ABSTRACT
The study of exactly solvable models has been always an active area of research in the field of statistical physics. They help us to grasp general principles governing the physical behavior of realistic systems that, due to the complicated interactions among their different constituents, cannot be predicted using standard perturbative techniques. Glasses in general are systems falling into this category. The slow relaxation of glasses observed in the laboratory is a consequence of the simultaneous interplay of its components that yields a very complex and rich phenomenology. Firstly, we want to get more insight into the glassy dynamics, in its vari-
ous aspects, exploiting the analytical solubility of the models we will discuss. Indeed, thanks to their simplicity, the features of the glassy materials can be connected in a direct correspondence with given elements of the models. We can even switch on and off certain properties or certain dynamic behaviors, tuning the model parameters or implementing a given - facilitated or constrained - dynamics in alternative ways. Furthermore, in some cases, the thermodynamic state functions, including the configurational entropy, can be computed as functions of the dynamic variables of the model. Our second goal is to check the principal applicability and generality of the concept of effective temperature, very often discussed in literature in many different approaches (see Chapter 2) and to verify whether the possibility exists of inserting such a parameter into the construction of a consistent out-of-equilibrium thermodynamic theory. The question whether there exists a single effective temperature encoding the aging dynamics and whether this coincides with the fluctuationdissipation ratio is still controversial (see Sec. 2.8.2). We believe that this and other related questions can be better addressed by analyzing simple models. Recent reviews of kinetically constrained models for glass are, e.g., Workshop [2002], Ritort & Sollich [2003], Leonard et al. [2007]. In this chapter, we propose a set of dynamically facilitated models un-
dergoing a Monte Carlo parallel dynamics. For these models the statics is trivial and, nevertheless, the exactly solvable dynamics exhibits interesting glassy aspects. Basically all of the relevant features of much more complicated real glasses are displayed, such as aging, diverging relaxation time to equilibrium (either Arrhenius or Vogel-Fulcher), configurational entropy, the Adam-Gibbs relation between relaxation time and configurational entropy,
of the
Kauzmann transition, off-equilibrium fluctuation-dissipation relation and Kovacs effect. In the next chapter, we present urn models whose dynamics proceeds through
entropic barriers and that can be exactly computed in a certain adiabatic approximation. The dynamic behavior describes the slow relaxation and many related properties typical of glasses. Also there we will face the case where the dynamical relaxation of different energy modes can be made explicitly clear. In Chapter 5 we present another solvable model based on directed polymers, that is simple enough to be analytically worked out and yet displays glassy behavior. The possibility of computing an exact solution for the dynamics allows for
a precise formulation of the two temperature picture presented in Secs. 2.3, 2.4, 2.5. Even though the physics of the class of models that we are going to discuss in this chapter is simple, we shall formulate general aspects of the results by analyzing them in thermodynamic language. This thermodynamic formulation also incorporates the interpretation of the fluctuation-dissipation ratio (FDR) as an effective temperature, as exposed in Sec. 2.8. Indeed, the relation between thermal correlation functions and responses to external drivings has become a central point in investigating out-of-equilibrium systems. The approach of Sec. 3.3, will show that the effective temperature that occurs in thermodynamics and the one that occurs in the FDR can be equal in the aging regime only if the relaxation is slow enough. We will discuss and quantify the latter expression in terms of the parameters of the models that will be presented in the following. The working hypothesis at the basis of the simplified models we will deal
with is shaped on one particular physical property of glasses: the exponential divergence of timescales around the glass transition.1 This induces an asymptotic decoupling of the time-decades (cf. Sec. 1.1). The reasonable assumption is then made that, in a glass system that has aged a long time t, all processes with equilibration time much less than t are in equilibrium (the β processes) while those evolving on timescales much larger than t (if existing) are still quenched, leaving the processes with a timescale of the order t (i.e., the α processes) as the only interesting ones. The asymptotic decoupling of timescales is the input for the family of models we are going to analyze in Sec. 3.2 and could be the basis for a generalization of equilibrium thermodynamics to systems out of equilibrium. We will see, specifically in Sec. 3.3, how this approach involves systems in which one extra variable (the effective temperature) describes the nonequilibrium physics and if and up to which extent the addition of this single variable is sufficient to yield a consistent thermodynamic theory, able to describe even typical off-equilibrium phenomena such as memory effects (Sec. 3.5). Always employing harmonic oscillators, in Sec. 3.6 we will discuss the problem of the direct measurement of an effective temperature and, in Sec. 3.7, its dependence on different frequency modes.
