Field-theoretical formalism

Kaplunovsky and Weinstein [kaplu85j develop a field-theoretic formalism that treats the topology and dimension of the spacetime continuum as dynamically generated variables. Dimensionality is introduced out of the characteristic behavior of the energy spectrum of a system of a large number of coupled oscillators. 

The use of the same analogy for the A + B - C reaction, described by a set of (2.3.67) is more problematic coupling of these equations results in a non-conserving number of particles in a system. This problem could be much easier treated in terms of the field-theoretical formalism. 

The use of an elegant field-theoretical formalism has been already discussed in Section 2.3.2 (see also review articles and a book [11, 41, 42] and... 

Note that another consisitent approach to the problem of mobile particle accumulation is based on the field-theoretical formalism [15, 37, 51]. However by two reasons this approach is not useful for the study of immobile particle aggregation (i) the smallness of the parameter U(t) = n(t)vq [Pg.414]

Equations (7.3.23) and (7.3.24) actually imply that one- and two-dimensional cases actually exhibit already macroscopic separation of the system into regions consisting of only A particles and only B particles. This is also confirmed by the fact that the integral over the spectrum of spatial fluctuations diverges in the cases at small k. On the other hand, to find the aggregation of particles in numerical experiments in the f/iree-dimensional case we must treat the deviations from the Poisson distribution in large volumes. More detailed field-theoretical formalism has confirmed this conclusion [15]. 

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. 

A general background on the field theoretical formalism for polyelectrolytes is presented in Section 6.4.1. Details of the commonlyused transformations in order to switch from a particle to the field description are presented in Section 6.4.2. DiEFerent kinds of charge distributions along the polydectrolyte chain and the well-known saddle-point approximation for computing the free energy are described in Sections 6.4.3 and 6.4.4, respectively. Numerical techniques to solve the nonlinear set of equations and one-loop expansions to go beyond the well-known saddle-point approximation are presented in Sections 6.4.5 and 6.4.6, respectively. 

Here, we present a general outline of the self-consistent field theory for polyelectrolyte solutions containing externally added salt ions. The theory is a generalization of the field theoretical formalism developed by Edwards [48-50] for neutral polymers to polyelectrolytes. We start from the path integral representation of a polymer chain and readers interested in the derivation of the path integral representation are referred to Ref [56]. 

Fluctuations of the local monomer concentration are of importance to the description of polymers at surfaces owing to the many possible chain conformations. These fluctuations are treated theoretically using field-theoretical or transfer-matrix techniques. In a field-theoretical formalism, the problem of accounting for different polymer conformations is converted into a functional integral over different monomer-concentration profiles [12]. Within transfer-matrix techniques, the Markov-chain property of ideal polymers is exploited to re-express the conformational polymer fluctuations as a product of matrices [22]. 

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it... 

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). 

In order to describe strong-field interaction of the five-state system in Figure 6.9 with intense shaped femtosecond laser pulses, the theoretical formalism prepared in Section 6.3.2.1 is readily extended. The RWA Hamiltonian H f) for the five-state system in Figure 6.9 in the frame rotating with the carrier frequency reads... 

In spite of simple theoretical formalism (for example, mean-field descriptions of certain aspects) structural aspects of the systems are still explicitly taken into account. This leads to the results which are in a good agreement with computer simulations. But the stochastic model avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed for obtaining good statistics for the reliable results. Therefore more complex systems can be studied in detail which may eventually lead to better understanding of real systems. In the theory discussed below we deal with a disordered surface. This additional complication will be handled in terms of the stochastic approach. This is also a very important case in catalytic reactions. 

Finally, for completeness in Appendix A 7.1 we consider the formal relation of the continuous chain model to a field theoretic Hamiltonian, used to describe critical phenomena in ferrornagnets. It was this relation discovered by de Genries [dG72] and extended by Des Cloizeaux [Clo75, which initiated the application of the renormalization group to polymer solutions and led to the embedding into the larger realm of critical phenomena. 

Electron transfer reactions and spectroscopic charge-transfer transitions have been extensively studied, and it has been shown that both processes can be described with a similar theoretical formalism. The activation energy of the thermal process and the transition energy of the optical process are each determined by two factors one due to the difference in electron affinity of the donor and acceptor sites, and the other arising from the fact that the electronically excited state is a nonequilibrium state with respect to atomic motion (P ranck Condon principle). Theories of electron transfer have been concerned with predicting the magnitude of the Franck-Condon barrier but, in the field of thermal electron transfer kinetics, direct comparisons between theory and experimental data have been possible only to a limited extent. One difficulty is that in kinetic studies it is generally difficult to separate the electron transfer process from the complex formation... 

P. Fevrier, O. Simonin, and K. Squires. Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution Theoretical formalism and numerical study. J. Fluid Mech., 533 1-46, 2005. 

The need to divide the system into multiple regions with quite different characteristics suggests that a flexible but unified theoretical formalism suitable for QM, MM, and continuum methods would be very useful. To this end, Boresch et al. [133] have introduced the dielectric field equation (DFE) for biomolecular solvation. The DFE is a general expression for the net electric field of the form ... 

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