Landau coefficients

For different systems, we have different signs of the real and imaginary part of Landau coefficient /. Here, we will keep our attention focused to flow past a circular cylinder, that works as a prototypical model for bluff-body flow instability. This instability begins as a linear temporal instability and its first appearance with respect to the Reynolds number is referred to as Hopf bifurcation. Thus, the Reynolds number at which the first bifurcation occurs is given by Rccr- Thus, above Rccr the value of <7 > 0 signifies linear instability. One of the most important aspect of this linear instability is the subsequent non-linear saturation that can be adequately explained by the Landau s equation, if only R is positive. We will focus upon this type of flow only in the next. 

Landau did not address the issue of phase angle (see Landau (1944)), it was later derived by treating the landau coefficient /, as a complex quantity-as given in the last section. Despite the nonlinearity of Eqn. (5.1.7), it is readily integrable to provide... 

In Eqn. (5.4.4), the Landau coefficient /(. is truly a constant independent of Re its dependence on Reynolds number comes from the factor in the denominator through a. Once again, one would need different values of fcjS for different Reynolds number ranges- as given in Table 5.1. 

The nonlinear effects on vortex shedding is also demonstrated in Fig. 5.5, where the present computed Strouhal number is compared with linear global stability results of Morzynski Thiele (1993). Having obtained cOe by direct simulation, one can estimate the imaginary part of the Landau coefficient by using Eq. (5.3.2) employing the linear stability results of Morzynski Thiele (1993). These results for the variation of /j with Re is shown in Fig. 5.6, in the limited range of Re molds numbers, for which... 

Figure 5.6 Variation of Imaginary part of Landau coefficient with...

In the case of quartz, the only strains are non-symmetry-breaking and would normally be expected to lead to renormalisation only of the fourth order Landau coefficient. Higher order coupling leads to renormalisation of higher order terms as well, however. A renormalised version of the Landau expansion for the 3 a quartz transition (Eqn. 28) would be... 

Figure 5. The dependence of the critical exponent p on the Landau coefficients. The critical exponents were calculated by least squares fitting of an equilibrium order parameter evolution from a Landau potential. The transition temperature was kept constant, while the b and c coefficients were varied thus changing the value of the a coefficient. The dark grey regions show critical exponents of less than 0.25, approaching a first order type of behaviour. The light grey regions indicate a zone were the critical exponent is between 0.25 and 0.5, and the transition is continuous.

Figure 6. The behaviour of two order parameters relative to each other. The numbers on the graph refer to the strength of the coupling. Both order parameters have the same value for the a Landau coefficient in the expansion of the free energy. The transition temperature for Q2 is twice that of Qi.

It follows from the continuity of the basis functions as a function of q that the Landau coefficients (9) are continuous functions of q and... 

It is a special layer formation susceptibility close to the phase transition the nematic is very sensitive to the spatially periodic molecular field, which induce the density wave with period 1. In order to study this phenomenon one is tempted to use an external spatially periodic force with the same period, but, at present, it is technically impossible. Therefore, we cannot find the Landau coefficient a above T va using some analogy with the Kerr or Cotton-Mouton effects. 

The ratio of fourth order Landau coefficients Pi IP2 — 2 was taken as evidence for the weak coupling nature of superconductivity. Therefore the question arises whether it is possible to calculate the synunetry of the order parameter within a microscopic weak coupling Hamiltonian starting from an on-site effective quasiparticle interaction which is repulsive. Such... 

Classical theories fail to find a stable fixed point (not even isotropic) and consequently cannot corroborate the experimental evidence that the transition can be second order. Two possible situations can be imagined upon moving away from the mean field tri-critical point towards negative values of the Landau coefficient, u either the transition is still first order but the discontinuities are... 

Here, we discuss the theoretical aspect of our discrete Hamiltonian to show nonlinear K-G equation and then deal with quantirm mechanical approach for phonon boimd state or QB state to explain the possible dependence of criticality on the Landau coefficient through quantum route. 

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