Replica methods

Very recently, the scientific interests of several leading theoretical laboratories have turned to studies of quenched-annealed fluids. To the best of our knowledge, there has not been a comprehensive review of the theoretical studies of quenched-annealed fluid systems. Our intention in this chapter is to fill, at least partially, an existing vacuum. Evidently, it is impossible to discuss the state of the art in this rapidly developing area in every detail in a single paper with restricted dimensions. We will omit, for example, the discussion of the fundamentals of the replica method for lattice systems, referring the reader to a monograph [1]. 

The logarithm of the partition sum of the annealed species under the sign of integration in Eq. (1) makes the analytical treatment difficult. Therefore, the replica method, see, e.g.. Ref. 1, is used, or, in other words the following identity is exploited... 

The size of the nanoparticles was determined with TEM after the final ion bombardment on the same sample. The samples for TEM investigation were prepared by extraction replica method. The particles were stripped off the substrate surface by means of collodion which was covered by carbon film. The collodion was dissolved after carbon deposition. 

Silica matrices MCM-48 and SBA-15, as well as carbons obtained by replica method are characterized by nitrogen adsorption at 77K and X-ray diffraction (Table 2). 

Yamaguchi M, Kondo I. Immunoelectron microscopy of Proteus vulgaris by the plasma polymerization metal-extraction replica method differential staining of flagellar (H) and somatic (0) antigens by colloidal golds. J Electron Microsc 1989 38 382-388. 

Tanaka A, Sekiguchi Y, Kuroda S. A new replica method for electron microscopic studies of a plasma polymerization film with a glow discharge. Seikagaku 1983 55 1212-1219. (in Japanese)... 

It should be noted that in our micrograph, in which the higher resolution replica method is used, the semispheres of about 200 A in diameter can not be observed (6, lA). 

The parallel-replica method [5] is perhaps the least glamorous of the AMD methods, but is, in many cases, the most powerful. It is also the most accurate AMD method, assuming only first-order kinetics (exponential decay) i.e., for any trajectory that has been in a state long enough to have lost its memory of how it entered the state (longer than the correlation time icorr, the time after which the system is effectively sampling a stationary distribution restricted to the current state), the probability distribution function for the time of the next escape from that state is given by... 

The parallel-replica method also correctly accounts for correlated dynamical events (there is no requirement that the system obeys TST), unlike the other AMD methods. This is accomplished by allowing the trajectory that made the transition to continue for a further amount of time Afcorr > Tcorr, during which recrossings or follow-on events may occur. The simulation clock is then advanced by Afcorr, the new state is replicated on all processors, and the whole process is repeated. 

Figure 1 Schematic illustration of the parallel-replica method. The four steps, described in the text, are (A) replication of the system into M copies, (B) dephasing of the replicas, (C) propagation of independent trajectories until a transition is detected in any of the replicas, and (D) brief continuation of the transitioning trajectory to allow for correlated events such as recrossings or follow-on transitions to other states. The resulting configuration is then replicated, beginning the process again. Reprinted, with permission, from ref. 6. Copyright 2002 by Annual Reviews, www.annualreviews.org...

Voter, A.F. Parallel replica method for dynamics of infrequent events. Phys. Rev. B 1998, 57, 13985-88. 

Figure 1.2. Electron micrograph of a RI FE fracture surface using the two-stage replica method. Large lamellae are visible with smaller lamellae perpendicular to them. Specimen was cooled from the melt at 0.12 deg/min. Scale bar = 1 pm.

Among the methods utilizing the TST, the parallel replica method is the simplest and most accurate dynamics techniques, because the only assumption for the method is that of infrequent events obeying first-order kinetics (exponential decay). In practice, replicated MD simulations are started in a particular basin state in search for a transition to an adjacent basin state on M non-correlated processors. Whenever a transition is detected on any processor, the... 

Figure 3. Electron-micrograph of polystyrene-rubber blend by the solvent-etch-double replica method...

Cottin (C50) earlier studied pastes of Ciment Fondu, largely by the TEM replica method. The microstructures of pastes of low w/c ratios hydrated for 7 days at 12°C were too compact to show more than occasional detail, but with pastes of w/c = 0,7 much detail could be observed in cavities exposed by fracture, which were lined with small crystals of CAH,o and sometimes contained plates of C,AHg. The crystals of CAH[q, which were hexagonal prisms up to 1 pm long and 0.2 pm wide, were present in rounded aggregates. Hydrous alumina was seen in a paste of w/c = 1.0 hydrated at 30"C it occurred in the lining of a cavity, mainly as rounded masses of randomly oriented platelets. 

The replica method can be also applied to express some of the thermodynamic properties of a confined fluid in terms of the correlation functions of the system. The details of the procedure can be found elsewhere [6, 26], In our system, we need to rearrange some expressions so as to avoid the difficulties in the numerical calculations due to the long range behavior of some correlation functions. 

Jackson, M. L. (1969). "Soil Chemical Analysis Advanced Course," 2nd edn. 7th printing, 1973. Department of Soil Science, University of Wisconsin, Madison, Wisconsin. Jackson, M. L. and G. D. Sherman (1953). Chemical weathering of minerals in soils. Adv. Agron. 5, 219-318. Jackson, M. L., S. Y. Lee, F. C. Ugolini, and P. A. Heimke (1976). Age and uranium content of soil micas from Antarctica by the fission track replica method. Soil Sci. 123, 241-248. 

In both cases an expression for the free energy results which is identical with tq. (37) derived by the replica method From the congruent results obtained by different theoretical methods for physically equivalent models, it can be concluded that Eq. (37) is representative for network models with harmonic constraining potentials. Equation (37) will be used extensively in all further discussions concerning the properties of the tube model of polymer networks. 

The parameter replaces the macroscopic deformation ratio only in the contributions to the free energy by the topological constraints. This can be shown rigorously using the replica method. It tmns out that the ideal contribution to the free energy of the network caused solely by the deformation of the boundary of... 

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 ... 

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... 

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