Periodic pinning models

Localization and delocahzation for a periodic pinning model may be characterized once again by looking at the free energy. As we will explain in Chapter 3, it is possible to generalize the renewal theory approach introduced in Section 1.2, however the algebra is substantially more complex and the point process hidden behind periodic models is not a standard renewal, but rather a Markov renewal (see Chapter 3). This will allow precise computations, but for the moment we observe that ... 

While a quantitative analysis of the periodic pinning model of Section 1.6.1 is not immediate, on a qualitative level the mechanism of the transition is not new with respect to the corresponding homogeneous model. Now instead we are going to present a case in which the inhomogeneous character of the charge distribution is at the base of the localization mechanism. 

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. 

We include in this section also another result that is in the spirit of Proposition 3.7. Consider the general model (3.1), but set = 0 for every n, that is there is no copolymer interaction (we denote by its free energy). The charges oj are instead periodic but not necessarily centered. The result we are going to state says that the periodic pinning model localizes more than the corresponding homogeneous model. 

Fig. 3.2 The free energy of the periodic pinning model compared to the one of the associated homogeneous model.

Fig. 4.1 A (possible) plot of the free energy for a disordered pinning model (thick line) and for three approximating periodic models, at f3 fixed. Even if lim7 ->oo =...

Fig. 12. The rotation periods To offree spiral waves (crosses) and r ofthe spiral waves pinned by the minimal hole (black dots) obtained by numerical simulation of the reaction-diffusion model (60)-(62) at different values of the parameter s. Curve 1 gives the dependence of To on in the kinematical approximation when the recovery effects are neglected. Curve 2 shows the dependence of the rotation period T of the spiral wave pinned by the minimal hole in the same approximation. Curve 3 gives the dependence of T on e calculated using the kinematical approximation including the effects of recovery. The dashed curve shows the computed dependence of the minimal period Tmin of stable propagation of pulses in the same model. (From [27])...

The radius R of the minimal hole that is still able to maintain a pinned spiral wave and the rotation period T of this pinned wave are important properties of an excitable medium. Figure 12 shows (curve 3) the rotation period T as a function of e, computed in [27] using the kinematical description. We see that it fits well the data (black dots) of the numerical simulation of the respective reaction-diffusion model. 

Spiral waves have been studied in most detail on Pt(l 10), both under conditions of excitability and of double metastability [108]. Under identical external conditions the spirals did not exhibit a fixed period and wavelength, rather a continuous distribution of these quantities was observed (Figure 18). It is well known that spirals can be pinned to artificial nonexcitable cores [109, 110]. Such artificial cores can be formed by surface defects consequently the continuous distribution of rotation periods simply reflects the fact that defects of various size exist on the surface. Comparison of the dispersion relation computed from the model [111] with the experimentally observed periods and wave velocities (Figure 19) allows conclusions to be drawn about the size distribution of surface defects by using the Tyson-Keener formula [112], which gives the relationship between core size and period. For the data... 

Fig. 3.1 The (qualitative) phase diagram of a periodic copolymer with pinning strength — A, that is the model with energy given in (3.34). The system with parameters chosen above the half-line (Ac, oo) has g = limAr, oo(l/ ) 5Zn=i = + > while it...

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