Replica integral equation

From a practical point of view, however, it is clear that the procedure described above is highly nontrivial Apart from the necessity to deal with mixtures of n-l-1 components, the way to carry out the limit n — 0 in practice is by no means straightforward. One method to deal with these difficulties is the replica integral equation formalism, which we will introduce Section 7.5. However, before doing this we first introduce the key concepts of the replica integral equations, which are the two-particle correlation functions of the QA system. 

The earliest applications of the replica integral equation approach date back to the beginning of the 1990s. They focused on quite simple QA systems such as hard-sphere (HS) and LJ (12,6) fluids in HS matrices (see, for example. Refs. 4, 286, 290, 298, 303, 312, and 313 for reviews). Fiom a technical point of view, these studies have shown that the replica integral equations yield accurate correlation functions compared with parallel computer simulation results [292, 303, 314, 315]. Moreover, concerning phase behavior, it turned out that the simple LJ (12,6) fluid in HS matrices already displays features also observed in experiments of fluids confined to aerogels [131, 132]. These features concern shifts of the vapor liquid critical temperature toward values... 

Motivated by this success, a series of more recent replica integral equation studies has focused on the effects of more realistic features of both the adsorbed fluid and its interactions with matrix. Examples are studies of the influence of templated matrix materials [316], associating fluids [317], LJ mixtures [114, 311], and QA systems with ionic interactions [305, 306, 318-320]. However, until recently, only one study [309] has been available on QA systems with angle-dependent (specifically anisotropic steric) interactions. 

The effect of the variation of on the stability limits of a polar Stock-mayor fluid, which implies a variation of the dipolar fluid matrix coupling, is illustrated in the upper part of Fig. 7.4. All results correspond to dilute matrices that do not suppress the gas liquid transition but lead to a significant shift of that transition. In particular, the critical temperature decreases with increasing p.,n, whereas the critical density increases. This characteristic effect is referred to as preferential adsorption in other contexts. The replica integral equation results thus demonstrate, at a microscopic level, that polar... 

A replica Omstein-Zemike integral equation approach... 

A adsorption model for a templated porous material may be posed in terms of seven replica Ornstein-Zernicke (ROZ) integral equations [58-60] that relate the direct and total correlation functions, c, (r) and hij (r), respectively, of the matrix-adsorbate system... 

Use of the term replica for a virtual system in statistical physical ensembles should not be confused with the same term used in conjunction with integral equation theories in Chapter 7. 

In this chapter we will deal with this problem by starting from integral equations for the (n-l-l)-component mixture and assuming then permutation symmetry between the replicas. Thereby the Ji-dependence in the equations becomes isolated, which finally allows us to take the limit n - 0 relatively easily. 

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-... 

An important exception are atomic simulations of proteins in explicit solvent, where individual particles (e,g, water molecules) represent the solvent Monte Carlo updates of the system would too frequently collide with solvent particles which causes unacceptably high rejection rates of trial moves. In this case, the cooperative many-body motion is more efficiently simulated by integrating the equations of motion of each particle step by step. It can even be most efficient to combine Monte Carlo and nwlecular dynamics. The most prominent hybrid method is replica-exchange molecular dynamics (REMD), where Langevin simulations ran in different threads at various temperatures and the replicas are exchanged after some time steps with the exchange transition probability (4.92), which effectively is a Monte Carlo step. 

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