Disorder quenched

The presence of quenched disorder naturally leads to the following questions  

An important early step toward answering these questions came from the work of Harris who considered the stability of a critical point against disorder. He showed that if a clean critical point fulfills the exponent inequality 

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Here, the second term describes the change of the hopping amplitudes due to the displacement of the atoms parallel to the chain [cf. Eq. (3.5)] and the third term is a random contribution resulting from the conformational disorder (chain twists). While the lattice displacements u are dynamic variables, the fluctuations dl +t due to disorder are assumed to be frozen ( quenched disorder). 

Already at the very beginning of his studies in this field [ 1 ] I.M. Lifshitz insisted that the most important stage of the work to follow should be developing a theory of heteropolymers with a frozen (or quenched) disordered sequence of chemically different units. Many years passed before an essential progress was made in this direction despite the motivation originating from protein physics, that is the very problem of protein globules that I.M. Lifshitz had in mind while starting to study polymers. 

J. P. Bouchaud and M. Mezard, Self induced quenched disorder a model for the glass transition. J. Phys. I (France) 4, 1109-1114 (1994). 

It was shown theoretically (D.P. Li et al., 2006) that the presence of quenched disorder might be responsible for the positive slope of the H2(T) line in Figure 53, whereas in the clean case the hexagonal-to-square transition line would be parallel to the temperature axis if thermal fluctuations are neglected. 

However, in all the papers mentioned above the authors analyzed only three-dimensional (3D) systems, while a two-dimensional (2D) case is also experimentally observed surfaces of various absorbers, heterogeneous catalysts, photocatalysts, etc. In [137], Fel dman and Lacelle examined the quenched disorder average of nonequilibrium magnetization, i.e., a free induction decay G(t) and its relative fluctuations for dipolar coupled homonuclear spins in dilute substitutionally disordered lattices. The studies of NMR free induction decays and their relative fluctuations revealed that the functional form of the disorder average (G(t))c depends on the space-filling dimentionality D of the lattice. Explicit evaluations of these averages for dilute spin networks with D = 1, 2, 3 were presented in [137] ... 

Two types of models have been suggested, namely, a diffusion approach and a random trap model. The measurements of Dahan s and Bawendi s groups [5,7], which show the universal power law a = 0.5, are consistent with the diffusion model (see details below). The fact that all dots are found to be similar [5] seems not to be consistent with models of quenched disorder [4,9,10] since these support the idea of a distribution of a . However, some experiments show deviations from the a+ a 0.5 and may support a distribution of a . It is possible that preparation methods and environments lead to different mechanisms of power-law blinking, along with different exponents [6]. More experimental work in this direction is needed in particular, experimentalists still have to investigate the distribution of a and need to show whether and under what conditions are all the dots statistically identical. We discuss the diffusion model below different aspects of the tunneling and trapping model can be found in Refs. 4, 6, and 10. 

Another variation on the theme of quenched disordered structures noted in Section ILA is to employ a removable template such as an organic molecule, colloid, or metal ion during the synthesis of a porous material [52-54], Following formation of the quenched material structure, as shown in Fig. 7, the template is removed, leaving behind a matrix of particles with a pore space that mimics, to some extent, the original template. Because templates of diverse size and shape are available, templating offers the prospect of designing porous materials whose architectures are tailored for specific applications. 

For coiiveiitioual fluids the outer (disorder) average of the first term on the right side is absent and each thermal average equals the singlet density. Thus, hb = 0 for systems without quenched disorder. In the presence of disorder, on the other hand, the blocked correlation function is usually nonzero, because the singlet density for a particular realization, (9 ) can be... 

For most of this book we consider cases of ideal confinement, that is, situations where the geometry of the confining substrates is simple. The most prominent example is that of a slit-pore where the confining substrates are planar and parallel to one another. In Chapter 7 we focus on the op>-posite extreme, that is, a fluid confined to a randomly disordered porous matrix. Experimentally this situation is encountered in aerogels. The simultaneous presence of both confinement and (quenched) disorder representing the nearly-random silica network renders the treatment of such s3rstems quite challenging from a theoretical perspective. In Chapter 7 we discuss one of the... 

It is a natural assumption, that local communities in habitats at different geographic locations should vary in their local growth conditions. Therefore, quenched disorder is imposed onto the system by assigning to each local model i an independent set of control parameters Xi =... 

The most important consequences of atomic disorder are observed in the permeability value, shown in Fig. 6.7 for Ni-Fe alloys. Permeability peaks at 75%Ni ( Ni3Fe) for quenched (disordered) samples for slowly cooled (ordered) samples, two small maxima are observed. These results can be explained in terms of the anisotropy behaviour. Fig. 6.8. Quenched samples show a monotonic decrease in anisotropy as the Ni content increases, crossing the Xj = 0 axis for 75% Ni, the composition with maximum permeability. Slowly-cooled samples exhibit similar behaviour overall, but anisotropy crosses the K, = 0 axis at 65% Ni and shows a minimum for NijFe. Accordingly, the permeability of... 

However, since f AT , where AT = T — Td/T, we can express AT as For a sharp transition the local fluctuations in T, is required to be much smaller than this critical temperature interval AT, and consequently we need AT, AT, or d/2 > or (2 — dv) < 0. Using the hyperscaling relation a = 2 — dv, the above condition becomes a < 0 indicating nontrivial effect (a new different sharp transition with negative value of a, or a smeared transition) due to quenched disorder for (pure) systems with a > 0 [8]. 

Let us note, that there are two classes of problems, dealing with disorder, utunely those with annealed and quenched disorder. In the first case, the different realizations of disorder are averaged simultaneously with a thermodynamical averaging over different conformations of the SAW. In the present review, we however focus on the case of quenched disorder [15], which is introduced such that it is not in thermodynamic equilibrium with the unperturbed system. The quantities of physical interest must then first be calculated for a particular configuration of disorder, followed by the average over all configurations of disorder. 

As far we consider the case of quenched disorder, the free energy of the system is obtained by averaging the logarithm of the partition function Z over the disorder distribution [15] this amounts to use so-called replica trick [52] writing the logarithm in the... 

There exists a second way to obtain the effective Hamiltonian (21). A weak quenched disorder term can be introduced directly into the effective Hamiltonian (13). The presence of non-magnetic impurities in a microscopic model (15) manifests itself in fluctuations of the local temperature of the phase transition. Introducing = V ( ) the field of loccd critical temperature fluctuations, one obtains the effective disordered Hamiltoni ln [53] ... 

Applying the replica method in order to average the free energy over different configurations of quenched disorder one finds the effective Hamiltonian of the m-vector model with long-range-correlated disorder [65] ... 

To perform the average of the free energy over the quenched disorder, the replica method is applied and for the n-replicated partition function we obtain ... 

Something similar happens if the s, (5-expansion is applied to study models of m-vector magnets with long-range-correlated quenched disorder [65,71] also in the case of magnets, as well as for polymers the first order e, (5-expansion leads to a controversial phase diagram. In order to obtain a clear picture and more reliable information, one should proceed to higher order calculations. 

Solving the SAW problem with quenched disorder is another interesting question. For the 3-simplex fractal, this corresponds to making the variables random variables, and one has to determine the probability distribution of this variable for large r. 

Note that, due to historical reasons, what is called quenched disorder is sometimes also referred to as annealed average in the literature, whereas annealed disorder is denoted as quenched average . 

The phase diagram for random copolymers with quenched disorder that gives the change in the critical adsorption potential, e, with changing percentage of the sticking A-monomers, p, has also been determined from extensive computer simulations carried out with the two employed models (cf. Fig. 11b). We observed... 

Guegan R, Morineau D, Loverdo C, Beziel W, Guendouz M (2006) Evidence of anisotropic quenched disorder effects on a smectic liquid crystal confined in porous silicon. Phys Rev E73(l) 011706-011707... 

At the nematic-isotropic transition, the response time 300 ns. Modulating the thermal grating into the isotropic phase results in a viscous limited decay associated with a quenched disordered nematic phase. These results could be of interest in the analysis of thermally written smectic display devices, where the picture element is pulsed into the isotropic phase. ... 

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