Mean field ideas

The step length a of the primitive path is assumed to be comparable to the tube diameter which is unknown. 

This prediction is borne out by many experiments. Similarly the viscosity was shown by de Gennes to be proportional to This does not agree with experiment in that rather than is well attested, and there is a large but inconclusive literature why one of de Gennes predictions is very accurate and the other only approximate. 

But the mean square displacement of a monomer is related to the displacement along the tube according to 

Thus Td is also the longest relaxation time. It can readily be shown that the mean square displacement of the center of mass of the chain is given by 

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In this paper we will not pursue such formal developments any further, and instead use mean field ideas and heuristic arguments to motivate the choice of the appropriate free energy functional. We represent the intrinsic free energy functional in the form of an effective 2D step Hamiltonian H and imagine on physical grounds that it has the... 

What happens in between For a long time, the only interpolation method available was a mean field theory due to Flory and Huggins. However (as realized early by these authors) the mean field idea is not adequate at low and intermediate concentrations. The question remained obscure for a long time. It has now been clarified experimentally (mainly through neutron scattering experiments see Fig. III.l) and theoretically (by certain manipulations as described in Ch q>ter X). Fortunately, the final picture is simple and can be explained without referring to abstract theory. 

Segregation effects are also important for fundamental studies. For each demixing process there is a critical point, near which certain fluctuations of concentration in the mixture become anomalously large. Essentially all the work on the theory of these effects in polymer systems has been based on mean field ideas. However in many cases critical phenomena are now known to be qualitatively different from mean field predictions. One of our tasks in this chapter is to classify the critical points some will be of the mean field type, others will be different. 

We conclude that, when 6, jc is small near the critical point, the fluctuations are not dangerous, and the mean field approach makes sense. But there is a critical dimensionality de = 6 below which the mean field idea is not self-consistent. The fact that our world has a value of d(d = 3)... 

It is natural to expect that the simplest analytical approach to the iV-body cooperative system (5.2.21) would be provided by something similar to the mean field idea of thermodynamic phase transitions. In the present problem, the in-... 

The canonical model for hard chains has been the tangent-hard-sphere freely jointed chain model, and several equations of state are available for fluids composed of these molecules. There are two categories of hard chain equations of state (i) those based on the mean field ideas of Flory and Huggins, and (ii) those based on the theory of associating fluids. ... 

In principle, MC algorithms can be tuned for particular systems and can thus be more efficient than MD for obtaining equilibrium distributions. An interesting idea is to use MC simulations to obtain accurate initial guesses for subsequent MD simulations. Already as early as 1993, Venable and co-workers [68] used a scheme for efficiently sampling configurations of individual lipids in a mean field. These configurations were then used to develop the initial conditions for the molecular dynamic simulations. 

The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a probe molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical... 

Lyotropic polymeric LC, formed by dissolving two aromatic polyamides in concentrated sulphuric acid, have been studied using variable-director 13C NMR experiments.324 The experimental line shapes at different angles w.r.t the external field were used to extract macromolecular order and dynamic in these ordered fluids. An interesting application of lyotropic LC is for the chiral discrimination of R- and S-enantiomers, and has recently been demonstrated by Courtieu and co-workers.325 The idea was to include a chiral compound 1-deutero-l-phenylethanol in a chiral cage (e.g., /1-cyclodextrin) which was dissolved and oriented by the nematic mean field in a cromolyn-water system. Proton-decoupled 2H NMR spectrum clearly showed the quad-rupolar splittings of the R- and S-enantiomers. The technique is applicable to water-soluble solutes. 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

Before we present our idea, we give a short summary of the mean-field theory. The core concept there is the velocity (v)-position (r) probability distribution of stars,/(r, v, t), a positive and integrable function that is a priori time-dependent. The number density p(r, t) is the integral over velocities of/(r, v, t) ... 

The present model may impose too strong an obstruction on a real system, and it will be of great interest to know whether the idea of a higher shell substitution effect can be extended and adjusted to the excluded volume problem. Certainly, nobody can characterize the cascade theory with substitution effects as a mean field theory. 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

The name, DLYO, originates from the first letter in the surname of the four authors (Derjaguin, Landau, Verwey and Overbeek) from two different groups, which originally published these ideas. The theory is based on the competition between two contributions, a repulsive electric double layer and an attractive van der Waals force [4,5]. The interaction in the electric double layer was originally obtained from mean field calculations via the Poisson-Boltzmann equation [Eq. (4)]. However, the interaction can also be determined by MC simulations (Sec. II. B) and by approximate integral equations like HNC (Sec. II. C). This chapter will focus on the first two possibilities. 

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