Field-theoretical models

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

The preceding results show that the equilibrium TFD is equivalent to the Matsubara imaginary-time formalism (for a detailed discussion, see the chapter by Santana et. al. in this Proceedings). Matsubara formalism has been used also to consider spatial compactification in field theoretical models (A.P.C. Malbouisson et.al., 2002 A.P.C. Malbouisson et.al., 2002 A.P.C. Malbouisson et.al., 2004). 

Kambara presented a ligand field theoretical model for SCO in transition metal compounds which is based on the Jahn-Teller coupling between the d-electrons and local distortion as the driving force for a spin transition [193]. The author applied this model also to interpret the effect of pressure on the ST behaviour in systems with gradual and abrupt transitions [194]. By considering the local molecular distortions dynamically this model turned out to be suited to account for cooperative interactions during the spin transition [195]. 

There have been few attempts to generalize mean-field theories to the unrestricted case. Netz and Orland [227] applied their field-theoretical model to the UPM. Because such lattice theories yield quite different critical properties from those of continuum theories, comparison of their results with other data is difficult. Outhwaite and coworkers [204-206] considered a modification of their PB approach to treat the UPM. Their theory was applied to a few conditions of moderate charge and size asymmetry. 

At this point it may help the reader to simplify the Green s functions by setting ri = r2. In this case the numerator is just the density of charge. So we find that the reaction of a system is related to the electron density. If ri r2 then this entity describes how an electron, injected at ri, is changing the physical situation at r2. It can therefore be said to propagate a cause at iq to an effect at r2. For this reason Green s functions are often called propagators, in particular in field theoretical models. 

Feng, E.H., and Fredrickson, G.H. "Confinement of equilibrium polymers a field-theoretic model and mean-field solution". Macromolecules 39, 2364-2372 (2006). 

It is easily seen that for such trial functions the minimization of the Hamiltonian K, Eq. (5.1), may be replaced by the minimization of a specified nonlinear functional 6(0) of the molecular states 0 alone. In the following we refer to either formulation as seems convenient. This argument also enables one to connect these field theoretical models with the earlier suggestion of mine that molecular structure states can be associated with those solutions of the Schrodinger equation for the full molecular Hamiltonian ft that satisfy certain subsidiary conditions3,35), if the latter are associated with the nonlinearity in the functional 6(0). As we shall see, it may happen that 6(0) has two degenerate minima and it is in this sense that the dynamics gives rise to a double-well structure. 

C.Steinhardt. Constraints on field theoretical models for variation of the fine structure constant. Physical Review, D71,043509 (2005). 

The first step in quantitative description of pure polyamorphic fluid is a selection of the model that can qualitatively describe a possible multiplicity of critical points in wide range of temperatures and pressures. A great many of explanations of multicriticality in monocomponent fluids (perturbation theory models semiempirical models lattice models, two-state models, field theoretic models, two-order-parameter models, and parametric crossover model has been disseminated after the pioneering work by Hemmer and Stell Here we test more extensively the modified van der Waals equation of state (MVDW) proposed in work and refine this model by introducing instead of the classical van der Waals repulsive term a very accurate hard sphere equation of state over the entire stable and metastable regions... 

Bridging Between Particle-Based and Field-Theoretic Models. 218... 

At this point, two approaches are available for calculating the 27t-exchange contribution to the NN interaction dispersion theory (Paris [21]) and field theory (Bonn [15]). Let us now compare the predictions by the two approaches with each other as well as with the data. For this purpose, it is appropriate to look into the peripheral partial waves of NN scattering. In Fig. 7, the curve BONN represents the predictions by the field-theoretic model of Fig. 6 the dotted curve labeled P 73 is the original Paris result [21] as obtained from dispersion theory, while the dotted curve... 

Fig. 6. Field-theoretic model for the 27c-exchange. Solid lines represent nucleons, double lines isobars, and dashed lines pions. The hatched circles are nrt correlations.

Fig. 7. Phase shifts of some peripheral partial waves as predicted by a field-theoretic model for the 2n exchange (solid line, BONN [15]) and by dispersion theory (dotted line labeled P 73 [21]). Both calculations also include OPE and one-to-exchange. The dotted line labeled P 80 is the fit by the parametrized Paris potential [14]. Octagons represent the phase shift analysis by Arndt et al. [23] and triangles the one by Bugg and coworkers [24].

Since Hagg (1962) had aigued the importance of the quasi-local structures in the field theoretical models, they have been applied to the quantum statistical mechanics. 

Feng EH, Lee WB, Fredrickson GH (2007) Supramolecular diblock copolymers a field-theoretic model and mean-field solution. Macromolecules 40(3) 693-702... 

The theoretical derivation of the existence of a coherently excited polar mode following from (11) can also be phrased in a field theoretical model, or in a more exact way be supported by numerical work as quoted in Ref. 7, as well as by more recent work—the present chapter is not meant to provide a complete review. It is of importance, however, that other models involving nonlinear coupling of oscillations can lead to coherent excitation of a single mode. " ... 

The mean-field theoretical model developed by Hao and Clem allows the anisotropic superconducting state thermodynamic parameters to be determined from measurements of M(H,T) outside of the linear Abrikosov regime (Hao et al. 1991) their model reduces to the Abrikosov linear region only in the vicinity of the transition. However, the determination of the mean-field upper critical field curve is fiirther complicated... 

Both the statics and the collective dynamics of composition fluctuations can be described by these methods, and one can expect these schemes to capture the essential features of fluctuation effects of the field theoretical model for dense polymer blends. The pronounced effects of composition fluctuations have been illustrated by studying the formation of a microemulsion [80]. Other situations where composition fluctuations are very important and where we expect that these methods can make straightforward contributions to our understanding are, e.g., critical points of the demixing in a polymer blend, where one observes a crossover from mean field to Ising critical behavior [51,52], or random copolymers, where a fluctuation-induced microemulsion is observed [65] instead of macrophase separation which is predicted by mean-field theory [64]. 

Finally, we mention an interesting recent study by Chandler that extended the Gaussian field-theoretic model of Li and Kardar to treat atomic and polymeric fluids. Remarkably, the atomic PY and MSA theories were derived from a Gaussian field-theoretic formalism without explicit use of the Ornstein-Zernike relation or direct correlation function concept. In addition, based on an additional preaveraging approximation, analytic PRISM theory was recovered for hard-core thread chain model fluids. Nonperturbative applications of this field-theoretic approach to polymer liquids where the chains have nonzero thickness and/or attractive forces requires numerical work that, to the best of our knowledge, has not yet been pursued. 

Field-Theoretical Modeling of Nanoparticle - Membrane Interactions 321... 

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