Weakly Inhomogeneous Models

In this chapter we consider polymer models with periodic charges. Precisely we consider the copolymer with adsorption model of Section 1.6.3, that is the measure defined by (1.66). This of course includes poljmier 

In this chapter the aim is above all to stress the basic steps that allow to reduce the weakly inhomogeneous case to the homogeneous one and convey the idea that in this class of models one gets as far as in the homogeneous pinning model treated in Chapter 2. This reduction is not without a price, since in reality formulas become substantially more complex in the details. However it should be stressed from now that new phenomena appear in this set up and one of our purposes is to point out in an informal way the phenomenological richness of periodic models. 

For the sake of conciseness we will work only with (p, g)-walks so K(n) Cjcn i.e. a = 1/2 (see Appendix A.6). 

1 A Formula for the Free Energy Reduction to a Finite Dimension Problem 

In this section we will make some algebraic manipulations on the partition function of periodic models that lead to a formula for the free energy. This is the generalization of Proposition 1.1 and of formula (1.6) to weakly inhomogeneous models. Very much like in that case, the computation of the 

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As we have already stressed, the analogy with (2.18) is evident and it is probably not surprising for the reader that from such a formula one can extract the sharp behavior of the partition function. It should however be noted that, while in the positive recurrent set-up ( > 1) the theory of Markov renewals is well developed, fewer results are available in the literature on the mass renewal function of null recurrent Markov renewals. Moreover the results, even only at the level of sharp asymptotic behavior of the partition function, are more involved. As we shall see, this complexity is not only of a technical nature, but it really reflects a substantially larger variety of phenomena that can be observed in weakly inhomogeneous models, with respect to homogeneous ones. 

The result we want to prove is that a quenched sequence of charges, even if they are centered and so on the average the charge is zero, plays successfully in favor of localization (the analogous result for weakly inhomogeneous models is proven in Proposition 3.8). This is of course due to the fact that the typical polymer trajectories target positive charges. We introduce the... 

The difficulty in proving the superior limit statement is the same as the one we have encountered in proving the convergence of the critical curve of weakly inhomogeneous models to the one of disordered models, see Figure 4.1. With the (considerable) difference that in this case one can actually show that limsup ) o c(A)/A < ihc-... 

A general consideration on this chapter is that the results presented are sensibly weaker than those in Chapter 2 and Chapter 3 for homogeneous and weakly inhomogeneous models. It is then natural to ask what should we expect to observe in the delocalized regime of disordered models This question is particularly relevant also because weakly inhomogeneous models have been studied as caricatures of disordered models. 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

This initial hypothesis was later revised, since some researchers (such as Walker et al., 1983) were able to show that, according to the model of inhomogeneous accretion, metallic iron was removed from the Earth s crust in a very early phase and accumulated in the core. These results led to the now generally accepted theory that the young Earth was surrounded by a weakly reducing atmosphere. 

Although Ba and heavier elements seem to fit the solar r-process pattern, lighter elements show wide varieties [5]. In particular, a large dispersion has been found in [Sr/Ba] at low metallicity[l], suggesting that lighter elements such as Sr does not come from a universal process, which produces Ba and Eu, but from weak r-process. An inhomogeneous chemical evolution model suggests that the dispersions in [Sr/Ba] are well-explained, when weak r-process produces 60% of Sr but only 1% of Ba in metal-poor stars. Furthermore, intermediate mass elements such as Pd must provide clues to understand the weak r-process yield. 

Deviation from the ideal exchange kinetic dependencies introduced by selectivity effects can arise in any ion-exchange system in vsiiich the resin phase ions can exist in two different states i.e., relatively free (condensed) and bound (complexed) as assumed in the model projected [45-50]. This is true for complex forming, weakly dissociating, chemically and structurally inhomogeneous ion exchangers. 

Figure 12. ISS data from CSj liquid at 165 K recorded in transient grating experimental configuration with V- and H-polarized excitation pulses. Weakly oscillatory signal indicates that orientational motion is librational in character. Inset fits to data based on models of homogeneous dephasing (broken curve) or inhomogeneous dephased (dashed curve, under data).

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