Glassiness in a model without energy barriers

We propose a microscopic model without energy barriers in order to explain some generic features observed in structural glasses. The statics can be exactly solved while the dynamics has been clarified using Monte Carlo calculations. Although the model has no thermodynamic transition it captures some of the essential features of real glasses, i.e., extremely slow relaxation, time dependent hysteresis effects, anomalous increase of the relaxation time and aging. This suggests that the effect of entropy barriers can be an important ingredient to account for the behavior observed in real glasses.

The nature of the glass transition is a longly debated question of much theoretical interest [1]. It is widely believed that the glass transition is mainly a dynamical process where the system can remain trapped in a metastable phase of a finite lifetime depending on the rate of the cooling process. If glasses are slowly cooled from the high-temperature region then it is possible to reach a crystal phase of very low entropy. But if the system is fastly quenched then it reaches a non-equilibrium regime characterized by the existence of very slow relaxation phenomena. Usually, the origin of these very slow relaxations is explained by the existence of a large number of metastable states separated by energy barriers [1].
The heights of the energy barriers are widely distributed and the system gets trapped in this metastable phase during its time evolution.
Recently, there have been developments towards a mean-field theory of glasses. In those cases one studies systems without quenched disorder with the aid of replica theory [2]. One finds the existence of a dynamical transition T D where the correlation time diverges. Below that temperature the system is always off equilibrium and relaxes towards a dynamical state of higher energy. At a lower temperature replica symmetry breaks and a large number of states dominate the statics. Below the dynamical transition temperature the system remains trapped in a very complex free energy landscape with huge free energy barriers. In meanfield theory, the height of these free energy barriers increases exponentially fast with N and metastable states have an infinite lifetime. Generally speaking, free energy barriers get contributions from an energetic part and an entropic part. In real structural glasses we expect the effects of energy barriers to be substantially different from that in mean-field theory because of the existence of nucleation processes [3]. Nevertheless, the effect of entropy barriers should be not so dependent on the range of the interaction. Then we expect entropy barriers to be a relevant mechanism in mean-field as well as in short-range models.
The purpose of this work is to understand the role of entropy barriers in the behavior observed in structural glasses. By entropy barriers we mean the existence of a very small number of directions in phase space where the energy decreases. We propose a mean-field model with purely entropy barriers and without metastable states. The phase space of this model is very simple and resembles to a golf-hole landscape. It mainly consists of flat directions in energy with a very small number of channels where the energy decreases.
Although the model has no thermodynamic transition it shows a behavior reminiscent of real glasses.
The model. Let us suppose N distinguishable identical particles which can occupy N different states. The Hamiltonian is defined by and the energy per site is given by the fraction of occupied states. The N r are the number of particles which occupy the state r ( r runs from 1 to N) and they satisfy the constraint The model defined in eq.(1) can be mapped into a Potts model with a large number of states N with Hamiltonian where m r is the magnetization of state r, m r = N r −1. Looking at the simple Hamiltonian of eq.(1) we observe that there is a trivial ground state with energy per particle e GS = −2(1 − 1 N ) = −2 in the large N limit. In this ground state all the N particles occupy one state, its degeneracy being equal to N. The evolution of the system as a function of the temperature is as follows. At high temperatures we expect all configurations to have the same probability and the energy per particle in this limit is −2( N −1 N ) N = −2/e. As the temperature is decreased the number of occupied states decreases while the occupied states increase their occupation numbers N r . Let us suppose we introduce a dynamics for the model eq.(1) (which gives the equilibrium Boltzmann distribution). The time evolution of the system as the temperature is decreased is as follows. The rate of variation of the number of particles in one state increases due to the particles which reach that state and decreases due to the particles which leave that state. The energy decreases when one state is emptied during the dynamical process. Because the total number of particles is conserved, as the number of occupied states decreases the time the system needs to empty a further state also increases. In this off-equilibrium situation the occupation numbers N r of the occupied states perform a random walk and the energy eq.(1) decreases very slowly to the static value at that temperature. This model has no dynamical phase transition but it shows the onset of very slow relaxations in the low temperature region (below T ≃ 0.2 close to the maximum of the specific heat). We will see that the main characteristics of this model are: strong dependence of the energy of the system with the cooling rate, hysteresis effects, anomalous increase of the relaxation time and presence of aging.
Statics of the model. In order to solve the statics of this model we have to compute the partition function. We will suppose that we have N particles and each particle i is associated with a variable σ i which can take N possible values according to the state the particle occupies.
The partition function is given by, where the factor N! in the denominator is a normalization constant in order to make the free energy extensive with N. The ocupation numbers N r satisfy the constraints r N r = N and N r = N i=1 δ σ i r . Eq. (4) can be rewritten in term of occupation numbers as We use the integral representation for the delta function, where m is an integer. Substitution into eq.(5) leads to, The integral in the previous equation can be readily evaluated by the saddle point method in the large N limit. The saddle point λ = iz gives the free energy βf = −Max z A(z) with The saddle point equation is e 2β − 1 = (y − 1)e y where y = e z . The solution to this equation gives a value y * . The free energy is given by f = −y * /β and the internal energy . We have checked that the first orders in the high-temperature expansion of eq.(4) for the energy coincide with the previous expression. The energy goes to −2 at zero temperature (see fig.1). The specific heat (first derivative of the energy) increases approximately like 1/T 2 as the temperature is decreased and shows a maximum at T ∼ 0.20. Numerical computations for a larger number of particles (N = 10 5 ) show that finite-size effects are negligible. Below T ≃ 0.17 we observe a strong dependence of the energy on the cooling rate and a slow relaxation of the energy to its equilibrium value. Figure 1 also shows the strong dependence of the energy on the temperature change rate during the heating process. The numerical data merges to the static result at a certain temperature. This is also the temperature at which the energy departs from the static value in the cooling procedure.
The dependence of this merging temperature on the time spent on the cooling rate is an estimate of the relaxation time.
We want now to show that the energy converges to its equilibrium value. We have studied the relaxation of the energy at zero temperature starting from an initial random distribution of particles. Because there are no metastable states the system can reach the ground state.
We have measured the time the system takes to reach the ground state at zero temperature as a function of the number of particles. We have computed log(τ ) for different values of N ranging from 5 to 20 (the average ... means average over different random initial conditions). We find the typical time very well described by τ ≃ 0.39 exp(0.67N). This means that the system takes an exponentially large time to reach the ground state. We have not succeded in deriving an exact expression for the decay of the energy at zero temperature in the infinite N limit. The problem, being highly non trivial, can be approximated taking into account the previous result for the exponentially growing time. We argue that the time dt the system needs to decrease the fraction of occupied states in a quantity d( Noc N ) scales like For finite values of N this expression yields an exponentially large time for reaching the ground state. The previous expression means that for a small number of occupied states the rate of decrease of the energy is also small (there are less states with more particles per state to be emptied). Using u = −2(1 − N oc /N) we get for the decay of the energy where u 0 is the initial energy at time zero. We have measured the decay of the energy as a function of time. While this expression is only approximate it shows a remarkable agreement with the numerical data especially in the large time region over several decades of time. The relaxation of the energy, far from being of a logarithmic or algebraic type, is extremely slow as eq. (10) shows. The fact that the energy converges to the static result at zero temperature suggests that a dynamical transition at a finite temperature is absent. In what follows we will check this point by computing the relaxation time.
The relaxation time and aging. To fully characterize the dynamics of this model we have computed the relaxation time. We can define two types of correlation functions, one for the σ i variables, the other one for the energy state variables. In the regime of low temperatures the system performs a random walk changing particles from one state to another and the appropiate correlation function is given by the state to state energy function, where e r (t) = δ Nr(t) 0 and u(t) is the mean energy per site at time t. We have normalized it in order to have C e (t, t) = 1. For times t larger than the correlation time τ (T ) the C e (t, t ′ ) should depend only on the time difference t ′ − t (t ′ < t) and decay exponentially with time While a pure Arrhenius divergence τ ∼ exp(A/T ) does not fit enough well (data present a systematic curvature in figure 2) we find that a Vogel-Fulcher law τ ∼ exp(A/(T − T 0 )) [4] describes extremely well the increase of the relaxation time. The value of T 0 ≃ 0.02 is stable to including more points in the fit and is definitely better than the Arrhenius one. We do not attach special physical meaning to the value of T 0 but the indication of an anomaly in the divergence of the relaxation time. This is different to the case of models where metastability is present where T 0 can be identified as a thermodynamic transition temperature [5].
The dynamical transition in this model takes place at T = 0 and we expect for t finite the correlation function of eq.(11) to display aging. Introducing the waiting time t w and redefining the times t w = t, t ′ = t + t w in eq.(11) we find that C e (t w , t w + t) is pretty well described by the scaling law Similarly to other mean-field models [6,7], it is very plausible that this scaling behavior is exact in the large t w limit at least at zero temperature. Results are shown in figure 3.
Data collapse in a single curve, the scaling function f of eq.(12) scales like f (x) ≃ 0.78 x −0.70 for large values of x. If we define u EA = lim tw→∞ C e (t w , 2t w ) we find that this value jumps discontinuously to a finite value ≃ 0.58 at T = 0 being zero at any finite T in agreement with the absence of any finite T dynamical transition.
We can summarize now our results. We have introduced a simple model without energy