Renormalization coefficients

At large scales and long times the effects of the velocity field can be modeled in terms of a reaction-diffusion process with renormalized coefficients [13],... 

The application of direct variational method to Eq. (3.51) yields, similarly to the previous case of cylindrical particles, the free energy function with renormalized coefficients... 

Euler-Lagrange Equation and Free Energy with Renormalized Coefficients for Antiferroelectric Films... 

For the case of second order phase transitions, the renormalized coefficients in Eq. (3.60) are the following ... 

Here, the variational parameter Py can be obtained by minimization of conventional fi-ee energy, but with renormalized coefficient before Py, namely ... 

Here we introduce the renormalized coefficient g and extrapolation length X ... 

Application of the direct variational method [8, 78] for the Euler-Lagrange Eq. (4.20) leads to the conventional form of the free energy with renormalized coefficients... 

Displacements Aw of sublattices differ finm zero only in the presence of internal stress, in particular, in magnetic and electric fields. The parameter matrices B T) and B T) represent renormalized coefficients of the linear magnetoelastic coupling ... 

The first approach is based on conventional linear elastic theory. It predicts that one of the five elastic coefficients of an uniaxial elastic medium, C5, is renormalized by the coupling between the strain and the director. The renormalized coefficient, called C5 (in the literature this coefficient is sometimes called C44), is smaller than C5, but nevertheless exhibits a value similar to those of the other elastic coefficients (de Gennes 1980). 

The contracted basis set created from the procedure above is listed in Figure 28.3. Note that the contraction coefficients are not normalized. This is not usually a problem since nearly all software packages will renormalize the coefficients automatically. The atom calculation rerun with contracted orbitals is expected to run much faster and have a slightly higher energy. 

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

Besides the crust and the hadron shell, the hybrid star contains also a quark core. Both the nucleon shell and the quark core can be in superconducting phases, in dependence on the value of the temperature. Fluctuations affect transport coefficients, specific heat, emissivity, masses of low-lying excitations and respectively electromagnetic properties of the star, like electroconductivity and magnetic field structure, e.g., renormalizing critical values of the magnetic field (/ ,, Hc, Hc2). 

Comparing Equations 5 and 6, it is evident that the stress distribution in the x-y plane can be calculated simply by using the ordinary plane strain model with the thermal expansion coefficient a in Equation 5 replaced by a renormalized expansion coefficient... 

Because large magnitudes of broadening were observed in PTFE, even for slowly cooled specimens, it was necessary to use line-width standards. The two materials used were annealed LiF and a diluted solid mixture of ammonium hydrogen phosphate. Data analysis proceeded by Fourier analysis of multiple orders, the well-known Warren-Averbach procedure. " Values of the domain size as measmed experimentally and with a correction using renormalized cosine coefficients (RCC), are given in Table 1.2. 

The fact that there are now two equations, viz. (8.12) and (8.9), implies that no longer is y(t) determined by /(0) alone, but by the initial vaue of a2 as well. One might hope that, after a short initial transient time, o2 adjusts itself by rapidly approaching an asymptotic value depending on the instantaneous y(t) alone, so that a renormalized equation for / holds after the initial transient. However, this is not the case the time scale on which a2 approaches (8.10) is determined by the coefficient a in (8.9) and is therefore comparable to the rate at which / itself varies, see (8.7). There is no separation of time scales and therefore no single equation for / by itself. 

The diffusion in a velocity field with a wide velocity spectrum supposedly describing turbulence is considered in the spirit of cascade-renormalization ideas. For the latter case of isotropic turbulence, we construct an ordinary differential equation for the turbulent diffusion coefficient. 

MCT can be best viewed as a synthesis of two formidable theoretical approaches, namely the renormalized kinetic theory [5-9] and the extended hydrodynamic theory [10]. While the former provides the method to treat both the very short and the very long time responses, it often becomes intractable in the intermediate times. This is best seen in the calculation of the velocity time correlation function of a tagged atom or a molecule. The extended hydrodynamic theory provides the simplicity in terms of the wavenumber-dependent hydrodynamic modes. The decay of these modes are expressed in terms of the wavenumber- and frequency-dependent transport coefficients. This hydrodynamic description is often valid from intermediate to long times, although it breaks down both at very short and at very long times, for different reasons. None of these two approaches provides a self-consistent description. The self-consistency enters in the determination of the time correlation functions of the hydrodynamic modes in terms of the... 

In the formulations developed from the renormalized kinetic theory approach, these self-consistencies were avoided either by using values obtained from computer simulation and experiments or by using some exactly known limiting values for the transport coefficient. For example, in the treatment of Mazenko [5-7], and of Mehaffey and Cukier s [8] the transport coefficients are replaced by their Enskog values. In the theory developed by Sjogren and Sjolander [9], the velocity autocorrelation function is required as an input that was obtained from the computer simulated values. This limits the validity of the theories only to certain regimes and for certain systems where the experimental or computer-simulated results are available. 

We know that the additively renormalized bare theory (the left hand side in Eq. (11.10)) exists in that limit. Also the -factors attain finite limits Z(ti), Zu(u), Zn(u). Indeed they are constructed as power series in u, the coefficients for t > 0,6 > 0 taking the form of polynomials in / r) . Thus no problem results from setting t = 0. Since the renormalized theory is finite for d = 4, whereas the bare continuous chain model diverges for d —> 4, showing poles in 5, also the Z-factors must diverge for d —> 4. In the NCL we can therefore formulate the theorem of renormalizability as follows ... 

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