Geometrical properties of the Potts model during the coarsening regime

We study the dynamic evolution of geometric structures in a polydegenerate system represented by a $q$-state Potts model with nonconserved order parameter that is quenched from its disordered into its ordered phase. The numerical results obtained with Monte Carlo simulations show a strong relation between the statistical properties of hull perimeters in the initial state and during coarsening: The statistics and morphology of the structures that are larger than the averaged ones are those of the initial state, while the ones of small structures are determined by the curvature-driven dynamic process. We link the hull properties to the ones of the areas they enclose. We analyze the linear von Neumann-Mullins law, both for individual domains and on the average, concluding that its validity, for the later case, is limited to domains with number of sides around 6, while presenting stronger violations in the former case.

Figure 1

Illustration of various interfacial length definitions on the square lattice. The colored sites are all occupied by the same spin species, while spins on white squares are in a different state. The hull length (solid circles) is the external border of the domain. The hull perimeter (open circles) contains only the external nearest neighbors of the domain (not the internal ones). The domain perimeter (crosses) is the ensemble of nearest neighbors of the domain and includes the internal ones as well. The measurements presented here are for the hull perimeter only.

Figure 2

Snapshots of the $q=3$ ferromagnetic Potts model on a square lattice with linear size $L=1000$ after a quench, from initial states prepared at $T_0\to \infty$ (left column) and $T_0=T_c$ (right column), to the final temperature $T_f=T_c/2$. Times shown are, from top to bottom, $t=0$, $2^5$, and $2^{10}$ MCs, respectively. In panels (g) and (h) the interfaces in the configurations at the latest time $t=2^{10}$ MCs [(e) and (f)] are shown, along with some small thermal fluctuations.

Figure 3

Sequence of snapshots for the $q=8$ Potts model. The left column displays results obtained by using an equilibrium initial configuration at $T_0\to \infty$ (a) and its time evolved after the quench [(c) and (e)]. The right column shows configurations found by using an equilibrium initial state at $T_0=T_c$ (b) and after the quench [(d) and (f)]. From top to bottom the measuring times are $t=0$, $2^7$, and $2^{10}$ MCs, respectively, and $T_f=T_c/2$. In panels (g) and (h) we show the domains with areas $A<R^2$ (green, or light gray), $R^2\left(t\right)<A<10R^2\left(t\right)$ (white), and $A>10R^2\left(t\right)$ (red, or dark gray) for the configurations at $t=1024$ MCs.

Figure 4

Examples of typical domain structures obtained in the simulations. In (a) a regular domain with area $A\sim p^{{\alpha }_h}$ and ${\alpha }_h=2$. The domain in (b) has a rougher morphology and a smaller exponent ${\alpha }_h$ certifies this feature, $1<{\alpha }_h<2$. In (c) a domain with very rough and stretched morphology, $A\sim p$; the exponent ${\alpha }_h$ reaches it minimum value, ${\alpha }_h=1$, in this case.

Figure 5

Scaling relation between hull-enclosed areas and hull perimeters at several measuring times for $q=3$ after a quench from $T_0\to \infty$ to $T_f=T_c/2$. The inset shows the raw data at several times from $t=0$ to $2^{14}$ MCs (bottom to top). The extent up to which the law $A\sim p^2$ (red solid line) applies increases with time, demonstrating that the domains become more regular in the course of evolution.

Figure 6

Scaling relation between hull-enclosed areas and hull perimeters at $t=2,...,2^{10}$ MCs for different models (various $q$ values) and initial conditions. The curves labeled (a) correspond to $q=2,3,8$ and $T_0=T_c$, those labeled (b) are for $q=2$ and $T_0\to \infty$, the ones labeled (c) have been obtained using $q=3$ and $T_0\to \infty$ and, finally, the ones carrying the label (d) are for $q=8$ and $T_0\to \infty$. All quenches are to $T_f=T_c/2$. When the initial state is at the critical point, case (a), the relation $A\simeq p^{{\alpha }_h}$ presents a similar exponent for all $q$, ${\alpha }_h\simeq 1.44$. Instead, after a quench from the $T_0\to \infty$ uncorrelated state the exponent ${\alpha }_h$ decreases from its $q=2$ value ${\alpha }_h\simeq 1.15$ to ${\alpha }_h\simeq 1.06$. Notice that for the case $q=8$ and $T_0=T_c$, case (a), the observed exponent is 1.44 because the measure was taken at a relatively small time, in which the finite correlations present at the initial state are still large compared with the characteristic length. At later times, this exponent will decrease, eventually approaching 1.06, as the characteristic length grows beyond those initial correlations.

Figure 7

Rate of change of the individual hull-enclosed area at time $t=1024$ MCs after a quench from $T_0\to \infty$ to $T_c/2$ ($q=8$). The data are grouped by the number of sides $n=4$ (top) and $n=8$ (down). The vertical dashed line indicates $R^2\left(t\right)\propto t$.

Figure 8

(Top) The area rate of change averaged over small areas, $A_0=8<A<c{R}^2$, for (a) $q=2$ and $T_0\to \infty$ (red), (b) $q=3$ and $T_0=T_c$ (blue), (c) $q=3$ and $T_0\to \infty$ (green), (d) $q=8$ and $T_0=T_c$ (pink), and (e) $q=8$ and $T_0\to \infty$ (cyan). In all cases the averaged rate of change seems to asymptotically reach a negative constant. (Bottom) The time-dependent average number of sides, ${\langle n\left(t\right)\rangle }_{<}$ for the same parameters is shown with the same color and label code. See the text for the discussion.

Figure 9

The same as in Fig. 8 but averaging over large areas, $A>c{R}^2$, for the same choice of parameters and color code used in that figure. Notice the order of the labels in the top panel.

Figure 10

Total number of hulls with $n$ sides for $q=3$, $T_0\to \infty$ (solid symbols, upper group of curves) and $T_0=T_c$ (open symbols, lower group of curves). The dotted lines indicate $N_h=1$ (horizontal) and $n=6$ (vertical). The different symbols correspond to different measuring times as given in the key. The distribution has weight well above $N_h=1$ for $n>6$ when the initial conditions have finite correlation length.

Figure 11

The full averaged area rate of change as a function of the number of sides at $t=64$ MCs for the parameters given in the key. The solid lines correspond to the function $f\left(x\right)\simeq x^{\gamma }$, with $\gamma =1,1.5,2$, from bottom to top, and serve as guides for the eye.

Figure 12

Rescaled hull perimeter distribution for $q=3$ at several times ($t=2,2^2,...,2^{14}$ MCs), after a quench from $T_0=T_c$ to $T_f=T_c/2$. The solid black line represents the theoretical prediction Eq. (13) valid for small areas, $A/R^2<10$, with ${\alpha }_h=2$, and for large areas, $A/R^2>10$, with ${\alpha }_h\simeq 1.44$.

Figure 13

Hull perimeter distribution for $q=8$ at several times given in the key after a quench from equilibrium at $T_0\to \infty$ to $T_f=T_c/2$. The declivity of the envelope, ${\zeta }_h\sim 3.0$, is a direct consequence of dynamical scaling.

Figure 14

Hull-enclosed area distribution for $q=8$ at several times after a quench from equilibrium at $T_0\to \infty$ to $T_f=T_c/2$, rescaled by $R^2\left(t\right)$. The solid line is a fit with Eq. (14) and $a\simeq 1.21$, $b\simeq 1.22$, $c\simeq 0.6$. (Inset) The same data in a linear scale, showing that for small areas, the collapse improves at large times.

Figure 15

Hull perimeter distribution for $q=8$ at several times after a quench from equilibrium at $T_0\to \infty$ to $T_f=T_c/2$, rescaled by $R\left(t\right)$. The solid line is Eq. (15) with ${\alpha }_h=2$ for $p<R$ and ${\alpha }_h\simeq 1.06$ for $p>R$. The fit parameters, however, besides being different from those of Fig. 14, differ also for $p<R$ and $p>R$. (Inset) The same data in a linear scale.