Many-body problem/effects

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

Next, the effect of z on A IT through the transition matrix element Hoj is considered as follows for rigorous determination of IToi, all electrons in the system should be treated. However, for the sake of simplicity, we devote our attention only to the transferring electron the other electrons would be regarded as forming the effective potential (x) for the transferring electron (x the coordinate of the electron given from the ion center). This enables us to reduce the many-body problem to a one-body problem ... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

Thus far, these models cannot really be used, because no theory is able to yield the reaction rate in terms of physically measurable quantities. Because of this, the reaction term currently accounts for all interactions and effects that are not explicitly known. These more recent theories should therefore be viewed as an attempt to give understand the phenomena rather than predict or simulate it. However, it is evident from these studies that more physical information is needed before these models can realistically simulate the complete range of complicated behavior exhibited by real deposition systems. For instance, not only the average value of the zeta-potential of the interacting surfaces will have to be measured but also the distribution of the zeta-potential around the mean value. Particles approaching the collector surface or already on it, also interact specifically or hydrodynamically with the particles flowing in their vicinity [100, 101], In this case a many-body problem arises, whose numerical... 

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

The many-body problem is reformiilated here by using a system of equations involving only first order Reduced Density Matrices. These matrices correspond to all the states of the spectrum of the system and to the transitions among the different states. Some results concerning the correlation effects are also reported here. 

Reactions between species which are present in comparable, and large, concentrations are complicated to analyse because any one species may react with one of several reactants. This competitive effect is one form of the many-body problem and these cannot be solved exactly. 

One problem of the measurement of weak absorption in the far IR is that short absorption paths must be used. At wavelengths comparable to the beam apertures diffraction effects lead to beam divergence [252]. (The combination of high gas pressures and short absorption paths may not be useful if many-body induction effects must be avoided, and an accurate measurement of a under conditions of weak absorption, I/Iq 1, is difficult [368].)... 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

Computation of the grand resistance matrix R for each possible particle configuration provides a major numerical challenge (see Section VIII). The various methods cited in Section II for dealing with the many-body problem are potentially useful in this context. Detailed calculations must be performed for a number of accessible configurations, and the configurational evolution determined by interpolation in order to effect the requisite time averaging. 

The X-ray singularity problem was originally solved in the asymptotic limit and the complicated many-body problem was turned into an effective one-particle problem (219). For the X-ray photon frequency threshold frequency (o0, the absorption spectrum g(m) for the process in which a deep, structureless core electron is excited to the conduction band by the absorption of an X-ray of frequency w is expressed by the power law... 

We refer the interested reader to our previous report1 for a review of the literature on many-body perturbation theory studies of relativistic effects molecules upto 1999. Here the background to the relativistic many-body problem in molecules was given in Section 2.1 and a review of the relativistic many-body perturbation theory was given in Section 2.3. 

Note that at this point we have turned the original (hopeless) many-body problem into a series of effective single particle Schrodinger equations. 

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