Microscopic fluctuations transformations

The basic idea is to develop expressions for common thermodynamic quantities in terms of fluctuations in a system open to all species. The key lies in the fact that the fluctuating quantities characteristic of the grand canonical ensemble can then be transformed into expressions, which provide properties representative of the isothermal isobaric ensemble. Using an equivalence of ensembles argument, one can then consider these fluctuations to represent properties of small microscopic local regions of the solution of interest. This approach can be used to understand many properties of isobaric, isochoric, or osmotic systems in terms of particle number (and energy) fluctuations. 

Upon increasing distance from the critical point, this crossover model provides a continuous transformation from Ising-like behavior asymptotically close to the critical point to mean-field behavior far away from the critical point. Due to the critical fluctuations, the position of the actual critical temperature is shifted with respect to the mean-field critical temperature Tc. The critical temperature shift Tg = (Tc — Tc)/Tc can be estimated from different properties such as the inverse susceptibility or the order parameter. These different estimates of Tg are all proportional to a unique combination of the crossover parameters (rg uA /ct). The transformation is generally controlled by two physical parameters the coupling constant u and the ratio A/k or, equivalently, by the ratio of the correlation length over the microscopic characteristic length d-... 

Fig. 5.9 Illustration of macroscopic enhancement of fluctuation from the initially microscopic one. Fluctuations in the initial and final regime can be well described by Gaussian approximation. In the transient regime fluctuation enhances macroscopi-cally, as can be calculated based on a generalised scale transformation of time, (a) Initial regime, (b) Scaling regime, (c) Final regime.

A more fundamental but much more complex numerical approach is provided by the hierarchical reference theory (HRT). °° In the HRT the numeral implication of the renormalization transformation is applied on a microscopic model of the fluid. One starts from a reference system with short-range repulsive interactions and then formulates a hierarchy of integral equations accounting for successively longer-range fluctuations. The theory has also been extended to fluid mixtures. " The HRT provides estimates for both... 

In addition to the deformation of the chains associated with the alteration of their chain vectors under strain, it is necessary to consider the action of the junctions on the constraint domains surrounding them. The junction and its domain are coupled elastic elements. Non-affine transformation of the fluctuation domains of junctions in the presence of chain connectivity creates an additional strain field in the medium around each junction. A omain deformation tensor can be introduced in analogy with A. (h) relates the mean deformation (according to ensemble distribution) of the junction constraint domains in the deformed, connected network to that in the absence of connectivity. Consequently, the state of microscopic deformation in a real network is the sum of the molecular deformation tensor. A, and the junction domain deformation tensor, Qj). ... 

The pair correlations between atoms or sites in a molecular fluid pertain to the microscopic spontaneous fluctuations that occur in a macrosCOpically homogeneous fluid. Under certain circumstances, these fluctuations conspire collectively or in concert to form ordered phases such as crystals. The description of these transformations of phase is beyond the scope of the linear (i.e., Gaussian) theory we have outlined thus far. The incorporation of nonlinearities is the subject we turn to... 

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