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Anankeon theory and viscosity of liquids: a toy model

A simplistic model, based on the concept of anankeon, is proposed to predict the value of the viscosity of a material in its liquid phase. As a result, within the simplifications and hypothesis made to define the model, it is possible to predict (a) the existence of a difference between strong and fragile liquids, (b) a fast variation of the viscosity near the glass transition temperature for fragile liquids. The model is a mechanical analog of the Drude model for the electric transport in metal.


I. A DEEP UNSOLVED PROBLEM
This article is proposing a simplistic model to account for the behavior of the viscosity of liquids as a function of temperature. The best illustration of this behavior is the data from Angell's review paper [1], that can be found below in Fig. 1. It shows on one graph the behavior of the logarithm of the viscosity η as a function of inverse temperature, in order to reveal whether or not an Arrhenius law is valid η ∼ e W/k B T . The excitation energy W is represented by the slope of the curve in such coordinate axis. The graph actually shows that the Arrhenius law is valid for a category of materials called "strong liquids". For most other materials, called "fragile", the slope changes as the temperature approach the glass transition. The glassy and liquid states of a material share many features. It is understood that the difference between them cannot be account for by using equilibrium Thermodynamics: the glass transition is not a phase transition, it is a dynamical process in which the viscosity becomes so large that the typical time scales involved in the material become macroscopic. The viscosity changes by 15 orders of magnitude over a very small temperature range near the transition. The reason why viscosity changes so rapidly, while the atomic structure does not show signs of this change, has been a mystery for a long time. It has been considered as one of the deepest unsolved problems in science [2,3] by the most prominent experts.
If there is only one atomic species in a condensed material, the local clusters are likely to have the shape of a regular icosahedron. Since regular icosahedra cannot tile the 3D-space, there is either a geometrical frustration or the local cluster deform in the solid phase to lead to a crystal.
The most likely symmetry of such a crystal is fcc. To get a glass, frustration is necessary. In the case of metallic glasses, frustration is obtained by mixing several species with very different atomic radius in order to prevent the systems to crystallize in a periodic order. However, Egami insisted for a long time that this geometrical description of the atomic distribution, while useful, is insufficient at describing the material. He insisted that an atom is not a hard sphere, but exhibits a bit of elasticity. Consequently, an atom can be seen as having a local stress tensor, in a way similar with particle having spins. This representation is powerful enough to describe the liquid phase in the high temperature regime [8]. In this view, a liquid appears as a free gas of objects represented by non interacting atomic stress tensors. Numerical simulations [8] suggest that this atomic stress tensor is a random Gaussian variable with zero average and with a covariance matrix given by a Maxwell-Bolztman factor using the expression of the elastic energy. It is even quantitatively better to use an effective elastic energy by taking into account the long distance effect of stress propagation using the mean field theory developed by Eshelby [9], in analogy with the Clausius-Mossoty theory of effective electric permittivity in dielectric. The equilibrium state of such a free gas can then be explicitly computed using Statistical Mechanics leading to a law of Dulong and Petit for the heat capacity, an observation made for a long time, which found an explanation with this theory.

C. A Toy Model
What happens at lower temperature ? The atomic vibrations, leading to acoustic waves, which will be called abusively phonons here, whereas quantized or classical, are still existing in liquids as in solids. However the acoustic waves with short wave length, namely comparable to the interatomic distance, are strongly damped in the liquid phase at high temperature. Whenever the typical time scale characterizing the jump dynamics is comparable to the phonon period, the phonon has no time to oscillate and it is damped. The present model investigates this damping more quantitatively. To proceed, the same intuition will be followed as in the original paper of Drude on electronic transport in metals [10]. In order to make the model computable, it will be assumed that it describes the mechanical fate of a given atom: (i) as long as no jump occurs, this atom is located near the minimum of a local potential well, and it will oscillate, harmonically, with a frequency given by the local curvature of the well, (ii) then at random times (Poissonian distribution) the atom jumps, finding itself, after the jump in a new potential well, with a new local curvature and a new initial location relative to the new equilibrium position and a new initial velocity (phase-space position). Hence, the curvature of potential well (expressed in terms of the frequency of oscillation) will be considered as a random variable with a fixed average and a given covariance σ which will be shown to be the most important parameter. Similarly, the new relative initial phase-space position, after the jump, will also be chosen randomly according to a Maxwell-Boltzmann distribution. To make the model even more computable, it will be assumed that only one degree of freedom of oscillation is allowed. It will be seen that, even though this model is so simplistic, it exhibits a transition between two time-scales as σ decreases. The computation of correlation functions, like the viscosity, shows that this change of scale may affect the viscosity in an essential way.

D. Anankeons in Glass: the STZ-Theory
For a very long times, engineers investigating the mechanical properties of materials measured the strain as the material is submitted to an external stress. The strain-stress curves shows usually a line at small stress, corresponding to the elastic response. This response is usually reversible: increasing then decreasing the stress follows the same curve. If the stress applied becomes large most materials just break. But some materials, like bulk metallic glasses, exhibit a plastic region in which the sample can be strained irreversibly, without much of additional stress. In a series of seminal papers [11][12][13][14][15], Langer and his collaborators developed a theory of what they called shear transformation zone (STZ), to account for the plasticity region in bulk metallic glasses. The idea is that in the plastic region, the local clusters exhibit droplets in which the solid really behaves like a liquid, called an STZ. Considering the size of such droplets as a random variable, acting as a local lubricant, Langer et al. proposed a system of effective equation at the macroscopic scale that are modeling the behavior of the material under stress with an amazing accuracy [16]. At the atomic scale though, numerical simulation show that these STZ's are the result of a cascade or avalanche of atomic swaps, namely of anankeons, [17], starting at a random site, called the germ. This initial anankeon induces other in its neighborhood, because of the disruption created by the propagation of the stress induced by such swaps. The cascade looks like a local catastrophic event, occurring during a very short time, leading to the building of the local liquid-like droplets. The time scale representing the time necessary to create an STZ is however long in comparison with the typical time scale of the anankeon-swap. A model for such a cascade of event was proposed in [18,19].

II. LIQUIDS AT LARGE TEMPERATURE AS A PERFECT ANANKEON GAS
The numerical simulation produced by Egami and Srolovitz [8] can be interpreted as a description of the liquid phase, far from the liquid-solid transition, in terms of a free gas of anankeon.
The degree of freedom associated with each anankeon will be given in terms of an atomic stress tensor. Their Gaussian distribution observed in the numerical results can be interpreted in terms of a Gibbs state describing the thermal equilibrium in a statistical mechanical approach. This Section is dedicated to describe this in detail.

A. The Delone Hypothesis and the Ergodic Paradox
An instantaneous snapshot of the atomic arrangement can be described as a set of points L, representing the position of atomic nuclei. With a very good approximation L is efficiently described as a Delone set. Namely it is characterized by two length scales 0 < r ≤ R < ∞ defined as follows: (i) any open Euclidean ball of radius r contains at most one point of L, (ii) any closed Euclidean ball of radius R contains at least one point of L. The first condition is equivalent to having a minimum distance 2r between any pair of distinct atoms. The second condition is equivalent to forbidding holes a diameter larger than 2R.
However, the principles of Equilibrium Statistical Mechanics and the ergodicity with respect to translations contradict such an hypothesis. This is the content of the so-called Ergodicity Paradox [20]. This is because in the infinite volume limit, the ergodic theorem implies that with probability one, given any > 0, somewhere in the atomic configuration two atoms are at a distance shorter than . Similarly, with probability one, given any > 0, somewhere in the atomic configuration there is a hole of size larger than 1/ . However, such local arrangement are rare, so that dynamically their lifetime is so short that they are unobservable. Hence restricting the atomic configurations to make up a Delone set is a reasonable assumption called the Delone Hypothesis x of radius R [21,22]. The closure T (x) of the Voronoi cell is called a Voronoi tile. Two such tiles can only intersect along a common face. In particular two Voronoi cells centered at two distinct points of L do not intersect. The entire space is covered by such tiles, so that the family of Voronoi tiles constitute a tiling, the Voronoi tiling. The vertices of the Voronoi tiles will be called Voronoi points. Their set L * is usual called the dual lattice. For a generic atomic configuration, a Voronoi point has not more than d + 1 atomic neighbors in space-dimension d (3 in the plane and 4 in the space). In particular, the Voronoi points are the center of a ball with d + 1 atoms on its boundary, a property called by Delone (who signed Delaunay) the "empty sphere property" [23]. It means that it is at the intersection of d + 1 Voronoi tiles. The d + 1 atoms located at the center of those tiles are the vertices of a simplex (triangle for d = 2, tetrahedron for d = 3) that generates the so-called Delaunay triangulation.
Two points x, y ∈ L will be called nearest neighbors whenever their respective Voronoi tiles are intersecting along a facet, namely a face of codimension 1. The pair {x, y} will be called an edge.
In practice, though, using physical criteria instead of a geometrical one, an edge might be replaced by the concept of bond, to account for the fact that the atoms associates with x and y are strongly bound [24][25][26]. The term "strongly" is defined through a convention about the binding energy, so as to neglect bonds between atoms weakly bound to each other. Whatever the definition of an edge, geometrical or physical (bonds), this gives the Delone graph G = (L, E), where L plays the role of the vertices and E denotes the set of edges or of bonds. At this point the Delone graph needs not being oriented.
Along a graph, a path is an ordered finite set of edges, each sharing a vertex with its successor or with its predecessor. The number of such edges is the length of the path. The graph-distance between two vertices is the length of the shortest paths joining them. A graph-ball centered at x of radius n will be the set of all vertices at graph distance at most n from x. Such graph balls correspond to local clusters. Two graph balls are called isomorphic if there is a one-to-one surjective map between their set of vertices, that preserve the graph distance. As a result, if the atomic positions is slightly changed locally in space, the graph balls might still be isomorphic.
Therefore graph ball, modulo isomorphism are encoding what physicists called the local topology of the atomic configuration.
The transition between two graph balls can be described precisely through the concept of Pachner moves [21,22,27]. This concept was created by experts of computational geometry, to describe the deformation of manifolds via a computer. A Pachner moves can only occur if at least one Voronoi point becomes degenerate, namely it admits at least d + 2 atomic neighbor at some point during the move. Such moves correspond to a local change in a graph ball, namely an edge disappear another may reappear. A Pachner move corresponds exactly to the concept of anankeon.
It can be described through using the Delaunay triangulation as explained in Fig. 2. In 3D-space such a move involves at least five atoms moving quickly to deform the Delaunay triangulation, changing the nature of the Delone graph, namely the local topology. This number five has been seen in numerical simulations [28] as can be seen in Fig. 3.

C. Atomic Stress Tensor
The definition of stress tensor in solids [29,30], using the hypothesis of Continuum Mechanics, gives an intuition upon how to build a stress tensor at the atomic level [31]. The present presentation will use a different route, but leads to a similar result. It will be assumed that interatomic forces are described by a two-body isotropic potential W (r), where r denotes the distance between the two atoms. In such a case, given a pair e = (x, y) of atoms bound by an edge, let n e denotes the unit vector along the line generated by x, y oriented from x to y. By isotropy, the force acting upon the atom located at x coming from y is F y (x) = W (r)n e . The total force acting on x is therefore given by the sum F (x) = y n e W (r), where the sum includes the nearest neighbors of the atom at x. In Continuum Mechanics, the stress tensor is defined so as to satisfy F = ∂Λ σ(s) · n(s)ds for any volume Λ with boundary ∂Λ. At the atomic level though, the integral will be replaced by the previous sum. The role of the volume Λ is played by the Voronoi tile T (x). Each nearest neighbor of x corresponds to a unique facet of T (x), and then the unit vector n e is nothing but the normalized unit vector perpendicular to the facet oriented outwardly w.r.t. the tile. This suggests the following formula for the stress tensor: each bond e gives it a value σ e so that where r e is the Euclidean length of the bond and A e is the Euclidean area of the facet associated with e. It ought to be remarked that the ratio tr(σ e ) = p e = W (r)/A e represents the pressure felt by the atom coming from its neighbor in the direction of the edge e. The deviatory stress is the traceless part σ e − p e /d where d is the dimension of the space in which this material lies. The atomic stress at x is just the sum of these contributions, namely for an atom located at the vertex In this expression, ∂ 0 e = x means the edges with origin at x. The formula above is very similar to the one obtained in [31], using an argument from Continuum Mechanics, apart from a possible numerical constant close to 1.
The numerical simulation in [8] suggests that in the liquid phase, far from the liquid-solid transition, the statistical distribution of the atomic stress tensor follows a Maxwell-Boltzmann-Gibbs law of the form In this expression, β = 1/k B T , Λ represents a volume in the space of stress tensors, Z 1 is a normalization factor, p is the pressure given by the normalized trace of the stress tensor, and τ deviatory stress (traceless part of σ). The coefficients B, G are respectively the bulk modulus and the shear modulus. The expression in the exponential represents the elastic energy associated with the atomic stress.
The domain of integration is the space of all real symmetric d × d matrices, representing a possible stress tensor. This Gaussian integral is easy to compute giving Z 1 (T ) = Z T d(d+1)/2 where Z is a constant depending on B, G, but independent of the temperature. The Clausius entropy is given by S = k B ln(Z(N, T )) leading to the following expression of the heat capacity where c d = d(d + 1)/2, which is 6 in 3D. In particular, the system follows a Law of Dulong-Petit like the contribution of phonons in crystal. However, the physical origin is different in liquid as the main degree of freedom associated with atoms are the atomic stress. Hence a liquid can be seen as a perfect gas of anankeons. This prediction gave a satisfactory answer to the experimental observation [32] of the saturation of the heat capacity at high temperature (see Fig. 4).

III. PHONON-ANANKEON INTERACTION
What happens at lower temperature, near the solid-liquid transition ? The following model will show that as the temperature decreases, the vibration modes are constraining the shape of local clusters, so as to change the viscosity in a non trivial way.
where η is the viscosity and G ∞ is the high frequency shear modulus. The where m is the mass of the atom and k is the spring constant, expressible in terms of the second derivative of the potential energy. This equation can equivalently be written as The solution of this equation, with initial position u(0) and initial velocity v(0), is given by The mechanical energy of this oscillator is given by It ought to be stressed that the spring constant k depends in an essential way upon the local environment seen by the test-atom. In particular, if the local configuration changes due to a Pachner move, or an anankeon, this constant will change as well. This will be described mathematically in the next paragraph.

C. The toy model for anankeon-phonon interaction
When an anankeon strikes, the atom is ejected from its position to jump on a position nearby, in a new harmonic potential. To describe such a situation, following the strategy described in Section I C, it will be assumed that the anankeons strikes at random times {· · · < τ n < τ n+1 < · · · }.
Here the integer index n varies from −∞ to +∞. To be more precise, the randomness of those times will be assumed to be Poissonian, namely the random variables τ n+1 − τ n are i.i.d., with average and an exponential distribution The phase-space position X(t) after the jump will also be relative to the minimum of the new potential well. In particular, there is a need to update this position right after the jump in terms of the position right before, namely X(τ n + 0) = X(τ n − 0) + ξ n .
The vector ξ n represents the result of the anankeon kick on the initial phase-space position, relative to the potential minimum. It is reasonable to assume that ξ n is also a random variable and that the family (ξ n ) n∈Z is made of i.i.d.'s. In addition it will be assumed that the common distribution is provided by the Gibbs state at the temperature T . Hence At last, the pulsation ω will depend upon the potential well in which the atom fell. Hence between the times τ n and τ n+1 it will be given by ω n . Since the atom this model is describing should be typical, the ω n 's should be considered as random variables, and again, the family (ω n ) n∈Z will be made of i.i.d.'s. At this point, it is unwise to fix the common distribution. It enough to fix the average ω and the variance σ ω = E{ω n } , σ = Var(ω n ) = E{(ω n − ω) 2 } 1/2 .
The viscosity is given by the fluctuation-dissipation theorem as where V is the volume of the fluid, T the temperature, and C f is a correlation functions defined by where f is a function on the phase space giving the expression of the stress tensor. For the purpose of this calculation, it is sufficient to consider f as a scalar valued function, because it does not change the conclusion of the computation. The goal will be to show that, as t → ∞, the This correlation function can also be computed using the dissipative evolution defined by the operator P t as the following conditional expectation Then, the correlation function is given by the phase-space integral In order to compute τ M precisely, it is convenient to compute the Laplace transform of C f instead, namely, given a complex variable ζ It turns out that the asymptotic behavior of the correlation at t → ∞ can be interpreted as the position of the nearest singularity of LC f in the complex plane. Namely the Laplace transform is analytic in the half-plane ζ > −1/τ M . Hence, in order to compute the correlation function, it is sufficient to compute the value f (X(t)) taken by the function f along the random phase-space trajectory of the atom under scrutiny.

IV. COMPUTING THE LAPLACE TRANSFORM OF A CORRELATION FUNCTION
The formula (11) suggests to compute the Laplace transform of the dissipative evolution operator P t instead, acting on some function space describing the relevant functions on the phase-space. In order to do so, it will be more efficient to compute the actions of the various part of the dynamic on functions rather than on the phase-space trajectory. As will be seen this calculus is very similar to the operator calculus used in Quantum Mechanics.

A. Operator Calculus
The first part of the evolution is provided by the phase-space mouvement of the harmonic oscillator, as described in Section III B. Namely where x = (u, v) is the phase-space position and J is very similar to the phase-space angular momentum but for the missing ı in front (namely J is anti-Hermitian instead) and its 2D character in phase-space.
Similarly, the impact of the anankeon on the atom is to translate the phase-space position of the atom after a kick, by a vector ξ = (a, b), leading to the translation operator Hence ∇ plays the role of a phase-space momentum operator apart for the missing ı in front (namely ∇ is anti-Hermitian instead).
Let X(t) denote the phase-space trajectory of the atom at time t with initial condition X(0) = x.
From the description of the model, an harmonic motion takes place between time τ 0 = 0 to τ 1 at pulsation ω 1 , then a phase-space translation by ξ 1 , then another harmonic motion between time τ 1 to τ 2 at pulsation ω 2 , then a new phase-space translation by ξ 2 , and so on, until the time t is reached. It becomes possible to describe it from the functional point of view as follows where the product is ordered from left to right, with j = 1 on the left and j = n on the right, and where n is the integer such that τ n ≤ t < τ n+1 . From the assumptions made on the model, the (τ j − τ j−1 ) are i.i.d., as well as the ω j 's and the ξ j 's. It follows that n is also a random variable.
Computing the distribution of n though, is hard enough to try to avoid it. The Laplace transform will allow to avoid such a calculation.

B. Laplace transformation
After averaging and taking the Laplace transform the result is Thanks to the linearity of the averaging process, this integral over time can be decomposed, under the averaging, into the time intervals [τ n , τ n+1 ], to give Using eq. (15), this gives Since τ 0 = 0, a telescopic sum give t = (t − τ n ) + (τ n − τ n−1 ) + · · · + (τ 1 − τ 0 ). This leads to distribute the first term e −tζ inside each terms of the product leading to replace each ω j−1 J by The first operation consists in evaluating the integral over t. The change of variable s = t − τ n gives the following operator valued integral over the time Poisson process, before averaging over the random phase-space positions ξ j 's or the random pulsations ω j−1 's. In addition, the stochastic independence of the (τ j − τ j−1 )'s permits to reduce the averaging to each factor in the product in eq. (17). Using the the time-scale assumption in Section III A on page 12, this averaging reduces to the following integral, where A is an operator where E τ denotes the average over the Poissonian times. The stochastic independence of each factor in the product, leads to a factorization of the averaging, since, whenever two random variable X, Y are stochastically independent the average of their product is given by the product of their average, . After a little algebra, using eq. (18,19), the eq. (17) leads to Similarly, since the ξ j 's and the ω j−1 's are stochastically independent, it is enough to compute the following averages where E ξ denotes the average over the phase-space position, using the Gaussian integral in eq. (6).
Similarly, the average over the ω j−1 's leads to the operator-valued function Evaluating this integral requires to know the distribution of the ω j−1 's. At this point, no further assumption will be made yet. Using eq. (21,22), the eq. (20) leads to The last series converges only if A(ζ) e k B T ∆/2m < 1 where · denotes the operator norm. At this point though, no accurate description of the space on which these operators act has been done, so that the norm condition is still meaningless. It will be the purpose of the discussion of the next Section. With this cautionary statement in mind, the problem is reduced to analyzing the properties of one operator at the end, giving the Laplace transform as a function of ζ.
In order to make sense of the calculation made in the previous Section IV B, as was remarked at the end, the function space on which these operators act must be made more precise. In order to do so, the first question is to know which type of function f are physically relevant. In the calculation of viscosity, the ideal function should be the atomic deviatory stress tensor as given in eq. (1). Using the finite sum decomposition over the edges linking the atom at x with its neighbors, it boils down to find a space containing the functions of the form (iii) the symmetry properties of the function f used in the calculation of the viscosity must reflect as much as possible the ones of the realistic stress tensor given in eq. (1).
In particular this eigenspace is infinite dimensional. Let Π denote the orthogonal projection from H onto this space. Moreover, these spaces are mutually orthogonal and then span the entire Hilbert space H.
The analogy between the realistic situation and the phase-space modeling, suggests to choose the eigenspace of J with angular momenta = ±2. It will be seen though, that this restriction does not matter in terms of the qualitative features of the model. Similarly the operator ∆ becomes self-adjoint with nonpositive generalized eigenvalues. In particular, the operator exp{k B T ∆/2m} is selfadjoint with its spectrum given by the interval [0, 1], so that 0 ≤ exp{k B T ∆/2m} ≤ 1 implying exp{k B T ∆/2m} ≤ 1.
The previous discussion shows that, while a choice has to be made for H, it will always be partly arbitrary. In particular, the discussion of symmetries might not be totally realistic. But the choice of H is not innocent. The properties of the unbounded operators J and ∆ depend crucially upon which choice is made. Hence, the main criterion is whether or not the model is effective at describing the physical properties of the system under study.

D. Analyticity Domain
Using the assumptions made in Section IV C, the operator {1 + τ (ζ + ω j J)} −1 has a spectrum of isolated eigenvalues (each which infinite multiplicity) given by {1 + τ (ζ + ıω j )} −1 with ∈ Z. In particular, it has poles at As the real valued random variable ω j varies, though, the pole position describes a subset of the vertical line ζ = −1/τ , the extend of which depends upon the support of the probability distribution of ω j . Hence after taking the average over ω j , this line becomes a cut in the ζ-complex plane. It is convenient to introduce the following functions In view of the definition of the Laplace transform (see eq. (12)), it implies that only the part (ζ) > −1/τ is relevant for this analysis. Hence

Result 1:
A(ζ) and the a (ζ)'s are complex analytic in the ζ-complex plane in the domain Since J is anti-selfadjoint, the operator A(ζ) is normal, namely it commutes with its adjoint. It can actually be expressed in terms of the eigenprojections Π using its spectral decomposition Consequently its norm is given by the maximum of the modulus of its eigenvalues, namely, using |E(X)| ≤ E(|X|), .

V. INTERPRETING THE RESULTS
From the mathematical computations made in the previous Section IV E, it follows that there are two regimes concerning the Maxwell time, both being controlled by the dimensionless parameter which is proportional to the variance of the distribution of the pulsations ω n , representing the curvature of the local potential energy, near its mechanical equilibrium, where the atom is located.
In order to draw practical predictions for a liquid material, some physical hypothesis are required.
In particular, it is highly expected that the parameters τ LC and σ depend on the temperature of the liquid. If so, the following conclusion can be made Result 4: Depending upon the material, there may or may not exist a cross-over temperature T co , such that (a) for T ≥ T co , the anankeons dominate and the Maxwell time coincides with the local configuration time, while, (b) if T < T co , the phonons restore some order, survive for a longer while, so that the Maxwell time becomes larger than the local configuration time.
In view of the results showed in Fig. 5 [37], it seems natural to assume that the time dependence of τ LC be given by an Arrhenius Law, namely In this expression, the energy W can be interpreted as the height of the potential barrier to be passed over as the anankeon strikes. How is the variance σ depends on the temperature ? So far no clues have been given on that. So we are reduced to speculate. By analogy and without further justification, though, the same hypothesis can be made on σ, If so, it is legitimate to ask the following question: Question: what is the meaning of the energy scale W v ?
It is only possible to see that if W v is small, then the variance of the frequency distribution is weakly dependent of the temperature. If, on the other hand, W v is large enough, then this variance is strongly depressed at low temperature. It is important to remember that such an Arrhenius Law, while not justified by physical arguments, needs to be valid only in a temperature range of moderate size above the glass transition. In such a case, it is only a convenient way to parametrize the temperature behavior of the variance using only one parameter, the energy scale W v .
Once these assumptions are made, it follows that the parameter K behaves like The condition required for the existence of a cross-over temperature T co is that, K(T ) decreases as the temperature decreases and that K(T co ) = π/2, leading to The first consequence of these hypothesis is that the difference between strong and fragile glassy materials [1] can be expressed by the conditions The second consequence is that the ratio between the Maxwell and the local configuration times diverges exponentially fast below the cross-over temperature (see eq. (34)) Such a law would explain why the viscosity diverges so fast near the glass transition temperature.