Landau description

In the macroscopic and phenomenological Ginzburg109-Landau description of superconductivity, a complex "order parameter" P(r) = j/(r) exp(i) is proposed, which equals zero above Tc and whose magnitude determines... 

Again one concludes that the scaling relations eqs. (80), (90) and (91) are satisfied, while the hyperscaling relation [eq. (93)] would only be satisfied at d — 8. Indeed, using the Lifshitz exponents in the Ginzburg criterion [eqs. (52)—(55)] one does find that the Landau description of Lifshitz points becomes self-consistent only for d > 8. Thus it is no surprise that the behavior at physical dimensionalities (d = 2, 3) is very different from the above predictions. In fact, in d = 2 one does not have Lifshitz points at non-zero temperature (Selke, 1992). 

As the experimental investigations have focused predominantly on static properties of the nematic-isotropic transition, most of the theoretical papers have used a Ginzburg-Landau description involving an expansion in the nematic order parameter to describe static properties [4, 17, 22-25]. 

The polydomain-monodomain transition in nematic LCE was investigated [26] by incorporating a local anchoring interaction into the Ginzburg-Landau description of the nematic-isotropic transition in LCE [4, 17,25]. 

The first step toward a Ginzburg-Landau description of oil-water -amphiphile mixtures was taken byTeubner and Strey [40], who noted that the scattering intensity in bulk contrast. 

Although the Landau description was established for continuous phase transitions it is also used for describing discontinuous transitions, though special care should be taken because of a jump in the order parameter. In the case of LCs, the phase transitions are only weakly discontinuous and the description within the Landau theory is useful. 

Optical antipodes possess the same magnitude of the spontaneous polarization Ps in their smectic-C phase but show opposite directions of Ps with respect to the molecular tilt direction in the racemate Pg equals zero. One thus expects a simple linear relation between Pg and the enantiomeric excess Xee- Experimentally, the behavior of the spontaneous polarization has been studied in [74], [75], [78], [82]. In [74], [78] a strictly linear Ps xee) dependence at constant temperature difference to the smectic-C-smectic- transition temperature Tac was observed (see Figure 8.11). The linear Ps xee) dependence is in accordance with the simple Landau description which uses the Landau free energy g = gg + gpi [(8.3) and (8.5)] and predicts Pgccd [(8.7)]. Proportionality between the tilt-polarization coupling constant C and Xee then results in the experimentally observed proportionality PsCCXee, provided the shift of the transition temperature Tac with varying Xee is taken into account. 

The Landau description, although in essence a mean field theory, is a phenomenological description of far greater generality... 

The electron acceptors discussed so far resemble these terminal polar compounds in their structure, or are even identical. Cladis, for example, has studied the phase behavior of mixtures of butyloxybenzylidene octyl-aniline with cyanooctyloxybiphenyl showing stabilized smectic phases of the A- and B-type, as well as an induced SmE phase the results are summarized in a Landau description [23 g]. However, the phase behavior of this particular system, strongly related to that studied by Park et al. [8], is discussed in terms of dipolar pair formation. Furthermore, the phase behavior and macroscopic properties, e.g., densities and rotational viscosities, in mixtures of polar 4-cyano derivatives of biphenyl with apolar azoxy compounds, were found to differ significantly from those comprising the relat-... 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Zhu I, Wdom A and Champion P M 1997 A multidimensional Landau-Zener description of chemical reaction dynamics and vibrational coherence J. Chem. Phys. 107 2859-71... 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

The basic Landau-Ginzburg model is valid only for relatively weak surfactants and in a limited region of the phase space. In order to find a more general mesoscopic description, valid also for strong surfactants and in a more extended region of the phase space, we derive in this section a mesoscopic Landau-Ginzburg model from the lattice CHS model [16]. 

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

At the mesoscopic level of description the Landau-Ginzburg model of the phase transitions in diblock copolymer system was formulated by Leibler [36]... 

Comparing J with the thermal energy, one can differentiate between the two processes. For adiabatic surfaces the coupling J kbT, while in the non-adiabatic case J kbT. A quantitative description of the two types of process can be made using the Landau-Zehner parameter g (Zulicke, 1985 Onuchic et al., 1986) defined in (5). Here, o)c is the frequency of the vibrational... 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

In principle, we already have in our disposal the SRPA formalism for description of the collective motion in space of collective variables. Indeed, Eqs. (11), (12), (18), and (19) deliver one-body operators and strength matrices we need for the separable expansion of the two-body interaction. The number K of the collective variables qk(t) and pk(t) and separable terms depends on how precisely we want to describe the collecive motion (see discussion in Section 4). For K = 1, SRPA converges to the sum rule approach with a one collective mode [6]. For K > 1, we have a system of K coupled oscillators and SRPA is reduced to the local RPA [6,24] suitable for a rough description of several modes and or main gross-structure efects. However, SRPA is still not ready to describe the Landau fragmentation. For this aim, we should consider the detailed Iph space. This will be done in the next subsection. 

The crudeness of Skyrme forces has certain consequences. For example, the Skyrme functional has no any exchange-correlation term since the relevant effects are supposed to be already included into numerous Skyrme fitting parameters. Besides, the Skyrme functional may accept a diverse set of T-even and T-dd densities and currents. One may say that T-odd densities appear in the Skyrme functional partly because of its specific construction. Indeed, other effective nuclear forces (Gogny [40], Landau-Migdal [41]) do not exploit T-odd densities and currents for description of nuclear dynamics. 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

I2) Landau-Stanyukovicb Ze/ dovicb-Kompaneets (LSZK). Derivation of this equation from that of Landau Stanyukovich is described in the book of Zel dovich St Kompaneets (Ref 12b, pp 223 28). A later description is given by Lutzky (Addnl Ref O) ... 

It should be emphasized that this way of including fluctuations has no other justification than that it is convenient and bypasses a description of the noise sources, compare IX.4. It may provide some qualitative insight into the effect of noise, but does not describe its actual mechanism. For instance, fluctuations in the pumping should give rise to randomness in the coefficient a, rather than to an additive term. Yet the equation (7.6) has been the subject of extensive study and it is famous in statistical mechanics under the name of generalized Ginzburg-Landau equation. It may well serve us as an illustration for a stochastic process.510... 

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