Weakly inhomogeneous

Of course, the above discussion apphes only to systems exhibiting domain wall structure, i.e., to weakly inhomogeneous phases formed on surfaces with low corrugation of the gas-solid potential and characterized by the presence of more then one type of equivalent sublattices. When this is not the case, i.e., when the dense incommensurate phase can be considered to be... 

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]. 

To approach this problem from our theory, we extend our Feh Eq. (3.6), to the case in which the crosslink density is weakly inhomogeneous as v(jc0) = v0 + <5v(jt0) P3]. The correlation of the deviation <5v(jc0) s short-ranged in the relaxed state as... 

In optical terms the Lagrange manifolds are the same as what Hamilton called ray systems (a system of normals to a smooth surface may serve as an example). In this case, the caustic is the point of the concentration of light. Thus, Ya.B. s pancake theory in particular predicts that the first caustics in a weakly inhomogeneous system of rays have the form of saucers (in a dispersive medium the saucers may become visible). 

We now consider the simple case of a spin echo in which two rectangular linear gradient pulses are applied as shown in Fig. 1. In this pulsed gradient spin-echo (PGSE) experiment, the time t used in Eq. (41) will refer to the position of the spin-echo formation. In analyzing this experiment, we will assume the weak inhomogeneous field limit and the uniform... 

Of practical interest is the ensemble averaged quantity X which is supposed to be weakly inhomogeneous for highly developed turbulence and crucially dependent on boundary conditions. Averaging (21) yields ... 

Here amplitudes 77, can be considered as order parameters the last expression includes also term with r)p = c and kp = 0 and for simplicity both parameters and factors exp(iksri) are supposed to be real which is the case, in particular, for the B2, Llo and Ll i-type order. For weakly inhomogeneous states, amplitudes r/p in (2) are not constants but smooth functions of coordinates rj. Thus, functions f cj in (1) can be expanded in powers of differences Sc.j = IZp[Pg.402]

Definition 1.9 A charge sequence u>n n is weakly inhomogeneous or periodic if there exists T G N such that u n+r = for every n. The minimal such T is called the period and it is denoted by T(w). A periodic sequence is said centered if = 0. 

Out of the class of weakly inhomogeneous charge distributions, the application of coarse graining ideas becomes much less evident. 

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. 

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... 

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. 

If > 1 the proof follows easily as in the case of Theorem 2.5(1), applying of course the estimate in Theorem 3.4(1), and choosing a = c or a = f makes no difference. In particular one shows that (2.39) holds also in this weakly inhomogeneous context. 

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