Replica interaction

Figure 9. (a) Inter-replica interaction in the RANI model. The dotted wavy line indicates the r -type interaction between the pairs (1,2) and (3,4). (b) indicates a loop formed by the disorder induced interaction. 

When we consider the second cumulant of the partition function, we require four chjuns. On averaging over the disorder, a new inter-replica interaction is generated that couples... 

Here, the replica interaction vertex g x) is the correlation function with Fourier image (26). 

In the other method, the space is considered filled with an infinite lattice of replicas of the system charges. Considering that the system is neutral and that the charges and those in the replicas interact by the Coulomb potential, it is easy to write the formal expression of the total Coulomb energy ... 

According to some remarks concerning the physical interaction between the incident ultrasonic wavelet and the defects [4-6], we consider that an Ascan signal, may be described as a weighted sum of few delayed and phase-shifted replicas of the ultrasonic incident wavelet j(r). We can express this mathematically as ... 

Here the superscripts serve as replica indices they attribute the coordinates of particles to a given replica but do not alter interactions in the system... 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

Sayano et al. [97] generated the potential of mean force for interactions between amino acids and base pairs. Okamoto and coworkers [98, 99] have also developed replica-exchange multicanonical methods for proteins. Numerous additional biological examples can be found among the references within [94]. 

In the EA model the spin-spin interaction is only of the nearest-neighbor type. The Sherrington-Kirpatrick (SK) model [79] is the infinite-range version of the EA model. It is most useful as a basis for mean-field calculations. One such solution is the replica symmetry breaking theory of Parisi [80-82]. 

Defects with large central cell correction have very localized wave functions. The larger the correction, the more localized the wave function and the higher the probability of interaction between the core (or central cell) and the electron and/or the exciton bound to the defect. Hence the reason why the line is so much smaller than the Q line in the spectrum, as well as the reason why the phonon replicas to the Q-line, is simply a matter of probability, since the central cell correction is so much larger for a nitrogen defect on a cubic site than a hexagonal site. 

Consider a molecule prepared in the absolute ground state in the absence of the field and subjected to microwave field of frequency . If collided with a structureless atom in the absence of the field and at collision energies below the first excitation threshold, the molecule can undergo only elastic scattering. In the presence of the field, the ground state of the molecule becomes a field-dressed state X). And for every field-dressed state X), there is an infinite number of replica states 2 - A ), lower in energy. The states 2 - A ) and X) are coupled by the anisotropy of the atom-molecule interaction potential, so collisions can induce... 

Such treatment of the CIDNP results produced serious objections. Lippmaa et al. (1973), investigating the same reaction, revealed a strong 15N, 13C, and H CIDNP effect. The 13C nuclei in the phenoxyl C6-ring of the azo dye were not polarized. At the same time, the polarization of 15N nuclei of the azo bond and 13C nuclei at positions 1 and 2 of the phenyl ring connected with the diazo link was an exact replica of the polarization of the same nuclei in the diazonium salt. This has led to the conclusion that the diazo component polarizes as a result of the side reactions and that it is the diazo component that brings it to the azo dye. Thus, the CIDNP effect does not support the ion radical mechanism presented earlier. Several explanations for the observed CIDNP effect have been proposed. We want to discuss one of them here because it seems to explain a whole range of interactions of diazonium salts with oxyanions, an interaction that is accompanied by a pronounced polarization of nuclei. 

Now consider a fractional replica of type 215 11, which is the 1/2048-replica of a FUFE. It is pointless in this case to write down all aliased/confounded estimates as their number is enormous. As an example, linear effects are aliased/confounded with 105 even interactions. The design matrix of 215 11 is shown in Table 2.96. 

The reaction takes place with solvent E being present. Five factors, as shown in Table 2.121, were varied in lab conditions. In designing the experiment it has been assumed that interaction effects are not significant. The experiment was therefore done by 2s 2 or 1/4-replica each. The design matrix with experimental outcomes is shown in Table 2.122. 

Let us analyze the previous case by taking into account the third factor X,. The outcomes of FUFE 23 and the results of application of method of steepest ascent are given in Table 2.216. Thirteen trials were necessary to reach the maximal yield of 85.2%. The outcomes of the simplex method are in Table 2.217. Maximal yield after 14 trials is 85.0%. Approximately the same number of trials has been necessary by both methods to reach the optimum. It should be stressed once again that FUFE requires replications, so that to reach optimum by the method of steepest ascent, we need at least twice as many trials. Evidently, a half-replica instead of FUFE in the basic experiment may reduce the number of trials. However, there is a possibility of wrong direction of the movement to optimum due to the possible effects of interactions. 

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