Replica trick

The mean field treatment of such a model has been presented by Forgacs et al. [172]. They have considered the particular problem of the effects of surface heterogeneity on the order of wetting transition. Using the replica trick and assuming a Gaussian distribution of 8 Vq with the variance A (A/kT < 1), they found that the prewetting transition critical point is a function of A and... 

Therefore, the central technical problem is the calculation of averaged value of F = — InZ this can be done using the so-called replica trick ... 

This replica-trick method was used in the Refs. [60,61] for the polymer network free energy calculation. For averaging of Z"-value over the ensemble of realizations the probability distribution should be chosen. The authors of Refs. [60,61] have used the condition of the thermodynamical equilibrium for the Gibbs probability distribution corresponding to the conditions of network preparation. To our mind, it is a beautiful idea but it should be considered more deeply because not all the types of networks can be described in such a way - many networks cannot be prepared under any equilibrium conditions. Using some additional tube-like approximations, the authors have obtained rather simple results for network elastic constants and for some other parameters. 

Dipolar fluid excess chemical potential. A closed expression for the evaluation of this quantity can be easily derived in the HNC approximation using Lee s star function technique [27] and the replica trick [18], leading to... 

Application of the replica trick of Edwards which is based on the identity... 

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... 

Averaging over disorder The replica trick The numerical results in the previous section reveal that a polymer in a random potential can be viewed as a glassy system. In order to develop this point of view, wc apply the replica method, which has been successfully applied in the study of glassy systems [21]. The replica approach was invented in order to compute quenched averages such as as given in Eq. (10), which is difflcult since it entails averaging a ratio of the... 

In this equation, y is the interaction strength, c(r) the crosslink concentration, the smectic order parameter, and Vz (r) the relative displacement of the rubber matrix. Witkowski and Terentjev [132] evaluated (15) for (r) = 1, which is valid deep in the smectic phase, i.e., far below the smectic-nematic transition. Using the so-called replica trick, they integrated out the rubbery matrix fluctuations and obtained an effective free-energy density that depends only on the layer displacements M(r). Under the restriction that wave vector components along the layer normal dominate over in-layer components, q q, and considering only long-... 

In order to avoid averaging In Z, EA use the so-called replica trick, where one averages quantities such as Z" when n is an integer. The relevance of averaging Z" instead of In Z lies in the relationship... 

The averaging of a logarithm is a difficult problem but with the use of the replica trick we can define F(n)... 

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. 

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