Order parameter space

The grand free energy at a particular temperature can be calculated by numerically integrating equation (7) over the order parameter space. 

Fig. 9. Two-parameter ordering phase diagram for a system of 500 identical hard spheres (Truskett et ai, 2000 Torquato et ai, 2000). Shown are the coordinates in structural order parameter space (r, ) for the equilibrium fluid (dot-dashed), the equilibrium FCC crystal (dashed), and a set of glasses (circles) produced with varying compression rates. Here, r is the translational order parameter from (26) and is the bond-orientational order parameter Q( from (25) normalized by its value in the perfect FCC crystal ( = Each circle...

Figure 38. Contour plot of P( il>t, (calculated using only intact bonds) for the time-averaged 3584-particle WCA liquid at p = 0.83. The letters indicate the approximate locations of the corresponding vertex types in order-parameter space, (a) Raw distribution. The contour levels are 0.125, 0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, and 128. (b) Rationalized distribution (see text). The contour levels are -16, -8, -4, -2, -1, -0.5, 0,0.5, 1, 2, 4, 8, and 16. Positive contours are indicated by solid lines, negative contours by dotted lines, and the zero contour by a heavy solid line.

Figure 68. Contour plot of PdiA, , I / J) (calculated using only intact bonds) for the DRP system. The letters indicate the approximate locations of the corresponding vertex types in order-parameter space. The contour levels are the same as for Fig. 38. (a) Raw distribution, (b) Rationalized distribution (see text).

In this case, there is a symmetry against the change of sign of 0 (fig. 10 shows that this simply corresponds to an interchange of sublattices) and hence a term 03 cannot occur. The fourth order term, however, now contains two cubic invariants rather than a single term [(02)2 =

rotational invariance in the order parameter space (0i, 02), since all directions in the (01, 02) plane are equivalent. No such rotational symmetry applies to the (2x1) structure, of course. So the expansion eq. (26) results, which defines the universality class of the X Y model with cubic anisotropy . Of course, in this approach not much can he said on the phenomenological coefficients r, u, u, R in eq. (26). 

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