Long wavelength approximation

In the presence of diffusion C(r,t) obeys the standard diffusion equation 

Here po r, t) is the random Gaussian function with the correlator 

Let us examine now the set of equations controlling the creation and annihilation of neutral A and B particles in the Euclidean space which have equal diffusion coefficients D = Dq = D [93]. It has the form of equations (2.2.20) to (2.2.21). Here K is a reaction rate of bimolecular recombination in particular, it can be equal to K = SirDro. Also, and 

For completely mixed particles in large enough volume, the square of the fluctuation of the number of particles SN = N — N obeys the ordinary statistical relationship 

Yet if the correlator of the number of particles looks like (2.2.28), we have stronger level of fluctuations  

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Because k = and the scales of rj and Ra are of the dimension of the molecule, k rj and k Ra are less than unity in magnitude, within this so-called "long-wavelength" approximation. 

The PDMS and the PIB chains consist of around 78 monomers rendering the usually applied long wavelength approximation of the Zimm model... 

The Coulomb attraction of the electrons in the intermediate state and in the final Volkov states by the ion, as well as the Coulomb repulsion between the two final electrons are missing from the theory developed thus far. A rigorous treatment of the first issue has so far resisted any efforts. It is much easier to deal with the Coulomb repulsion of the two electrons in the final state, depicted in the right-hand panel of Fig. 4.1, since the product of two Volkov states (4.2) can be extended exactly to incorporate the Coulomb repulsion [22]. This is possible because in the long-wavelength approximation the laser couples to the sum rq I rq of the two electron coordinates while the Coulomb repulsion affects their difference r = rq — r2 just like in the absence of the laser held. The state reads [22]... 

For this problem already the simple mean field approximation becomes rather involved [197,213]. Therefore, we describe here only an approach, which is even more simplified, appropriate for wavenumbers q near the characteristic wavenumber q, but strictly correct neither for q—>0 nor for large q the spirit of our approach is similar to the long wavelength approximation encountered in the mean field theory of blends, Eq. (7). That is, we write the effective free energy functional as an expansion in powers of t t and include terms (Vv /)2 as well as (V2 /)2, as in the related problem of lamellar phases of microemulsions [232,233],namely [234]... 

The following calculations are based on a linear stability analysis taking into account only the first order of. Furthermore, we use two approximations. One is the long wavelength approximation the wavelength of perturbation of the solid-liquid interface is much larger than the mean thickness of the liquid film, then we can define a small dimensionless wavenumber The other is the quasistationary approximation we... 

Equations (6-83) and (6-85) govern the dynamics of any time-dependent changes in the shape of the thin film, subject of course to the long-wavelength approximation, (6-86). Hence, if we wish to study the stability of the thin film to some perturbation of shape from the uniform ho, we consider some specified initial shape that we can represent symbolically (in dimensional terms) as... 

Let us calculate the small contributions of nanometer layers to ellipsometric parameters. In long-wavelength approximation in the first order with respect to the small parameter djX for<54 = 4 - Lo and M = A-A, where 4, A and 4 o> o te the ellipsometric angles of an ultrathin film and a bare substrate (d, = 0), respectively, we obtain the following approximate formulas ... 

When the radiation wavelength is much longer than the "atomic dimensions," i.e., when kr << 1, the "long-wavelength approximation" (LWA) is valid, meaning that (with V A = 0)... 

Y. Konminos, Th. Mercouris, C.A. Nicolaides, Long-wavelength approximation in on-and off-resonance transitions, Phys. Rev. A 71 (2005) 023410. 

Note that C t) is essentially the solvation time correlation function. Both C(t) and J co) are molecular properties and can be related to the dielectric properties of the solvent, in the long wavelength approximation (see [39] for a comprehensive discussion). In terms of the dielectric permittivity (o)) [7,21],... 

Thus, if the wavelength of the light is much greater than the grain size, the long wavelength approximation and effective medium theory can be applied to determine the effective value of the composite dielectric constant and, consequently, describe composites optical properties. However, if the size of the structure is of the order of tens and even units of nanometers, then the effective medium approach is not applicable. Indeed, within this approximation, the effective permittivity of a composite is determined as a function of the permittivity for each composite component and, in turn, the nanocomposite components are characterized by the same tensor of permittivity as those used for bulk media. ... 

Equation (4.71) works well for low k values (long wavelength approximation). P is the equilibrium polarization averaged over the nanoparticle volume. It satisfies... 

Because the extent of localization of the dressed photon is equivalent to the nanometric particle size, the long-wavelength approximation, which has always been employed for conventional light-matter interaction theory, is not valid. This means that an electric dipole-forbidden state in the nanometric particle can be excited as a result of the dressed photon exchange between closely placed nanometric particles, which enables the operation of novel nanophotonic devices. Details of such devices will be reviewed in Sect. 1.4. 

Because the long-wavelength approximation is not valid due to the localized nature of the dressed photon (refer to Sect. 1.2), an electron in the nanometric particle can be excited even to an electric dipole-forbidden energy level as a result of the dressed photon exchange between closely spaced nanometric particles, which enables novel nanometer-scale wireless optical devices with dimensions beyond the diffraction-limit, low energy consumption, and resistance to non-invasive attacks. 

Invoking the weak-field approximation (neglecting the term involving A A) and the long wavelength approximation (e 1), we arrive at the familiar dipole approximation expression,... 

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