The introduction of replicas

To introduce the replica concept we consider an arbitrary physical quantity expressible in the form of Eq. (7.8), such as the internal energy of the adsorbed fluid 

Using Elqs. (7.6) and (7 8), the double average over one pair term appearing on the right side of Eq. (7.9), that is, for example, the term corresponding to = 1, j = 2, can be wTitten as 

We now multiply both the numerator and the denominator of the last term in the integrand by Zq. where n is an arbitrai y integer. Because Zq involves an integral over the Nf fluid particles coordinates [see Elq. (7.7)], this multiplication implies that we are introducing n - 1 copies (i.e., replicas) of the fluid particles. Assigning an arbitrary index a to the variables ( 1,92) appearing in the numerator of Eq. (7.10), and inserting Eq. (7.1) for Eq. (7.10) may be recast as 

Inspection of Eq. (7.12) indicates that the replicated system introduced by the multiplication trick is a (n + l)-component mixture composed of the matrix particles and altogether n identical copies (i.e., replicas) of the fluid particles. Each of the copies interacts with the matrix particles, but there are no interactions among different copies. 

Returning to Eq. (7.11), one now uses the fact that the denominator appearing on the right-hand side, ZmZg, which still depends on the realization Q , becomes independent of in the limit n —+ 0. More specifically, one has 

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In a next step we make ase of the fact that the n copies of the fluid particles are identical (this is obvious from the introduction of the replicated system described in Section 7.3). As a consequence, there is permutation symmetry between the replica indices. This implies for the fluid fluid and fluid matrix/matrix fluid correlations... 

We suppose, for sake of simplicity, that the proposed representation is stiU vahd for HSS, by assuming that different replicas of the user profile database are available on the same hardware. In real settings, different hardware parts and some load sharing equipments may be necessary. However, the methodological approach we are going to propose still holds and requires only the introduction of some new elements. 

The structure of the chapter is as follows. First, we start with a brief introduction of the important theoretical developments and relevant interesting experimental observations. In Sec. 2 we present fundamental relations of the liquid-state replica methodology. These include the definitions of the partition function and averaged grand thermodynamic potential, the fluctuations in the system and the correlation functions. In the second part of... 

Because of the nature of this templating synthesis, which involves introduction of a carbon precursor into pores of the template followed by carbonization and template dissolution, inverse carbon replicas of the OMS templates are obtained. Although carbon replicas of MCM-48 often undergo a symmetry change ° (Figure 12.7) and the MCM-41 replication results in disordered carbon rods, the SBA-15 replication permits exact inverse replicas (Figure 12.8). The observed differences in replication of the aforementioned OMSs are caused by structural differences of these templates. For instance, the MCM-41 and MCM-48 (P6 mm and Ia3d symmetry, respectively) OMSs are... 

A server replica has two operational modes, primary or backup. We define in this section the micro-aspects needed to implement them. The first action is to initialize the operational mode of the replica. An AspectJ field introduction is used to set up the mode for each replica. A micro-aspect named AServerMode is responsible for the initialization of the replica s mode. It offers a unique advice AServerMode.init. 

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