Ordering parameters

At r = 0, all the spins are either aligned up or down. The magnetization per site is an order parameter whieh vanishes at the eritieal point. Along the eoexistenee eiirve at zero field... 

Assume that the free energy can be expanded in powers of the magnetization m which is the order parameter. At zero field, only even powers of m appear in the expansion, due to the up-down symmetry of the system, and... 

Figure A2.5.16. The coexistence curve, = KI(2R) versus mole fraction v for a simple mixture. Also shown as an abscissa is the order parameter s, which makes the diagram equally applicable to order-disorder phenomena in solids and to ferromagnetism. The dotted curve is the spinodal.

Figure A2.5.19. Isothemis showing the reduced external magnetic field B. = P Bq/ZcTj versus the order parameter s = for various reduced temperatures J = TIT. ...

In general the width of the coexistence line (Ap, Ax, or AM) is proportional to an order parameter s, and its absolute value may be written as... 

For the kind of transition above which the order parameter is zero and below which other values are stable, the coefficient 2 iiiust change sign at the transition point and must remain positive. As we have seen, the dependence of s on temperature is detemiined by requiring the free energy to be a miniimuii (i.e. by setting its derivative with respect to s equal to zero). Thus... 

A feature of a critical point, line, or surface is that it is located where divergences of various properties, in particular correlation lengths, occur. Moreover it is reasonable to assume that at such a point there is always an order parameter that is zero on one side of the transition and tliat becomes nonzero on the other side. Nothing of this sort occurs at a first-order transition, even the gradual liquid-gas transition shown in figure A2.5.3 and figure A2.5.4. 

Another problem can be the choice of an order parameter for the detemiination of p and of the departure from... 

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

Figure A3.3.6 Free energy as a function of the order parameter cji for the homogeneous single phase (a) and for the two-phase regions (b), 0.

For critical quench experiments there is a synnnetry < )q = 0 and from equation (A3,3,50) S( ) = ( ), leading to a syimnetric local free energy ( figure A3,3,6) and a scaled order parameter whose average is zero, 8v / = i /. For off-critical quenches this synnnetry is lost. One has 8( ) = ( ) -t ( ) which scales to 5 j/ = with... 

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... 

For a conserved order parameter, the interface dynamics and late-stage domain growth involve the evapomtion-diffusion-condensation mechanism whereby large droplets (small curvature) grow at tlie expense of small droplets (large curvature). This is also the basis for the Lifshitz-Slyozov analysis which is discussed in section A3.3.4. 

Again consider a single spherical droplet of minority phase ( [/ = -1) of radius R innnersed m a sea of majority phase. But now let the majority phase have an order parameter at infinity that is (slightly) smaller than +1, i.e. [i( ) = < 1. The majority phase is now supersaturated with the dissolved minority species,... 

Global and local correlation times, generalized order parameter, S... 

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