Landau point

For the enthalpy discontinuity at 7 n i one obtains Hi H = 2aB I9C. When 5=0, there is no enthalpy jump or latent heat and one has a critical point (Landau point) on an otherwise first-order line. Because of the presence of this (small) cubic term in Eq. (34), the N-I transition should (normally) be weakly first order. This is in agreement with experimental observations. In Fig. 8, part of the enthalpy curve near N-I is shown for hexylcyanobiphenyl (6CB), a compound of the alkylcyanobiphenyl (nCB) homologous series [5]. The nearly vertical part of the enthalpy curve corresponds to the... 

Actually there are a few micellar nematic liquid crystals where the I-N transition is exactly of second order (T( = Tc). In these materials one generally finds a point is in the I-N transition line, where the two second-order transition lines end, and the three nematic phases and the isotropic phase become identical. This point is called Landau point. In the case of tire potassium laurate (KL)-l-decanol-D20 system, this point is shown in Figure 3.5. ... 

Phase diagram of the potassium laurate (KL)-l-decanol-D20 system, with the Landau point. Nc and Nl are uniaxial nematic phases with cylindrical and lamellar micelle structures, and Nj, denotes the biaxial nematic phase. (Figure reproduced from Reference [8] with permission... 

The shape of these mesogens is clearly biaxial. However, the stability of a biaxial nematic phase strongly depends on the inter-arm angle 6. A theoretical model based purely on repulsive molecular interactions [30] as well as another model including attractive and repulsive anisotropic interactions [31] both predict the Landau point, i.e., the point where a direct transition from the isotropic state to the biaxial nematic state occurs, to be achieved when the inter-arm angle adopts the tetrahedral value of 109.47°. On the other hand, the stability of the biaxial nematic phase is limited by the crystallization of the system. Therefore, the range of angles for which a biaxial nematic phase can be expected is extremely small, on the order of 2° from the tetrahedral value. 

Menon and Landau" point out a number of complications associated with modeling alloy deposition. First, the values for the kinetics parameters of the alloy species are likely to be different from those measured for the pure components. Furthermore, these parameters may vary with the composition of the alloy. Determination of such parameters can be done experimentally from measured alloy composition in well-controlled deposition experiments. A second issue has to do with the difficulty in determining the activities of the alloy components in the solid phase. The activity, which affects the electrode kinetics equation (82), is unity for single-component deposition however, in multicomponent alloy deposition it varies with the nature of the deposited alloy." ... 

This subject has a long history and important early papers include those by Deijaguin and Landau [29] (see Ref. 30) and Langmuir [31]. As noted by Langmuir in 1938, the total force acting on the planes can be regarded as the sum of a contribution from osmotic pressure, since the ion concentrations differ from those in the bulk, and a force due to the electric field. The total force must be constant across the gap and since the field, d /jdx is zero at the midpoint, the total force is given the net osmotic pressure at this point. If the solution is dilute, then... 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

The problem of branching of the wavepacket at crossing points is very old and has been treated separately by Landau and by Zener [H, 173. 174], The model problem they considered has the following diabatic coupling matrix ... 

In the vicinity of the critical point (i.e. t < i) the interfacial width is much larger than the microscopic length scale / and the Landau-Ginzburg expansion is applicable. 

Johnson D, Aiiender D, Dehoff D, Maze C, Oppenheim E and Reynoids R 1977 Nematio-smeotio A-smeotio C poiyoritioai point Experimentai evidenoe and a Landau theory Phys.Revs B 16 470-5... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Figure C3.6.10 Defect-mediated turbulence in tire complex Ginzburg-Landau equation, (a) The phase, arg( ), as grey shades, (b) The amplitude [A], witli a similar color coding. In tire left panel topological defects can be identified as points around which one finds all shades of grey. Note tire apparently random spatial pattern of amplitudes.

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

All Np states belonging to the Pth sub-space interact strongly with each other in the sense that each pair of consecutive states have at least one point where they form a Landau-Zener type interaction. In other words, all j = I,... At/> — I form at least at one point in configuration space, a conical (parabolical) intersection. 

At this point, we make two comments (a) Conditions (1) and (2) lead to a well-defined sub-Hilbert space that for any further treatments (in spectroscopy or scattering processes) has to be treated as a whole (and not on a state by state level), (b) Since all states in a given sub-Hilbert space are adiabatic states, stiong interactions of the Landau-Zener type can occur between two consecutive states only. However, Demkov-type interactions may exist between any two states. 

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

The transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

Landau proposed in 1944 that turbulence arises essentially through the emergence of an ever increasing number of quasi-periodic motions resulting from successive bifurcations of the fluid system [landau44]. For small TZ, the fluid motion is, as we have seen, laminar, corresponding to a stable fixed point in phase space. As Ti is... 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

Fig. 10.12. Vapor-liquid phase behavior for the Lennard-Jones fluid. Solid triangles and hollow squares indicate the results of the particle addition/deletion and volume scaling variants of the flat-histogram simulation using the Wang-Landau algorithm. Crosses are from a histogram reweighting study based on grand-canonical measurements at seven state points. The solid line is from Lotfi, et al. [76], Reprinted figure with permission from [75]. 2002 by the American Physical Society...

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