Expansion parameter

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

If we now include the anliannonic temis in equation B 1.5.1. an exact solution is no longer possible. Let us, however, consider a regime in which we do not drive the oscillator too strongly, and the anliannonic temis remain small compared to the hamionic ones. In this case, we may solve die problem perturbatively. For our discussion, let us assume that only the second-order temi in the nonlinearity is significant, i.e. 0 and b = 0 for > 2 in equation B 1.5.1. To develop a perturbational expansion fomially, we replace E(t) by X E t), where X is the expansion parameter characterizing the strength of the field E. Thus, equation B 1.5.1 becomes... 

In the original mathematical treatment of nuclear and electronic motion, M. Bom and J. R. Oppenheimer (1927) applied perturbation theory to equation (10.5) using the kinetic energy operator Tq for the nuclei as the perturbation. The proper choice for the expansion parameter is A = (me/M) /", where M is the mean nuclear mass... 

To carry out the normal form calculation, we introduce a formal expansion parameter s to identify the orders of perturbation theory via... 

Variations in fluid density on reaction can have significant effects on the size ratio, but the effects are secondary when compared to the variations in reaction order. For positive values of the expansion parameter SA, the volume ratio is increased, for negative values of 8a, the volume ratio decreases. However, the fact that in practice CSTR s are used only for liquid phase reactions makes this point academic. 

Since there is a change in the number of moles on reaction, the volumetric expansion parameter d will be nonzero. Consequently,... 

The volumetric expansion parameter S may thus be taken as 0.9675. The product distribution will vary somewhat with temperature, but the stoichiometry indicated above is sufficient for preliminary design purposes. (We should also indicate that if one s primary goal is the production of ethylene, the obvious thing to do is to recycle the propylene and ethane and any unreacted propane after separation from the lighter components. In such cases the reactor feed would consist of a mixture of propane, propylene, and ethane, and the design analysis that we will present would have to be modified. For our purposes, however, the use of a mixed feed would involve significantly more computation without serving sufficient educational purpose.)... 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

Here we have introduced an auxiliary expansion parameter 5 to be set equal to unity after calculations, similar to 5 expansion method (de Souza,2002).The first order term AVq b) in this equation will... 

Abstract For the case of small matter effects V perturbation theory using e = 2V E/ Am2 as the expansion parameter. We derive simple and physically transparent formulas for the oscillation probabilities in the lowest order in e which are valid for an arbitrary density profile. They can be applied for the solar and supernova neutrinos propagating in matter of the Earth. Using these formulas we study features of averaging of the oscillation effects over the neutrino energy. Sensitivity of these effects to remote (from a detector), d > PE/AE, structures of the density profile is suppressed. 

Having derived the symmetry relations between the expansion parameters in equation (55), we can proceed to fit the expansions through the ab initio dipole moment values. The expansion parameters in the expressions for and fiy are connected by symmetry relations since these two quantities have E symmetry in and so and fiy must be fitted together. The component ji, with A" symmetry, can be fitted separately. The variables p in equation (55) are chosen to reflect the properties of the potential surface, rather than those of the dipole moment surfaces. Therefore, the fittings of fi, fiy, fifi require more parameters than the fittings of the MB dipole moment representations. We fitted the 14,400 ab initio data points using 77 parameters for the component and 141 parameters for fi, fiy. The rms deviations attained were 0.00016 and 0.0003 D, respectively. 

Equation 13 can be extended to non-6 conditions by multiplying by an expansion parameter Since it is known that the expansion paramenter is greater than unity for good solvents and less than unity for poor solvents. Equation 13 would predict that the slope of the D/Dq versus c line for c >c would decrease in good solvents and increase in poor solvents, in agreement with predictions from... 

This can be done without loss of generality because of the arbitrariness in the choice of the small expansion parameter tj. One way to rationalize this would be to say that we expand the solution and the control parameter in the vicinity of bifurcation in powers of the amplitude of the main harmonics e T. This amplitude is thus assumed fixed, so that all corrections to it coming from the solution of homogeneous equations at higher orders are dropped. 

Here, the subscript (c) is short for the set of expansion parameters (c) = (2i, 22, A, L, oi, u2) r, is the vibrational coordinate of the molecule i R is the separation between the centers of mass of the molecules the Q, are the orientations (Euler angles a, jS y,) of molecule i Q specifies the direction of the separation / the C(2i22A M[M2Ma), etc., are Clebsch-Gordan coefficients the DxMt) are Wigner rotation matrices. The expansion coefficients A(C) = A2i22Al u1u2(ri,r2, R) are independent of the coordinate system these will be referred to as multipole-induced or overlap-induced dipole components - whichever the case may be. 

The expansion parameters that appear in Eqs. 4.7 and 4.8 as arguments of the Clebsch-Gordan coefficients satisfy the triangular relationship,... 

For molecules with inversion symmetry, like H2, the expansion parameter X must be even, Eqs. 4.15 through 4.17. (It also must be non-negative.) In order to relate the expansion coefficient Axl to the Cartesian dipole components calculated in a body-fixed frame, we choose the unit separation vector, R, to be parallel to the z-axis, hence M = 0, Yw = [(2L + 1)/4k] /2, and... 

Fig. 6.3. The significant free —> free components of the spectral functions of molecular hydrogen pairs at 77 K. For a given set of expansion parameters A1A2AL, a different line type is chosen. When two curves of the same type are shown, the upper one represents the free — free, the lower the bound —< free contributions their sum is the total FG al T). The extreme low-frequency portion of the bound — free contributions with the dimer fine structures is here suppressed [282],...

The anisotropy of the interaction couples the translational and rotational states of collisional systems. This in turn couples the various dipole components. Instead of computing for each set of expansion parameters X XiSL one general profile for all rotational components associated with that set, one now has a much more complex computational task to compute the induced absorption continua. Moreover, the energy level diagrams as well as the spectra of van der Waals dimers are much more complex when the anisotropy of the interaction is accounted for. 

In other words, the spectral function is written as a sum over rotational line profiles centered at the rotational transition frequencies ojr r,. For each set of expansion parameters (c) = A1I2AL, the quantities a, satisfy the selection rules appropriate for the (c) component, and are chosen such that... 

The situation can only be remedied by a systematic approximation method in the form of an expansion in powers of a small parameter. Only in that case does one have an objective measure for the size of the several terms. Our first task is therefore to select a suitable expansion parameter. It must be a parameter occurring in the M-equation, i.e., in the transition probability W. Furthermore, it must govern the size of the fluctuations, and therefore the size of the jumps. We denote this parameter by Q and choose it in such a way that for large Q the jumps are relatively small In many cases Q is simply the size of the system. 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

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