Expansions with respect to

For Landau-Ginzburg theory as for polymer theory, the space dimension d = 4 plays a pivotal role. Indeed, for d 4, the corresponding systems have a classical behaviour, and accordingly for d 4, the critical exponents v, y, and co have the classical values v = 1/2, y = 1, and co = 0. For d 4, the behaviour changes and with it, the values of the critical exponents which become functions of d. These functions can be expanded, in the vicinity of d - 4, in powers of 6 = 4 — d. 

Various methods were used to obtain such expansions in the framework of Landau-Ginzburg theory, but all of them are founded on common principles the existence of critical systems described by renormalized theories and the necessity of transforming the expansions with respect to the interaction, into 

Interpreting the fixed point is not obvious, if one adopts the point of view and the methods originally developed by K. Wilson. On the contrary, the interpretation is transparent, if one uses the most recent renormalization methods. Then the expansion parameter is a number defining the intensity of a physical quantity this quantity itself is expressed by using the characteristic length of the system as the length unit. Therefore, if, as we believe, there exists a limiting critical system, the expansion parameter which is a physical observable must have a finite limit. 

In the following, we shall not explain the technical points more precisely. In fact, in the preceding section we already described a renormalization technique in detail, and at the end of this chapter we shall show, also in detail, how we can obtain e-expansions by direct renormalization of polymers. Consequently, only results will be given here, and the reader will be referred to the relevant original articles. 

Wilson was the first to obtain an exact expansion28 of the critical exponents to first- and second-order in e. From his results, (and also by using the relation 

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Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

This is the result expected if r is effectively a control parameter. Taylor expansion with respect to 1 /k introduces the correction terms as derivatives. To first order in l/k, we obtain... 

The double trace term is due to the absence of the condition for the vanishing of the trace for the broken generator X5. It emerges naturally in the non linear realization framework at the same order in derivative expansion with respect to the single trace term. In the unitary gauge these two terms correspond to the five gluon masses [431. 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

The above relation is very similar to the first-order expansion with respect to Rew of the polynomial fitting of numerically obtained data for heat transfer in a square channel with one porous and three solid walls (Hwang et al., 1990). 

The usefulness of the Uhlenbeck-Ornstein weighting lies in situations where U(y) contains a substantial harmonic part V (x) is then considerably smaller than V(x), and an approximation scheme based on any sort of expansion with respect to V(x) is therefore much closer to its goal than one based on V(x) and the Wiener weighting. 

TABLE 3.1 Comparison of Reversible and Irreversible Gas Expansion with Respect to Equilibration, Reversibility, Rate, and Work Capacity... 

Note that the dependence of W(y r) on its second argument r is fully maintained an expansion with respect to this argument is not allowed as W varies rapidly with r. The first and fourth terms cancel. The other two terms can be written with the aid of the jump moments... 

Perturbative expansion with respect to the external electric field yields,... 

A method similar to the iterative, is the partial closure method [37], It was formulated originally as an approximated extrapolation of the iterative method at infinite number of iterations. A subsequent more general formulation has shown that it is equivalent to use a truncated Taylor expansion with respect to the nondiagonal part of T instead of T-1 in the inversion method. An interpolation of two sets of charges obtained at two consecutive levels of truncations (e.g. to the third and fourth order) accelerates the convergence rate of the power series [38], This method is no longer in use, because it has shown serious numerical problems with CPCM and IEFPCM. 

The problem of finding a vector is usually solved by representing the required vector as an expansion with respect to some natural set of basis vectors. Following this method one can expand the vector of the n-th order correction to the k-th unperturbed vector- 44 n terms of the solutions b p (eigenvectors) of the unperturbed problem eq. (1.51) ... 

CVFF [182] is the valence version of CFF. It uses only harmonic expansion with respect to displacements in the diagonal force fields and reduces cross terms selection to some extent. 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

Stolarczyk LZ, Piela L (1979) Invariance properties of the multipole expansion with respect to the choice of the coordinate system. Int J Quantum Chem 15 701-711... 

All these identities are useful in the evaluation of integrals in diagrammatic expansions with respect to interaction and also in the derivation of equations of motion. 

The 2 -order expanded density matrix, which is obtained by the n-order perturbation expansion with respect to the effective interaction Hamiltonian, is then given as... 

The second mode dynamics can be obtained by an expansion with respect to the eigenstates I > of a properly chosen zero order Hamiltonian h which leads to a set of equations of motion for the variables a , y, p, A, and y [18, 45, 46]... 

An alternative formulation examines the molecular energy expansion, rather than the previous dipole expansion, with respect to the field ... 

First-Order (NONMEM) Method. The first nonlinear mixed-effects modeling program introduced for the analysis of large pharmacokinetic data was NONMEM, developed by Beal and Sheiner. In the NONMEM program, linearization of the model in the random effects is effected by using the first-order Taylor series expansion with respect to the random effect variables r], and Cy. This software is the only program in which this type of linearization is used. The jth measurement in the ith subject of the population can be obtained from a variant of Eq. (5) as follows ... 

By virtue of the perturbation expansion with respect to C—>°°, Eqs. (101)— (103) are exact in high concentrations. This is fortunate, because gelation is a phenomenon typical of concentrated systems [91,92]. 

Only one unknown parameter with Eq. (102) is the relative cyclization frequency, /v. Note that Eq. (102) was derived by means of the perturbation expansion with respect to C— °° (y = 0). In this hypothetical limit, the cyclic production becomes negligible, and all sorts of excluded volume effects should vanish rigorously, thus realizing the ideal situation3. 

While a chain within a branched molecule is by no means Gaussian in this real world (d = 3 note that we are considering the end-to-end distance distribution, and not the segment-density distribution about the center of gravity), by virtue of the perturbation expansion with respect to one can apply the... 

Step 4. When we happen to develop a perturbation expansion with respect to the Hartree-Fock reference state, the number of diagrams is further reduced. In this case, all diagrams produced by a contraction within an operator, which results in Goldstone terminology to a bubble ... 

We therefore start assuming that the particles are compact and define the radius a of the smallest sphere containing the particle [44]. Thus, the condition r a is always fulfilled. The incoming plane wave can be expanded into spherical waves through a multipole expansion with respect to the parameter x = a/X T1. ... 

The first order term of Eq. (8) is reduced to the field r ), the electric dipole field, whereas the second order term, linear with the parameter order x = a/X, is the sum of tire fields ( >, ) and ( >, ), respectively the electric quadrupole and the magnetic dipole fields. The surface nonlinear polarization of the form of Eq. (3) is now a series expansion with respect to the parameter x = a/X too. Its general expression is ... 

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