Beyond Mean Field

The RPA method is based on a completely random organization of the polymer chains where the effects of interactions are estimated using a perturbation calculation. Such mean field calculations are only valid as long as the length scale of the fluctuations are small compared to characteristic lengths of the system (the 

The thermal composition fluctuations tend to stabilize the disordered phase, giving rise to a renormalized critical temperature. The relation between the mean-field value T and the real Tc is expressed by the Ginzburg rdation [8]  

Polymers are, by definition, molecules composed by a large number of small chemical units, the monomers. Above, we have discussed A- and B-homopolymers, assuming that all A-polymers are composed of the same single A-monomer, and B-polymers by another specific B-monomer. Such chemical equality is often the situation in synthetic polymers, as for example polyethylene purely composed of -CH2- ethylene monomers. Many natural polymers, on the other hand, are composed of several different monomers example of these include proteins, which are polymeric chains composed of different amino acids such polymers are termed copolymers. 

One important class of synthetic copolymers is composed of two different monomers which, in the general discussion, will be abbreviated A and B. The A and B segments may be positioned randomly within the chain, or in blocks of respectively A and B. Diblock copolymers composed of two linear polymer blocks linked covalently together are the most simple of this class. Diblock copolymers are closely related to blends of homopolymers both systems are composed of two linear polymer chains, but the covalent bond between the A- and the B-blocks has of course significant implications on the physical properties. 

Polymers of different chemistry are generally not miscible, as discussed above. The A- and B-molecules of an AB-diblock copolymer melt will therefore tend to cluster into domains rich in respectively A and B units. In opposition to polymer 

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The case of intermediate and strong electron-vibron interaction at intermediate coupling to the leads is the most interesting, but also the most difficult. The existing approaches are mean-field [131-133], or start from the exact solution for the isolated system and then treat tunneling as a perturbation [134-140]. The fluctuations beyond mean-field approximations were considered in Refs. [141,142]... 

To improve DFT-based models several approaches were suggested, including inelastic electron-vibron interaction [121,126,272-279] or Coulomb interaction beyond mean-field level [280], or based on the LDA+U approache [281]. The principally different alternative to DFT is to use an initio quantum chemistry based many-body quantum transport approach [282-285]. 

This is a very important result. It states that both dipole amplitudes from the RRPA calculation are modified by a common factor that reflects the influences of electron correlations in the initial and final ionic states which are beyond mean-field electron-electron interactions. The A0a0 2-value is called the spectroscopic factor (or the quasi-particle strength or the pole strength or the renormalization factor) and describes the weight given to the improved 2p photoionization cross section as compared to a calculation which does not include these specific electron correlations. The remaining intensity is transferred to satellite processes... 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

Beyond mean-field effects are then explored in a theoretical calculation of the spatial correlations in a boson gas where the interactions are no longer weak. The authors find an enhancement in the response of the system, related to a possible roton in such system, as observed in liquid 4 He. 

In order to resolve beyond mean-field processes in BEC, we must overcome inhomogeneous broadening, which masks homogeneous lineshapes we want to measure. One possibility is to resolve the radial modes (Sec. 1), or for appropriate parameters, the use of our echo spectroscopy scheme (Sec. 6). 

We calculate the roton [Steinhauer 2004] in atomic vapor BEC using a Jas-trow wavefunction, which allows us to calculate beyond mean-field correlations in the quantum state, giving rise to a roton peak in the structure factor at a momentum k oc a-1 (where a is the s-wave scattering length of the atoms). The relevant parameter e = J i t a (n is the density of the atoms in the BEC), is calculated over a range of values from e < 1 up to e 1. 

Correlation energy beyond mean-field approach. A rather different (but equivalent) way to understand correlation is to consider the following alternative form of the operator representing the Coulomb interaction, equivalent to Eq. (3),... 

This argument can be made precise [79] by a mapping of the free energy functionals of both problems onto each other. Note that the partition function (even beyond mean-field ) is given by a functional integral involving the free energy functional A F,... 

Anglin, J.R. and Vardi, A., Dynamics of a two-mode Bose-Einstein condensate beyond mean-field theory, Phys. Rev. A, 64, 013605, 2001. 

This chapter has also touched on approaches that go beyond mean field theories such as NSCFT, in Section VII. There is much more work to be done in this area, but it has the potential to provide insights into the effects of fluctuations in polymer systems in general, and the behavior of block copolymers near the MST in particular. 

Clearly there is need for spin-glass theories beyond mean-field. One approach in this direction is presented by Malozemoff et al. (1983) and Malozemoff and Barbara (1985). They propose a critical fractal cluster model of spin glasses which is able to describe the essential features of the phenomena occurring near the spin-glass transition and to account for the static critical exponents. The basic assumption of this fractal model is the existence of a temperature- and magnetic-field-dependent characteristic cluster size on which all relevant physical quantities depend and which diverges at the transition temperature Tj. It is related to the correlation length and the cluster fractal dimension D by More... 

A basic approach for the description of polymer chains in the continuum is the Gaussian thread model [26, 31]. Treating interactions among monomers in a mean-field-like fashion, one obtains the self-consistent field theory (SCFT) [11, 32-36] which can also be viewed as an extension of the Hory-Huggins theory to spatially inhomogeneous systems (like polymer interfaces in blends, nucrophase separation in block copolymer systems [11, 13], polymer bmshes [37, 38], etc.). However, with respect to the description of the equation of state of polymer solutions and blends in the bulk, it is stiU on a simple mean-field level, and going beyond mean field to include fluctuations is very difficult [11, 39-42] and outside the scope of this article. 

Going beyond mean field models, the next level of theoretical approximation is to allow for (usually thermal) fluctuations around the mean system configuration. These fluctuations are considered to be a weak perturbation with respect to the average order. Unfortunately, many soft materials are strongly fluctuating, especially near to a phase transition, and this approach then breaks down. This is not too much of a concern, however, because just as for atomic systems, more sophisticated perturbation theories are sure to be developed for soft materials. 

The first step beyond mean field and Gaussian approximations is to expand the partition function in powers of the interae-tion (i.e. terms of order higher than 2 in q) and integrate over Gaussian fluctuations. It is a standard result that such perturbation expansion presents infrared divergences (i.e. for wavenumber fe->0) in three dimensions [8]. It can be used, however, to estimate the critical domain be-... 

B. Beyond Mean-Field Theory Scaling Arguments for Good Solvents... 

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