Self- consistency condition

For the case of nonzero temperatures the vacuum averages in Eq.(7) should be replaced by thermal averages over phonon populations. Using (7) and (5) we obtain that the scattering of an exciton in the effective medium by the perturbation fi — v z)) is described by the following self-consistent condition... 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

In order to find the self-consistent condition, we have to determine the order parameter... 

In spite of its simplicity this approach, supplemented with Blatt s correction [.3] for lattice distortion, was applied successfully for decades [4, 5] in studies of systematics in the residual resistivity. Its power was the exact treatment of the scattering and the use of the Friedel sum rule [1] as a self-consistency condition ensuring a correct valency difference between impurity and host atom. 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

The partition function (Eq. 26) may be regarded as a functional of external field H = H(r]),..., H(rl), which can be determined from the self-consistence condition [30]. Of utmost importance in finding the chemical correlators... 

The surface CP oc(E) in the chemisorbed system is determined by a self-consistency condition, which is found by substituting Gc(l,l) into the general CPA equation (6.39), to give... 

The unambiguous identification of the extraction rate regime (diffusional, kinetic, or mixed) is difficult from both the experimental and theoretical viewpoints [12,13]. Experimental difficulties exist because a large set of different experimental information, obtained in self-consistent conditions and over a very broad range of several chemical and physical variables, is needed. Unless simplifying assumptions can be used, frequently the differential equations have no analytical solutions, and boundary conditions have to be detemtined by specific experiments. 

Since self-consistency condition (5) is also formulated in terms of fs, we get a closed system of two equations on a real function fs which is solved numerically. 

The relaxation of a local mode is characterized by the time-dependent anomalous correlations the rate of the relaxation is expressed through the non-stationary displacement correlation function. The non-linear integral equations for this function has been derived and solved numerically. In the physical meaning, the equation is the self-consistency condition of the time-dependent phonon subsystem. We found that the relaxation rate exhibits a critical behavior it is sharply increased near a specific (critical) value(s) of the interaction the corresponding dependence is characterized by the critical index k — 1, where k is the number of the created phonons. In the close vicinity of the critical point(s) the rate attains a very high value comparable to the frequency of phonons. 

On the other hand, the properties of the system as a whole can be calculated and the macroscopic dynamic modulus can be determined. Here the question of the relation between the postulated micro-viscoelasticity and the resulting macro-viscoelasticity appears. The answer requires a properly formulated self-consistency condition. Simple speculations show that equality of the micro- and macro-viscoelasticity cannot be obtained. Nevertheless, it is natural to require the equality of relaxation times of micro- and macroviscoelasticities. It will be shown in this section that this condition can be satisfied. 

When self-consistency conditions are achieved, we obtain from eq.(60) and eq.(58) the following expression for Gei... 

From the analysis described in the previous section, we look for the ground state density as the density n(r) which satisfies the self-consistency condition (58). Starting from a guess density n(r) = n° (r) we construct the associated Kohn-Sham potential ... 

With either nematic potential, the equilibrium orientation distribution T eq( ) can be computed via a self-consistency condition, and the isotropic-to-nematic transition has been analyzed in terms of nematic strength parameters. ... 

Suppose that a is sufficiently small, i.e., We is sufficiently large, that surface tension plays no role in determining the bubble shape, except possibly locally in the vicinity of the rim where the spherical upper surface and the flat lower surface meet. Further, suppose that the Reynolds number is sufficiently large that the motion of the liquid can be approximated to a first approximation, by means of the potential-flow theory. Denote the radius of curvature at the nose of the bubble as R(dX 6 = 0). Show that a self-consistency condition for existence of a spherical shape with radius R in the vicinity of the stagnation point, 0 = 0, is that the velocity of rise of the bubble is... 

The imposition of the self-consistency condition for S using Eq. (14) gives... 

Equation 6.13 is a self-consistency condition for p, since its right-hand side containsp through the presence of / , in F of (6.12), and / 2 2 of... 

The self-consistency condition (6.13) is the key to the glass transition. To make its content and meaning clearer, we derive it by an alternative procedure. Instead of considering all possible variations of P v) that leave it... 

The free volume Vf is given by the self-consistency condition, (6.13). The contribution to Rj be neglected because Q (p), the... 

To complete the calculation of C, we must know the temperature dependence of i5y, Vg, and most important, p. This can only be done by solving the self-consistency condition p = h(p), (6.13), in more detail. To reduce the number of free parameters, we use the viscosity data fitted by the parameters in Table II for the temperature dependence of Vy. For this reason, we limit our discussion to systems in which tj has been measured over a wide temperature range. The remaining parameters are then Oq, v, and K to describe/(c) andp, a,A,A, D, and in C. If we scale all volumes by vg taken equal to v , that leaves only v /vq and k = kvo as unknowns in f(v). The latter is constrained, since we know... 

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