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J. Phys. Chem. B 2001, 105, 11854-11858
Energy Landscapes Composed of Continuous Intertwining Equipotential Ribbons† Daniel Kivelson* Department of Chemistry and Biochemistry, UniVersity of California, Los Angeles, California 90095
Gilles Tarjus Laboratoire de Physique Theorique des Liquides, UniVersite Pierre et Marie Curie, 4 place Jussieu, Paris 75005, France ReceiVed: May 25, 2001; In Final Form: July 25, 2001
We develop a picture of energy landscapes which for large enough systems consists of continuous (unsegmented) intertwining equipotential ribbons. We discuss the significance of this construct, which differs from conventional topographical views of energy landscapes. Although analyses based upon energy landscapes are unlikely in and of themselves to lead to quantitative theories, they may lead to useful insights.
Landscape Composed of Continuous Equipotential Ribbons The energy landscape corresponding to a set of N interacting particles is the surface described by the potential energy in a space specified by the coordinates of all N particles. The energy landscape is a very complex structure, but nonetheless the concept has been useful in gaining insights into both thermodynamic and dynamical phenomena; in recent years it has been used in studying supercooled liquids,1-8 glasses,9-13 and protein folding.14,15 The picture underlying most applications of energy landscapes envisages them much like commonly encountered physical landscapes in three dimensions, i.e., a system of maxima, minima, and saddle points. We suggest, however, that the energy landscape should best be described by continuous (unsegmented), intertwined equipotential ribbons, which we shall denote “equi-ribbons”. In this picture there need not be local minima, and one can move from one point to another on an equi-ribbon without ever crossing other equi-ribbons; thus, if the system happens to be on a low-energy equi-ribbon, it can move to any other point on the equi-ribbon without ever crossing energy barriers. See Figure 1a. If, however, one projects the energy landscape onto a configuration subspace, the equi-ribbons are then segmented, i.e., cut into disconnected equi-ribbon segments; in such subspaces the minima are short equipotential valleys, and as the number of dimensions is reduced, these valleys become more and more constricted, i.e., more and more pointlike. See Figure 1b. Exceptionally, the very lowest energy regions on the energy landscape, those associated with the ground states of systems with long-range order (such as perfect crystals at 0 K), may correspond to disconnected points that do not lie on equi-ribbons. We provide justification for this view of the energy landscape, which we believe to be exact for large enough systems, but first we discuss its implications. The energy landscape is often envisaged in terms of separated deep “basins” with corrugated walls, i.e., walls with their own sequences of shallow minima.2 The basin minima are said to specify stable structures, known as “configurations” or “inherent structures,” and the basins form a complete partition of the configurational space.2 For an energy landscape composed of †
Part of the special issue “Howard Reiss Festschrift”.
Figure 1. (a, top) 2D energy landscape. Darker shading represents higher potential energy. Regions around A and B are two of the three equi-lakes that are connected by means of both equi-streams and activated paths. In this 2D plot we have drawn the lowest energy equiribbon as continuous (unsegmented); in 3D it would be easy to draw the two lowest energy ribbons as continuous, and the higher the dimensionality, the more continuous equi-ribbons that can be formed. (b, bottom) 1D slice along the AB activated path of the 2D landscape in (a).
equi-ribbons this picture must be modified because there need not be true local minima. This means that on the manydimensional energy landscape (in the thermodynamic limit) there exist low-energy equipotential routes that skirt around the energy hills or barriers. However, the equipotential route connecting two given points on the equipotential may sometimes
10.1021/jp0120209 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/04/2001
Energy Landscapes Composed of Equi-Ribbons be short and broad, while in other cases it may be long, tortuous, and narrow. Regions on the energy landscape in which a given equi-ribbon is broad (by which we mean it takes up a significant volume of configuration space) can be termed “equi-lakes”, and the connecting regions where the equi-ribbons are narrow can be termed “connecting equi-streams”. See Figure 1a. When the connecting equi-steams are very narrow, the equi-lakes may be quite distinct and localized, in which case they are similar to inherent structures, and the energy landscape is then similar to that described in terms of valleys and basins, saddle points, and maxima. But when the connecting equi-streams are significant (for whatever reason), then the concept of inherent structures loses some of its usefulness; furthermore, we propose that, with the exceptions indicated above, connecting equi-streams always exist, be they relevant or not. At constant volume the energy landscape is independent of temperature (T), and for this reason we should always take the volume to be constant, despite the fact that experiments are ordinarily carried out at constant pressure. (The concept of an energy landscape, or more precisely a potential energy landscape, is not to be confused with a free energy landscape, which is T-dependent.) If near-neighbor intermolecular interactions are of order , one might expect those parts of the landscape associated with liquids to have characteristic roughness of order , where > kT, although excluded-volume effects can give rise to very high blips. (If kT . , the system is a gas.) One can envisage the liquid and crystal as occupying widely different regions on the landscape: the liquid occupies a huge, high-energy region and the crystal a smaller low-energy one. The energy separation between regions (identified with a first-order phase transition) is on the order of N, where N is the number of particles, and the relative sizes of the regions are associated with their relative areas on the landscape. Although all intermediate potential energies between the regions will be represented, the theory of phase transitions teaches us that there will be vanishingly few of these, ergodicity being broken at the liquid-crystal phase transition. (Generally speaking, ergodicity-breaking in the equiribbon picture implies that the probability of finding a ribbon joining the regions of the configuration space associated with the different phases vanishes in the thermodynamic limit.) Thus, the equi-ribbons of interest lie within one phase or the other, and when referring to “low-energy equipotential ribbons” we shall mean low energy relatiVe to the appropriate “floor”. For the crystal at low temperature only the low-energy regions corresponding to long-range order are populated, and for such systems there are local minima not connected by equi-ribbons. Thermodynamics and Dynamics The concept of phase space, and of configuration space as well, can be of special interest in classical statistical mechanics where the position of a system is represented by a point, and the properties of an ensemble are represented by the distribution of points over the space. Thermodynamic properties can then be represented as averages over the distribution of points (systems) in an equilibrated ensemble, the equilibrated ensemble being one in which a system has the opportunity of visiting all allowed regions of phase space (and consequently of configuration space); this implies ergodicity, i.e., that there be sufficient time for a system to visit all (actually a representative number) of the accessible points in phase space and, therefore, also in configuration space. Both energy and entropy come into play. The classical entropy is proportional to the log of the number of states accessible to the system, i.e., to the volume in phase
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11855 space occupied by the systems in an ensemble; a factor of 1/N! is often included in the statistical counting. Because of the separation of kinetic and potential energies, one can specify a “classical configuration-space entropy” which is proportional to the log of the area on the landscape occupied by the systems at a given T. Alternatively, we can set the classical configurational entropy proportional to the log of the area coVered by those equi-ribbons that are actually occupied at a giVen T. If the occupied equi-streams are narrow and cover little area on the landscape, they contribute little to the entropy. Note that although the entropy is a statistical thermodynamic quantity that implies ensemble averaging, the concept of an energy landscape is a strictly mechanical concept which does not imply any statistical aVeraging. Dynamical properties can be studied by examining individual systems or groups of systems in an ensemble and by following their evolution in phase (or configuration) space oVer a limited length of time. Relaxation properties in liquids involve transport from representative low-energy (highly populated) volume elements in configuration space to similar final volume elements; the transport may be either along an equi-ribbon (or, allowing for corrugation, within kT of an equi-ribbon) or alternatively across high-energy equi-ribbons (with energies on the order of > kT); the paths or reaction coordinates along equi-ribbons can be said to specify “flow dynamics”, while the paths across energy barriers are said to specify “activated dynamics” with activation energies which are on the order of or more, but nonextensive. See Figure 1. A given relaxation process may involve both kinds of paths, and there need not be a clear separation into these two limiting classes of dynamics; however, there often is such a separation. Connections along equi-streams play a major role in flow dynamics, whereas they can be treated as too narrow and tortuous to be relevant to activated dynamics. One of the significant features of the equi-ribbon picture is that both flow dynamics (along or nearly along an equi-ribbon) and actiVated dynamics (across energy barriers corresponding to passage across high-energy equi-ribbons) are always possible. One should note that, in discussing transport phenomena, the relevant dynamics most likely inVolVe collectiVe motions of many particles and that activated dynamics in this sense involve barrier-crossing on a multidimensional, multiparticle energy landscape. The discussion must be altered somewhat to encompass quantum mechanics, and at low T quantum mechanics must always be considered. The energy landscape itself is not a dynamical concept, and so it is no different in classical or quantum situations, but the effect of the energy landscape on the thermodynamics and dynamics can be different classically or quantum mechanically. At equilibrium an ensemble can be described in terms of a distribution among eigenstates, an eigenstate not necessarily having the localization of a classical volume element in configuration space. Because the kinetic and potential energies are no longer separable, the very concept of obtaining meaningful information from considerations of configuration space (energy landscape) is compromised in quantum mechanics. The quantum states are delocalized over an equi-ribbon, actually over a spread of equi-ribbons. As T f 0 K and the systems occupy only the lowest-lying delocalized equi-ribbons, neither the area on the landscape occupied by the equi-ribbons nor whether the connecting equi-streams are narrow or wide is important in determining thermodynamic properties (entropy). In contrast, dynamical properties depend on prepared, highly localized initial and final states (equi-lakes) which need not be eigenstates, and the classical picture of transit from
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Kivelson and Tarjus
one localized region on an equi-ribbon to another, whether along the equi-ribbon or across energy barriers, may remain Valid eVen at Very low temperatures. Thus, although it is sometimes hypothesized to be so, there is no imperative that the entropy and the relaxation rate be directly related. An overall conclusion is that although the physics underlying the thermodynamic quantities is Very different at high and low temperatures, it may not be so for the physics underlying the dynamics. This suggests that although there may be strong coupling between thermodynamics and dynamics at high temperatures, this is not likely to be the case at low T. Of course, as T is varied the mechanisms of the dynamics could change, e.g., from activated (across equi-ribbons) to flow (along equiribbons that lie within less than kT of each other), or vice versa; the dominant dynamics are those (conforming to the equations of motion) that bring a system most rapidly between initial and final volume elements in configuration space. Justification of the Model We next seek justification of the picture of an energy landscape composed of continuous intertwining unsegmented equipotential ribbons. A trivial example of an intertwining continuous equipotential ribbon landscape picture is provided by a system of hard spheres. The corresponding landscape has only two equipotentials: infinite-energy peaks and zero-energy valleys. As long as the total volume (in the thermodynamic limit) allows for a free volume of at least one molecule, it should be possible to move through every allowed configuration so that the zero-energy valley is a continuous equipartition ribbon. Only for the perfect crystal at constant volume might one expect the free volume to be truly zero. Thus, it would appear that the equipotentials for hard spheres are continuous and consequently intertwined except if the volume is so low that the structure is possibly closepacked. An intuitive picture can be developed by noting that in one dimension an energy barrier between initial and final states cannot be circumvented, but in two dimensions many barriers can be bypassed. See Figure 1ab. In three dimensions it is still easier to bypass barriers, and as the number of dimensions increase, there are fewer barriers that cannot be bypassed. For a very large number of dimensions, there should then be ways of bypassing most barriers. The potential energy can be expanded about an arbitrary point in configuration space:
U ) Uo +
N
N
∑i a1(i) δqi + ∑ij a2(i,j) δqi δqj + N
a3(i,j,k) δqi δqj δqk + ... ∑ ijk
(1)
where δqi represents the vector displacement of the ith particle away from its position in the instantaneous configuration (which is not necessarily at a potential energy minimum). The an values are appropriate tensors which can readily be expressed in terms of the nth derivatives of U at the given point in configuration space. The question we pose is whether all points on the energy landscape with energy Uo can be connected without changing the potential energy, i.e., whether the δqi values can be taken such that all but the first term in the expansion vanish. Note that we do not ask that this be true for arbitrary displacements but merely that such displacements can be found; thus, we do not require that the coefficients {an} vanish but that for each n some an values have different signs than others. If this is true,
then for each n one can find an interrelationship among the δqi values that will guarantee that the term of nth order in eq 1 will vanish, and at least in the thermodynamic limit or for large enough systems for which there is enough flexibility in adjusting the δqi values, one can guarantee that all but the zeroth-order term in eq 1 will vanish. For the above to be true, one must ask whether indeed for each given n the an values have both positive and negative signs. Consider first the stationary points of the potential energy function, i.e., the points at which the linear terms a1(i) in eq 1 all vanish. The a2(i,j) values form the 3N×3N Hessian matrix, and its 3N eigenvalues can in general be positive or negative (the former corresponding to stable directions and the latter to unstable directions). The number of negative eigenvalues is called the instability index of the stationary point. If the index is equal to zero, the point is a local minimum; otherwise it is a saddle point or a local maximum. A corollary of the universality of equi-ribbons is that the index is different from zero for all but a few stationary points. It is expected8 that high-energy stationary points have an index on the order of N, i.e., a finite index density, and that this index decreases as the energy of the points decreases. However, except for special models with infinite range interactions,9,11 there is no evidence that for large enough systems the index becomes equal to zero (nor even that the index density becomes equal to zero) at energies above that of the deep crystalline minima. This in turn implies that a generic equipotential ribbon running through instantaneous configurations need not be stopped when reaching one of the stationary points of the potential energy hypersurface. More generally, most points in configuration space that correspond to amorphous material can be probed by examining a succession of randomly generated configurations for any given potential energy. What is needed is the distribution of interparticle separations for each configuration. If in any given configuration the distribution of near-neighbor interparticle distances is sufficiently broad, the an of each order will have varying signs, and the higher the n, the broader the requisite distribution. For a large system, the distribution of interparticle separations is similar to that in an ensemble of systems; for the moment we take this equality to be rigorous, and we take the pair distribution function to represent the distribution of interparticle separations in any given amorphous system. That the distribution of near-neighbor interparticle separations is “broad” is indicated by the breadth of the first broad peak in the structure factor of liquids and glasses. How broad the distribution actually is and how broad it must be for any given potential energy to satisfy the requirement of sign alternations in the an coefficients are not issues with which we have successfully dealt; it seems reasonable that for disordered systems there should always be a sufficiently broad distribution to ensure the possibility of finding displacements that eliminate all but the first term in eq 1. There may also be points in configuration space that represent statistically rare configurations (perhaps representing structures between amorphous and ordered systems) which may have been overlooked in the ensemble-averaged pair distribution function. It is possible that at such points there would not be a sufficiently broad distribution of interparticle separations to satisfy the requirements of sign variation for each order of an values. These points might therefore not lie on equi-ribbons, and if so, they would represent isolated points or segmented equi-ribbons. The arguments above are admittedly not rigorous, but they are sufficient, we believe, to support the plausibility of the equiribbon picture of the energy landscape.
Energy Landscapes Composed of Equi-Ribbons Comments For a finite system composed of N particles, not all the higher order terms in eq 1 can be made to vanish. In this case the equi-ribbons are truncated; i.e., some of the equi-streams connecting equi-lakes are blocked, and these equi-lakes can be connected only by passing over high-energy equi-ribbons, i.e., energy barriers. As more and more equi-streams get blocked, the landscape looks more and more like one dominated by pointlike inherent structures. Note that in simulations of models of finite-size liquid systems it has been found that the index density of the stationary points vanishes at a finite energy above the ground state,16-18 but this may well be due to the finite size of the systems studied. We expect that the larger the system, the lower in energy one can find equi-ribbons. In light of the above, it is worth pointing out that models having a topological transition in the energy landscape at which the index density goes to zero at an energy much above the ground state also have a Kauzmann-like transition at which the logarithm of the number of local minima (the configurational entropy) goes to zero.10,11,19 It has, however, been argued20,21 that the Kauzmann-like transition cannot exist in finitedimensional, finite-range systems, and one may similarly wonder whether the “topological transition” at which the index density vanishes is an artifact of the infinite range of the interactions. Because of continuity of the potential energy, all equi-ribbons are bordered by other equi-ribbons that are only slightly higher or lower in energy. Consequently, in discussing flow dynamics, one might not adhere strictly to a picture with passage along an equi-ribbon but one might consider a collection of equiribbons that lie within less than kT of each other. As T f 0 K, the perfect crystal falls into one of N! equivalent wells associated with the absolute minima. These are localized inherent structures. However, because the barriers between the inherent structures, though high, are not infinite, the localized inherent structures “interact”, and the absolute ground state (lying at the bottom of a very narrow band associated with delocalization or permutation over the inherent structures) is nondegenerate. Just how these N! permutations should be handled to obtain thermodynamic consistency between the crystal and liquid remains a point of discussion; however, it should be noted that, in discussions of energy landscape, one must include all N! permutations of identical crystal structures because the landscape is a static mechanical and not a statistical thermodynamical construct. Activated versus Flow Dynamics Although it seems unlikely that considerations of the energy landscape, whether formulated in terms of equi-ribbons or not, can lead to quantitative advances, they can, as suggested above, be useful in interpreting both thermodynamics and dynamics. The principal feature newly highlighted by the equi-ribbon model is that paths corresponding to both actiVated and flow relaxation always coexist even though it is possible that some flow paths may be sparse, even of measure zero. Gases relax by flow dynamics. The density is low, the particles are well separated, and the connecting equi-ribbons are very broad, so broad that one cannot distinguish equi-lakes or inherent structures. At the opposite extreme, only “hopping” dynamics exist in crystals at low temperatures: at the relevant low energies the only possible dynamics for systems with longrange order are activated barrier-crossing between distinct equilakes. For liquids the free volume is a small fraction of the total volume, and one can distinguish equi-lakes connected by equi-
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11857 streams (as discussed in the hard sphere example above). Whether the fastest route between moderately separated equipotential points (in distinct equi-lakes) is by passage along equistreams (flow dynamics) or by passage over barriers (activated dynamics) is questionable. Experimentally one might decide between the two by examining the empirical activation energy, Ea, for relaxation processes. If τ(T) is a relaxation time, one can define the empirical activation energy by
Ea(T,V) ) T ln[τ(T)/τ∞]
(2)
where τ∞ is the value of τ in the extrapolated high-limit. If
Ea(T,V) . kT
(3)
the process is taken as activated; such processes have strong T dependences and are said to be T-controlled. The empirical activation energies are taken at constant volume because if V is allowed to change, the energy landscape changes, and Ea is then not exclusively associated with the dynamics but also with the energy required to change the landscape. Although the actual activated process involves barrier-crossing at a given T, the quantities Ea and τ∞ in eq 2 can be evaluated only by making measurements at at least two temperatures, or equivalently, measuring τ and (∂τ/∂T)V at one T; it is for this reason that different values of Ea are obtained at constant volume and at constant pressure, the latter being larger than the former.22 On the other hand, if
Ea(T,V) e kT
(4)
barriers would seem not to be important to the dynamics, and one could infer flow dynamics. By these criteria it appears that above their melting points some liquids (atomic fluids and some hydrocarbons) relax via flow dynamics, whereas others (those with stronger, more directional intermolecular bonds) relax via activated dynamics (although Ea/kT is usually much smaller than for chemical reactions).22 Below their melting points most deeply supercooled liquids appear to relax via activated dynamics.23,24 For many supercooled liquids, termed “fragile glass formers”, the empirical activation energy Ea(T,V) increases appreciably with decreasing T;23,24 this behavior can be called “superArrhenius” to contrast it with Arrhenius behavior specified by constant Ea. Since the energy barriers on the landscape do not change as long as V is held constant, the super-Arrhenius behavior must be attributed to changes in the reaction path with decreasing T, presumably because the initial and final states are T-dependent. The inequality in eq 3 guarantees that activated relaxation will exhibit very strong T dependence, whereas the condition in eq 4 suggests that flow dynamics will be associated with weaker T dependence. Thus, activated dynamics are “temperature-controlled” and are limited by the lack of thermal energy needed to pass over the energy barriers, whereas flow dynamics (with hard spheres as the paradigm) might be envisaged as “Volume-controlled,” a decrease in volume leading to a decrease of “free volume” and a narrowing of the connecting equi-streams leading to blockage of flow dynamics. It is, in fact, found that relaxation in many supercooled liquids, though very strongly dependent upon T at constant V, is only very weakly dependent upon V at constant T.22 On the other hand, the relaxation of near-glassy colloidal suspensions (at a given T) is strongly dependent upon the concentration of the colloid,25 which can be interpreted as the replacement of the control variable V by
11858 J. Phys. Chem. B, Vol. 105, No. 47, 2001 concentration; one therefore expects relaxation in such systems to take place via flow dynamics. It has been observed23,24 that for many (perhaps all) deeply supercooled liquids the relatively slow relaxations seem to be separable into very slow “R-relaxations” and somewhat faster, secondary “β relaxations”. The R-relaxations appear definitely to be activated. The nature of the β-relaxations is far from having been clearly established. One usually distinguishes between “slow β-relaxations” and “fast-β-relaxations”; the former, first observed by Johari and Goldstein,26 seem to be activated and follow an Arrhenius T dependence. Stillinger2,7 proposed a mechanism for the slow β-relaxation which, in the formalism discussed here, attributes the elementary steps as activated passage between contiguous equi-lakes. The fast β-relaxation, whose presence has been stressed by mode-coupling theory,27 could very possibly be associated with flow dynamics. For liquids we propose the following as a working model based on an energy landscape composed of equi-ribbons. The relaxation process takes the system from one equi-lake on a low-lying equi-ribbon to another. Some of these equi-lakes lie close to each other and are well connected by navigable equistreams, but the equi-streams connecting these nearby equi-lakes to more distant ones are long, narrow, and tortuous, and these distant equi-lakes are, therefore, more accessible via alternative routes that pass over barriers. One might then expect the shorttime fast β-relaxation to be described by flow dynamics and the long-time R-relaxation to be described by activated dynamics. At temperatures below Tg the glass exhibits relaxation properties (called “aging”) that are quite different from those observed above Tg.10 Thus, although Tg is not, we believe, an indicator of a thermodynamic phase transition, but only of a point of dynamic arrest on the time scale of the measurements, it may be that below Tg the system undergoes a dynamical crossover, possibly from activated (above Tg) to flow (below Tg) dynamics. (The activated, slow β-relaxation is however observed both above and below Tg.) The reason one might anticipate flow dynamics in the glass is simply because as T is lowered toward Tg the activation processes become so slow that alternative reaction paths are sought by the system, even though these alternative paths may themselves lead to slow processes. Thus, although flow dynamics along the very narrow connecting equi-streams in the supercooled liquid are slow compared to the activated dynamics, it is possible that below Tg the activated dynamics become so slow that the flow dynamics win out, even though they, too, are getting slower. The above picture of flow-dominated aging in the glass can be related to the interpretation given by Kurchan and Laloux.9 In the latter, however, the system is considered quenched from a high-T phase and always stays at a relatively high energy level (higher than that characteristic of the equilibrated liquid at Tg); aging then proceeds via flow along gorges (equipotential ribbons) at the ridges separating basins, although slipping to lower equi-ribbons also occurs. In Cavagna’s picture8 of relaxation in supercooled liquids and glasses, flow dynamics are specifically forbidden for equilibrated systems with typical energies below an energy threshold which is more or less associated with the mode-coupling temperature Tc; such a picture does not seem compatible with our view that aging in glasses may proceed via flow dynamics because for real liquids Tg, the temperature at which the liquid falls out of equilibrium and becomes a glass, lies well below Tc and its energy lies well below that of the putative energy threshold.
Kivelson and Tarjus Finally we point out that the considerations developed in this section might explain why the mode-coupling theory,27 which is essentially a theory of flow dynamics where activated processes play no role, has been applied with some success to both the fast β-relaxations in supercooled liquids and the aging phenomena in glasses. We recognize that the validity of the above model, which evolved from considerations of equi-ribbon landscapes, has yet to be tested, but we believe that its simplicity, generality, physical appeal, and lack of obvious contradictions earn it a position of consideration. Acknowledgment. We thank Andrea Cavagna, Charles M. Kobler, Jorge Kurchan, Robert L. Scott, and Steven A. Kivelson for their input. We dedicate this work to Howard Reiss, a truly distinguished contributor to the science of metastable systems, whose work has greatly influenced our thinking in this field. References and Notes (1) Goldstein, M. J. Chem. Phys. 1969, 51, 3728. (2) Stillinger, F. H.; Weber, T. A. Science 1984, 225, 983. Stillinger, F. H. Science 1995, 267, 1935. (3) Angell, C. A. Science 1995, 267, 1924. (4) Keyes, T. J. Phys. Chem. 1997, 101, 2921. (5) Sastry, S.; Debenedetti, P. G.; Stillinger, F. H. Nature 1998, 393, 554. (6) Schroder, T. B.; Sastry, S.; Dyre, J.; Glotzer, S. W. C. J. Chem. Phys. 2000, 112, 9834. (7) Debenedetti, P. G.; Stillinger, F. H. Nature 2001, 410, 259. (8) Cavagna, A. Europhys. Lett. 2001, 53, 490. (9) Kurchan, J.; Laloux, L. J. Phys. A: Math. Gen. 1996, 29, 1929. (10) Bouchaud, J. P.; Cugliandolo, L.; Kurchan, J.; Mezard, M. In Spin Glasses and Random Fields; Young, A. P., Ed.; World Scientific: Singapore, 1998; p 161. (11) Cavagna, A.; Giardina, I.; Parisi, G. Phys. ReV. B 1998, 57, 11251; Cond. Mat/0104537, 2001. (12) Angelani, L.; Di Leonardo, R.; Parisi, G.; Ruocco, G. Phys. ReV. Lett. 2000, 84, 6054. (13) Kob, W.; Sciortino, F.; Tartaglia, P. Europhys. Lett. 2000, 49, 590. (14) Bryngelson, J. D.; Wolynes, P. G. J. Chem. Phys. 1989, 93, 6902. Bryngelson, J. D.; Onuchic, J. N.; Soici, N. D.; Wolynes, P. G. Proteins: Struct., Funct., Genet. 1995, 21, 167. (15) Becker, O. M.; Karplus, M. J. Chem. Phys. 1997, 106, 1495. (16) Broderix, K.; Bhattacharya, K. K.; Cavagna, A.; Zippelius, A.; Giardina, I. Phys. ReV. Lett. 2000, 85, 5360. (17) Angelani, L.; Di Leonardo, R.; Ruocco, G.; Scala, A.; Sciortino, F. Phys. ReV. Lett. 2000, 85, 5356. (18) Grigera, T. S.; Cavagna, A.; Giardina, I.; Parisi, G. Cond. Mat/ 0107198. (19) Kirkpatrick, T. R.; Thirumalai, D. Phys. ReV. B 1987, 36, 5388. Kirkpatrick, T. R.; Wolynes, P. G. Phys. ReV. B 1987, 36, 8552. (20) Biroli, G.; Monasson, R. Europhys. Lett. 2000, 50, 155. (21) Stillinger, F. H. J. Chem. Phys. 1988, 88, 7818. (22) Ferrer, M. L.; Lawrence, C.; Demirjian, B. G.; Kivelson, D.; AlbaSimionesco, C.; Tarjus, G. J. Chem. Phys. 1998, 109, 8010. Ferrer, M. L.; Kivelson, D. J. Chem. Phys. 1999, 110, 10963. (23) Tarjus, G.; Kivelson, D. In Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales; Liu, A., Nagel, S. R., Eds.; Taylor and Francis: London, 2001; p 20. (24) Ediger, M. D.; Angell, C. A.; Nagel, S. R. J. Phys. Chem. 1996, 100, 13200. (25) See, e.g.: Pusey, P. N.; van Megen, W. Phys. ReV. Lett. 1987, 59, 2083. Rosenberg, R. O.; Thirumalai, D.; Mountain, R. D. J. Phys.: Condens. Matter 1989, 1, 2109. Pusey, P. N. In Liquids, Freezing, and the Glass Transition; Levesque, D., Hansen, J. P., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1991; p 763. Bartsch, E.; Franz, V.; Sillescu, H. J. Non-Cryst. Solids 1994, 172-174, 88. (26) Johari, J. P.; Goldstein, M. J. Chem. Phys. 1970, 53, 2372. (27) Go¨tze, W.; Sjo¨gren, L. Rep. Prog. Phys. 1992, 55, 241. Go¨tze, W. J. Phys.: Condens. Matter 1999, 11, A1.
