Order parameter theory

Wrack, B., Salje, E.K.H., Graeme-Barber, A. (1991) Kinetic rate laws derived from order parameter theory IV kinetics of Al, Si disordering in Na feldspars. Phys Chem Minerals 17 700-710 Yeomans, J.M. (1992) Statistical Mechanics of Phase Transitions. Clarendon Press, Oxford, UK Ziman, J.M. (1979) Models of Disorder. Cambridge University Press Cambridge, UK... 

A little work seems to have been carried out on the wavenumber-dependent orientational correlation functions C/m(, t). These correlation functions can provide valuable insight into the details of microscopic dynamics of the system. A molecular level understanding of C/m(, t) would first require the development of a molecular hydrodynamic theory that would have coupling between C/m(, t) and the dynamic structure factor S(k, t) of the liquid. A slowdown in C/m(, t) may drive a slowdown in the dynamic structure factor. This would then give rise to a two-order parameter theory of the type develops by Sjogren in the context of the glass transition and applied to liquid crystals by Li et al. [91]. However, a detailed microscopic derivation of the hydrodynamic equations and their manifestations have not been addressed yet. 

Ramakrishnan, T.V., and Yussouf, M. (1979) First-principles order parameter theory of freezing, Phys. Rev. B. 19, 2775. 

Ramakrishnan, T.V., and Yussouff, M. Theory of the liquid-sohd transition. 1977. Solid State Gomm. 21 389. Ramakrishnan, T.V., and Yussouff, M. First-principles order-parameter theory of freezing. 1979. Phys. Rev. B Gondens Matter and Mater. Phys. 9 2775. 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

Four Parameter Models. Two- and three-parameter theories are only accurate for simple, normal, and some slightly polar fluids. In order to accurately predict polar fluid behavior a fourth parameter is needed (80). The Stiel polarity factor, is one such fourth parameter and follows from the... 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

FIG. 13 Herringbone order parameter and total energy for N2 (X model with Steele s corrugation). Quantum simulation, full line classical simulation, dotted line quasiharmonic theory, dashed line Feynman-Hibbs simulation, triangles. The lines are linear connections of the data. (Reprinted with permission from Ref. 95, Fig. 4. 1993, American Physical Society.)... 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

MNDOC has the same functional form as MNDO, however, electron correlation is explicitly calculated by second-order perturbation theory. The derivation of the MNDOC parameters is done by fitting the correlated MNDOC results to experimental data. Electron correlation in MNDO is only included implicitly via the parameters, from fitting to experimental results. Since the training set only includes ground-state stable molecules, MNDO has problems treating systems where the importance of electron comelation is substantially different from normal molecules. MNDOC consequently performs significantly better for systems where this is not the case, such as transition structures and excited states. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

The large size of now is responsible for mean-field theory being reliable for large N Invoking the Ginzburg criterion one says mean field theory is self-consistent if the order parameter fluctuation in a correlation volume is much smaller than the order parameter itself. 

Recently efficient techniques were developed to simulate and analyze polymer mixtures with Nb/Na = k, k > I being an integer. Going beyond meanfield theory, an essential point of asymmetric systems is the coupling between fluctuations of the volume fraction (j) and the energy density u. This coupling may obscure the analysis of critical behavior in terms of the power laws, Eq. (7). However, it turns out that one can construct suitable linear combinations of ( ) and u that play the role of the order parameter i and energy density in the symmetrical mixture, ... 

TABLE I.a Comparison of Observed Activity Coefficients of Cu and Au with Values Calculated from Observed Short-Range Order Parameters by the First-Order Quasi-Chemical Theory... 

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