Macroscopic Landau free-energy

Let the order parameter Si be associated with the fraction active material, and be defined as S =f. For the second order parameter we choose S2 = Na (Na — 1 )Ka/X as a measure of the degree of polymerization of the active material and presume that the polymerization is highly co-operative, so K.a —> 0. The macroscopic Landau free-energy density F that describes the nucleated assembly now reads... 

In Landau theory, the information about the change of physical quantities is gathered in the order parameter Qo = V f d rQ(r) which is a macroscopic quantity that neglects spatial and temporal fluctuations. The basic concept of the description of phase transitions is the introduction of the Landau free energy, iF = J d rf, which takes into account the symmetry of the system through a power series expansion in terms of the scalar invariants of the order parameter, whereas the equation of state of the system reads... 

At the largest length scales, field theoretic and continuum mechanics models are able to predict the equiUbriirm stmcture of multicomponent systems and macroscopic flow response of a polymer system. However, these models do not contain molecular detail. They are based either on a phenomenological description of the free energy of the system, such as the Flory-Hu ns or Landau free energies, or on actual hydrodynamic parameters such as viscosity. 

Now — L is the Landau-Ginzburg free energy, where m2 = a(T — Tc) near the critical temperature, is a macroscopic many-particle wave function, introduced by Bardeen-Cooper-Schrieffer, according to which an attractive force between electrons is mediated by bosonic electron pairs. At low temperature these fall into the same quantum state (Bose-Einstein condensation), and because of this, a many-particle wave function (f> may be used to describe the macroscopic system. At T > Tc, m2 > 0 and the minimum free energy is at = 0. However, when T [Pg.173]

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

The temporal dependence of the fluctuations can be described by the overdamped equation of motion — the Landau-Khalatnikov equation [36, 37]. It can be understood as follows The equilibrium configuration of the order parameter is determined by the minimum of the free energy, 5T = f dVSf(T/,Vr/) = f dV 6f/6r])5r], which is satisfied for Sf/drj = 0. Here, S/Srj = d/dr] — V d/dVri). If the system is out of equilibrium, Sf/Sr] acts like a generalized elastic force which is balanced by viscous forces. When the macroscopic velocity can be neglected the viscosity is related solely to the rate of change of the order parameter Q,... 

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