Landau theory density

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

Figure 5. Temperature development of the electronic density of states in fee FeaNi with the temperature dependent input taken from the Ginzburg-Landau theory (magnetic moments are given per atom).

The properties of the two helium isotopes in the liquid state are strongly influenced by quantum effects. In Fig. 2.8, the specific heat of 3He, calculated from the ideal gas Fermi model (Tp = 4.9 K) with the liquid 3He density, is compared with the experimental data. The inadequacy of this model is evident. A better fit, especially at the lower temperatures, is obtained by the Landau theory [25]. 

One sfrategy is phenomenological and rooted in the Landau theory of phase transitions. The basic idea is to assert that the homogeneous free energy density... 

In the Landau theory (6), the change <5 > in the actual charge density as one moves away from the phase transition point is expanded in terms of a... 

Now the renormalized Hamiltonian can again be analysed in the spirit of the Landau theory, using once more Eq. (MO). This yields the Hartree free energy density [61]... 

It is always tempting to introduce the concept of a critical point to explain the increase of fluctuations as in the Landau theory. But we lack a clear physical picture at this point in time. It is certain that the free-energy surface becomes flatter with respect to local density fluctuations as the temperature is lowered, which is opposite to the occurrence in normal liquids. Enhanced fluctuations lead to the nucleation of the crystalline ice forms [8],... 

The electroclinic or soft mode effect could be explained within the framework of the Landau theory of phase transitions [4]. We can write the expansion of the free energy density g of smectic A up to the 0 term as follows ... 

Four principal theories describing the transformation of the isotropic fluid into a spontaneously organized nematic fluid, the I N transition, have been developed (1) Onsager s density expansion of the free energy of anisometric particles, (2) Rory s estimate of the insertion probability for a rod-like (multisite) solute into a lattice, (3) Maier and Saupe s construction of apotential of mean torque experienced by mesogens (or solutes) in a nematic environment, and (4) de Gennes transposition of Landau theory to the I N transition. We briefly examine each of these in reverse chronological order because each is relevant to more-recent theoretical descriptions of polymer mesophases. 

Brownian dynamics Dissipative particle dynamics Lattice Boltzmann Time-dependent Ginzburg-Landau Dynamical density functional theory... 

To describe interfacial forces caused by orientation correlations, Landau theory in combination with a mean field approach has been applied. In Landau theory, the free energy density of a system is expanded in a power series of the order parameters and their derivatives. A description of the theory is beyond the scope of this book and the reader is referred to textbooks on the statistical physics of liquids. It was first applied to describe hydration forces (see below) [1135,1136]. De Gennes applied it to liquid crystals [1137], and sometimes the theory is referred to as Landau-de Gennes theory. 

The Landau equation in plasma theory is a nonlinear variant, but there P is a particle density rather than a probability. L.D. Landau, Physik. Z. Sovjetunion 10, 154 (1963) = Collected Papers (D. ter Haar ed., Pergamon, Oxford 1965) p. 163. The same is true for the nonlinear Fokker-Planck equation in M. Shiino, Phys. Rev. A 36, 2393 (1987). 

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