Landau’s theory

From the point of view of statistical mechanics there are many problems, such as strongly anharmonic lattices, to which the theory can be applied.14 It appears as a natural generalization of Landau s theory of quasi-particles in the case when dissipation can no longer be neglected. The most interesting feature is that equilibrium and nonequilibrium properties appear linked. The very definition of the strongly coupled anharmonic phonons depends on their lifetime. 

L.V. Schubnikov and I.E. Nakhutin [24] have corroborated Landau s theory by electroresistivity measurements parallel and perpendicular to the direction of the external magnetic field on reaching the critical field intensity Hc, and called such a domain structure the intermediate state . A.G. Meshkovskyi and Yu.V. Scharvin [25] have demonstrated the existence of the intermediate state during its scanning by drawing a bismuth wire, well known for the remarkable sensitivity of its electroconductivity to magnetic field. 

The Ginzburg-Pitaevskii theory [134] for bulk liquid He near the X point rests on Landau s theory of second-order phase transitions [133]. This theory... 

Since the identification of universality classes for surface layer transitions needs the I-andau expansion as a basic step, we first formulate Landau s theory (Toledano and Toledano, 1987) for the simplest case, a scalar order parameter density

phase transition and slowly varying in space. It can be obtained by averaging a microscopic variable over a suitable coarsc-graining cell Ld (in d-dimensional space). For example, for the c(2x2) structure in fig. 10 the microscopic variable is the difference in density between the two sublattices I (a and c in fig. 10) or II (b and d in fig. 10), ,- = pj1 — pj. The index i now labels the elementary cells (which contain one site from each sublattice I, II). Then... 

The three ordered stales of the Potts model correspond to a preferential occupation of one of the three sublattices a,b,c into which the triangular lattice is split in the (-/3x-v/3)R30° structure. In the order parameter plane (0x.0r), the minima of F occur at positions (1, 0)MS, (—1/2, i/3/2)yWs, (—1/2, -yf3/2)Ms, where Ms is the absolute value of the order parameter, i.e. they are rotated by an angle of 120° with respect to each other. The phase transition of the three-state Potts model hence can be interpreted as spontaneous breaking of the (discrete) Zj symmetry. While Landau s theory implies [fig. 13 and eqs. (20), (21)] that this transition must be of first order due to the third-order invariant present in eq. (34), it actually is of second order in d = 2 dimensions (Baxter, 1982, 1973) in agreement with experimental observations on monolayer ( /3x /3)R30o structures (Dash, 1978 Bretz, 1977). The reasons why Landau s theory fails in predicting the order of the transition and the critical behavior that results in this case will be discussed in the next section. 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

In order to understand why Landau s theory is inaccurate, let ns recall the justification of eq. (14) in terms of the coarse-graining eq. (13), where short wavelength fluctuations of a microscopic model [such as the Ising model, eq. (1)] are eliminated. In fact, if L in eq. (13) would be the lattice spacing a,... 

This neglect of fluctuations in general is not warranted. One can recognize this problem in the framework of Landau s theory itself. This criterion named after Ginzburg (1960) considers the mean square fluctuation of the order parameter in a coarse graining volume Ul and states that Landau s theory is selfconsistent if this fluctuation is much smaller than the square of the order parameter itself,... 

One easily verifies that all these scaling relations hold for the critical exponents of Landau s theory, eq. (48), as well as for the spherical model [eq. (61)] and Ising and Potts models (see sect. 2.3 below). 

Thus Landau s theory predicts the following set of tricritical exponents (Griffiths, 1970 Sarbach and Lawrie, 1984)... 

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

Calculations for finite nuclei will be discussed which demonstrate that the distribution of sp strength in the experimentally accessible energy region can be qualitatively understood. In addition, it becomes possible to interpret both theoretical and experimental results in terms of quasiparticle excitations, the basic concept of Landau s theory of Fermi liquids [19-21]. In contrast to an infinite liquid, the sp basis must be appropriate for the finite system under study and is not composed of the sp momentum states. Apart from this obvious requirement, most notions carry over rather straightforwardly. The ability to calculate the sp strength distribution and compare to experimental data presents an advantage over the approach initiated by Migdal [22,23]. 

The agreement between the critical indices in both Landau s theory and the mean field approximation attests to the equivalence of the two approaches. 

According to Landau s theory, the binodal is symmetrical about 4>c near the critical point, and... 

The transition from water to ice at 1 atmosphere pressure is a first-order transition, and the latent heat is about 100 J/g. The isotropic-nematic transition is a weak first-order transition because the order parameter changes discontinuously across the transition but the latent heat is only about 10 J/g. De Gennes extended Landau s theory into isotropic-nematic transition... 

The lattice gas approach is valid within certain limits for typical metallic hydrides, binaries as well as ternaries. Deviation from this idealized picture indicates that metallic hydrides are not pure host-guest systems, but real chemical compounds. An important difference between the model of hydrogen as a lattice gas, liquid, or solid and real metal hydrides lies in the nature of the phase transitions. Whereas the crystallization of a material is a first-order transition according to Landau s theory, an order-disorder transition in a hydride can be of first or second order. The structural relationships between ordered and disordered phases of metal hydrides have been proven in many cases by crystallographic group-subgroup relationships, which suggests the possibility of second-order (continuous) phase transitions. However, in many cases hints for a transition of first order were found due to a surface contamination of the sample that kinetically hinders the transition to proceed. 

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