards health, I have nothing useful to say since I have little experience of illness. I eat and drink whatever I like, and sleep when I cannot keep awake. I never do anything whatever on the ground that it is good for health, though in actual fact the things I like doing are mostly wholesome. 3. Psychologically there are two dangers to be guarded against in old age. One of these is undue absorption in the past. It does not do to live in memories, in regrets for the good old days, or in sadness about friends who are dead. One’s thoughts must be directed to

大学英语精读5_课后答案

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