Aging urn models

Urn models consist, in their generic definition, of one or more sets of balls (particles, pawns, . . .) and a number of urns (boxes, cells, states, . . .) where the balls can be placed or taken from, according to a given extraction law regulating the evolution of the model. They are a classical issue in probability theory [Feller, 1993; van Kampen,

1981], and have been used, as well, to build relational databases in learning theory [Gardy & Louchard, 1995; Boucheron & Gardy, 1997; Drmota et al., 2001], to determine the efficacy of vaccines [Hernandez-Suarez & CastilloChavez, 2000], to study epidemic spreading [Daley & Gani, 2001; Gani, 2004], population genetics [Hoppe, 1987] and to represent evolutionary processes [Schreiber, 2001; Bena¨ım et al., 2004], just to mention a few applications. The urn models have, furthermore, played a very important role in formulating fundamental concepts of statistical mechanics such as the approach to equilibrium and fluctuations out of equilibrium [Kac & Logan, 1987]. The prototype of the dynamic urn models in statistical mechanics and one

of the most intensively studied ones has been the Ehrenfest model, otherwise called the “dog-flea” model, introduced with the aim of (critically) analyzing the H-theorem of Boltzmann [Ehrenfest & Ehrenfest, 1907]. In this model a given number N of fleas are randomly distributed over two dogs. The dogs stay near enough to each other so that the fleas can freely jump from one dog to the other one. In a probabilistic language, we have N distinguishable balls distributed between two urns. The dynamic rule of the model is elementary: at each time step a randomly chosen flea is “called” and changes dog. All fleas are equivalent. Even though extremely simple, the model has been a stimulus for many decades in physics and mathematics, even after it was exactly solved by Kac [1947], Siegert [1949] and Hess [1954] (see also [Kac, 1959; Emch & Liu, 2002]). To solve the problem means finding the evolution law for the number of fleas on each one of the two dogs. Since no interaction, nor energetic cost, nor constraint is involved, the occupation numbers at equilibrium are determined according to the requirement of maximum entropy. The dog-flea model has been generalized in various ways in the course of its century-long life. According to the classification of Godre`che & Luck [2001], we can identify a whole class of models whose common origin is the dog-flea model: the ”Ehrenfest class” of dynamic urn models. More generally, one can define a dynamic urn model specifying

of the

1. the components (urns and balls)

2. their statistics

3. the cost function

4. the dynamic algorithm

5. the geometry

The base of the behavior of dynamic urn models fundamentally resides in the way the starting and the ending point of each dynamic step are chosen (the statistics, point 2.). In the case just considered, the ball-to-box choice of the statistics, actually defines the Ehrenfest class of models. Further specifications are the transition probability of ball-to-box moves and the energy function, but the discriminating ingredient is the statistics. We will see in the specific case of the backgammon model how this choice implies a Maxwell-Boltzmann statistics for the occupation probabilities at equilibrium (Sec. 4.1). A qualitatively different class of models can, for instance, be defined by a

box-to-box choice for the single move: we choose a box, we take any ball at random from that box, we put it in another box, randomly chosen. The class of models produced this way is called the monkey class [Godre`che & Luck, 2001]. The equilibrium statistics computed for monkey models turns out to be Bose-Einstein, even though nothing quantum is involved in the box-to-box update. Example models belonging to this class are the “B-model” of Godre`che &

Me´zard [1995] and the zeta urn model [Bialas et al., 1997]. The B-model is identical to the backgammon model but for the statistics, that is, however, crucial. The zeta urn model has a Hamiltonian H = ∑Ni=1 log ni + 1, where ni are the occupation numbers. Apart from nontrivial dynamic properties displaying aging and coarsening off-equilibrium behavior [Drouffe et al., 1998; Godre`che & Luck, 2001], it also owns a static transition at finite temperature to a condensed phase. The monkey models are, however, not good models for glassy materials and we will not consider them any further. Concentrating on Ehrenfest models, an important property that can be

encoded, introducing disorder, is the existence of collective modes, that is, modes connected to the slowest processes evolving in a glassy system and carrying on the structural α relaxation. Take a liquid well above the glass temperature, where correlations decay

exponentially with time. One may consider the resultant behavior of the liquid as the superposition of different and independent harmonic modes. Each one of these energy modes corresponds to a normal mode of a system and describes a collective oscillation of the atoms around their local minimum. This is the harmonic approximation, known to work quite well in liquids (cf. Sec. 6.1.5). Nevertheless, already as the temperature undergoes the dynamic glass temperature (Sec. 1.1.1), other collective modes, different from the standard

vibrational ones, become important. The nature of these modes is quite different from the usual harmonic normal modes because they do not represent oscillations around a given configuration within a metastable well, but transitions among different wells. As the characteristic energy of the collective modes depletes, the typical

barrier separating these modes increases, leading to the opposite behavior with respect to the harmonic modes and to super-activation effects: while high energy collective modes are separated by low barriers, low energy collective modes are separated by high barriers and relax more slowly. A simple schematic representation of this scenario in a one dimensional configurational space is shown in Fig. 4.1. Below the dynamic transition, relaxation dynamics proceeds by activation

over the barriers characterizing the cooperativeness of the molecules over largely extended regions (see Sec. 1.1.1). In the model that we will discuss in Sec. 4.3, the size of these cooperative rearranging regions (CRR) cannot be inferred, but they can be related to the collective modes. As the temperature decreases, indeed, collective modes at low energy can be thought of as the representation of collective rearrangements of large regions, requiring higher activation energy and taking place at a lower rate. We will tackle this problem at the end of the chapter by means of a gener-

alization of the backgammon model where each urn has a distinct, randomly distributed, weight (or energy) [Leuzzi & Ritort, 2002]. Because of the different random energy values that the holes acquire, an analysis of the behavior of a material presenting several modes can be carried out in the situation in which those modes at larger energy (in absolute value) have thermalized while those at low energy relax too slowly to reach equilibrium on a given timescale (in the first order approximation of uncorrelated modes). This is qualitatively similar to the experimental results found, e.g., by Bellon et al. [2001] for the relaxation at different frequencies, Sec. 2.8.