# Disentangling Growth and Decay of Domains During Phase Ordering

Suman Majumder<EMAIL_ADDRESS><EMAIL_ADDRESS>Amity Institute of
Applied Sciences, Amity University Uttar Pradesh, Noida 201313, India

###### Abstract

Using Monte Carlo simulations we study the phase ordering dynamics of a multi-
species system modeled via the prototype $q$-state Potts model. In such a
multi-species system, we identify a spin states or species as the winner if it
has survived as the majority in the final state, otherwise we mark them as
loser. We disentangle the time ($t$) dependence of the domain length of the
winner from losers, rather than monitoring the average domain length obtained
by treating all spin states or species alike. The kinetics of domain growth of
the winner at a finite temperature in space dimension $d=2$ reveal that the
expected Lifshitz-Cahn-Allen scaling law $\sim t^{1/2}$ can be observed with
no early-time corrections, even for system sizes much smaller than what is
traditionally used. Up to a certain period, all the others species, i.e., the
losers, also show a growth that, however, is dependent on the total number of
species, and slower than the expected $\sim t^{1/2}$ growth. Afterwards, the
domains of the losers start decaying with time, for which our data strongly
suggest the behavior $\sim t^{-z}$, where $z=2$ is the dynamical exponent for
nonconserved dynamics. We also demonstrate that this new approach of looking
into the kinetics also provides new insights for the special case of phase
ordering at zero temperature, both in $d=2$ and $d=3$.

##  I INTRODUCTION

Existence of a discrete number themodynamically degenerate ordered phases at
low-temperature is the characteristics of many systems, e.g., ordering alloys,
adsorbed atoms and rare gases on surface, intercalation compounds Moncton et
al. (1977); Berker et al. (1978); Bak and Domany (1979). Theoretically and
computationally such systems can effectively be understood via a multi-species
model. In this context, the prototype is the $q$-state Potts model Wu (1982).
It has been used extensively, especially, to investigate phase transitions
associated with such systems. Slight variation of the model makes it useful in
diverse applications, be it in biological cell sorting or in modeling active
matter system Graner and Glazier (1992); Chen et al. (2007); Szabó and Merks
(2013); Guisoni et al. (2018); Chatterjee et al. (2020); Mangeat et al.
(2020). In the context of phase transition, among all the studied problems one
is particularly intriguing – to understand the nonequilibrium kinetics in
reaching the final equilibrium state following a quench from an initial
disorder state to a temperature below the transition temperature, where the
equilibrium is an ordered state. Such a process goes by the name of phase
ordering or coarsening. Its dynamics is highlighted by the formation and
growth of domains of like-species Bray (2002); Puri and Wadhawan (2009). The
phenomenon is characterized by the scaling of various morphology-
characterizing functions, viz., the two-point equal-time correlation function
exhibits a scaling of the form

$\displaystyle C(r,t)$ $\displaystyle=$ $\displaystyle\tilde{C}[r/\ell(t)],$
(1)

where $\tilde{C}$ is a time ($t$)-independent master function, and $\ell(t)$
is the average linear domain size which is expected to grow as

$\displaystyle\ell(t)\sim t^{\alpha},$ (2)

with an exponent $\alpha=1/2$, refer to as the Lifshitz-Cahn-Allen (LCA) law
Lifshitz (1962); Allen and Cahn (1979). It turns out that $\alpha=1/z$ where
$z$ is the equilbrium dynamical exponent Bray (2002).

Most of the simulation studies on phase-ordering dynamics of the Potts model
were focused on estimating the value of $\alpha$ for $q\geq 2$. Early results
for smaller system sizes at temperature $T>0$ where there is no pinned or
frozen dynamics, showed that the exponent decreases from $\alpha=1/2$ as $q$
increases from $q=2$ Safran (1981); Sahni et al. (1983a). Later, using results
from relatively larger system sizes for $q$ as larges as $q=64$ it was shown
that $\alpha=1/2$ can only be realized in the long-time limit at subcritical
temperatures Sahni et al. (1983b); Kaski et al. (1985); Ferrero and Cannas
(2007). However, for square lattices the problem of presence of pinned
configurations exists at low but finite temperatures Ferrero and Cannas
(2007); Ibáñez De Berganza et al. (2007). On the other hand, due to the
presence of pinned or frozen dynamics for $q\geq d+1$, zero-temperature phase
ordering is always of special interest both in $d=2$ and $3$ Lifshitz (1962);
Safran (1981); Loureiro et al. (2010); Olejarz et al. (2013). There, the
domain growth is found to be significantly slower than the expected LCA law.
For that matter, arguments in favor of logarithmic growth were also proposed
Safran (1981). In $d=3$, even for the $q=2$ Potts model which is basically the
spin $1/2$ Ising model, the issue of domain growth at $T=0$ is not well
settled Lippiello et al. (2008); Olejarz et al. (2011a, b); Chakraborty and
Das (2016); Das and Chakraborty (2017); Vadakkayil et al. (2019). Simulation
studies of relatively larger system sizes even suggested a possible crossover
in the domain growth from an early time $\sim t^{1/3}$ behavior to the
asymptotic $\sim t^{1/2}$ behavior Lippiello et al. (2008); Vadakkayil et al.
(2019).

It must have been quite clear by now that in the sutdy of phase ordering the
primary quantity which one measures is the average linear size of the domains
$\ell(t)$ as a function of time. To calculate $\ell(t)$, scaling property of
any morphology-characterizing function is used, viz., from the decay of the
two-point equal-time correlation function, from the first moment of the
structure factor, and from the first moment of the domain-length distribution
function Bray (2002). The measured $\ell(t)$ from different methods are
proportional to each other. At this point it is worth noting that in phase
ordering the dynamics is nonconserved, i.e., the global order parameter, viz.,
the magnetization, does not remain fixed to its starting value during the
entire evolution of the system. In other words, even though one typically
begins with a uniform mixture having equal number of every possible spin
states or species, eventually ends up with one of the spin states or species
emerging as the majority or winner in the final state. All other spin states
or species which we refer to as losers, are present in negligible amount. So,
it is easy to perceive that among all the spin states or species, domains of
only the winner will always be growing before reaching the finite-size limit.
Alongside domains of losers must be declining after a certain time and almost
vanishing in the final state. When one calculates the average domain length
$\ell(t)$, this distinction of the spin states or species in terms of winner
or losers is not taken into account. Thus, the effect of decay of the losers
is always embedded in the measured $\ell(t)$. Nevertheless, till date no
attempts have been made to disentangled the kinetics of the winner from
losers.

Here, keeping the above issues in mind, by means of Monte Carlo (MC)
simulations we study the phase ordering dynamics of the $q$-state Potts model
with a focus on monitoring the kinetics of individual spin states or species
by categorizing them as the winner or losers. Our results for $2\leq q\leq 10$
at a finite temperature for domain growth of the winner show no early-time
correction to the expected scaling law. Apart from the growth of the winner,
the kinetics of the losers exhibit a non-monotonic behavior with a long-time
power-law decay $\sim t^{-2}$, independent of the total number of spin states
or species $q$. We also demonstrate that our approach of monitoring the
kinetics of the winner provides improved results for the special case of phase
ordering of the $q$-state Potts model at $T=0$ in space dimension $d=2$.
Furthermore in $d=3$, for $q=2$, that corresponds to the Ising model, we show
that all the features that were expected to be observed in large systems can
essentially be captured in relatively much smaller systems.

The remaining of the paper is organized in the following way. In Section II,
we describe the model and details of the performed simulations. Then we
present results in Sec. III, where as we go along we also describe the
calculations of different observables. At the end in Sec. IV, we provide a
brief summary, conclusion and outlook.

##  II Model and Method

The Hamiltonian of the $q$-state Potts model is given by

$\displaystyle H=-J\sum_{\langle
ij\rangle}\delta_{\sigma_{i},\sigma_{j}};~{}\sigma_{i}=1,2,\dots,q;~{}J>0,$
(3)

where $\sigma_{i}$ is the spin state or species type $\sigma$ of the $i$-th
site, and the summation is over all possible pairs of nearest neighbor
indicated by $\langle ij\rangle$. The model exhibits an order-disordered
transition as a function of temperature. For a square lattice, the
corresponding transition temperature is $T_{c}=J/k_{B}\ln(1+\sqrt{q})$, where
$k_{B}$ is the Boltzmann constant Wu (1982). For $q\leq 4$ the transitions are
second order whereas for $q>5$ they are first order.

Figure 1:  Time evolution snapshots depicting the general sequence of events
during phase-ordering dynamics of the $q$-state Potts model , at $T=0.6T_{c}$
in $d=2$ using a square lattice with linear dimension $L=128$, for two
different choices of $q$, as indicated. Different colors represent different
spin states or species.

We study the model (3) by performing MC simulations on square and cubic
lattices, respectively, for $d=2$ and $3$, having linear dimensions $L$ along
each Cartesian direction. We apply periodic boundary conditions in all
possible directions. Dynamics in our simulations is introduced via the
standard Glauber spin-flip mechanism that changes the global concentration of
each spin state or species during the evolution, and finally ends up with one
of the spin states or species as the majority Newman and Barkema (1999);
Landau and Binder (2014). A typical MC move consists of randomly choosing a
site having the state $\sigma_{i}$ followed by changing its state to any of
the remaining $q-1$ states. The move is then accepted in accordance with the
standard Metropolis algorithm with a probability

$p_{i}=\textrm{min}[1,\exp(-\Delta E/k_{B}T)],$ (4)

where $\Delta E$ is the change in energy due to the attempted move. $L^{d}$
such attempts form a MC sweep (MCS) setting the unit of time.

To start our simulations, as initial condition we prepare a random mixture
containing all possible ($q$) spin states or species in equal proportion,
mimicking a high-temperature ($T=\infty$) paramagnetic phase. Such
configurations are then suddenly quenched below the critical temperature
$T_{c}$ by fixing $T=0.6T_{c}$. In our simulations, $J/k_{B}$ sets the unit of
temperature, and for convenience $J$ and $k_{B}$ are set to unity. We also
perform simulations at $T=0$ where the criterion for the Metropolis update of
the system is slightly different. There, an attempted move is accepted if
$\Delta E\leq 0$. All the results presented are averaged over more than $100$
independent realization accounting for different starting configurations and
thermal noise during evolution.

##  III Results

The results are presented in two subsections. In the first subsection, we
discuss the results for $d=2$ at the finite temperature $T=0.6T_{c}$. In the
other one, we present results for $T=0$ in $d=2$ and $3$.

###  III.1 Phase Ordering at finite $T$ in $d=2$

When a disordered system containing homogeneous mixture of $q$ spin states or
species is quenched below $T_{c}$, it becomes unstable to fluctuations and
starts evolving towards an ordered state where only one of the spin states or
species survives as the majority. Snapshots for such an evolution using a
square lattice at $T=0.6T_{c}$ is presented in Fig. 1 for $q=4$ and $10$. In
both cases the process starts with formation of domains of like spins as
represented by the snapshots at $t=10$ MCS. These domains then start growing
with time, as evident from the snapshots at $t=10^{2}$ MCS and $t=10^{3}$ MCS,
respectively, for $q=4$ and $10$. Afterwards, once the domains of individual
spin state or species have attained a considerable size, one of the species
keeps on growing further while the others start decaying and eventually almost
vanish. This leads to the emergence of the winner species which stays as the
majority in the final configuration. For example, in the case presented in
Fig. 1, the yellow and purple states are winners, respectively, for $q=4$ and
$10$. There all other states are marked as losers. By arranging the spin
states or species in descending order in terms of the total number of sites in
the final configuration having that particular spin state or species, we
assign a ranking $\Re_{\sigma}$ to them where

$\Re_{\sigma}\in\\{1,2,\dots q\\}.$ (5)

The winner will have $\Re_{\sigma}=1$ and all other spin states or species
with $\Re_{\sigma}>1$ correspond to losers. $\Re_{\sigma}=q$ indicates that
the number of sites in the final configuration belonging to that spin state or
species is the least. Note that the condition $\Re_{\sigma}=\sigma$ is not
necessarily true.

We start with measuring the primary quantity of interest, the growing
characteristic length scale, which in the present case is the linear size of
the domains. However, in contrast to the usual practice, we measure the length
of the domains as a function time of each spin state or species and store them
during the simulation. The domain of an individual spin state or species is
measured by mapping the Potts spins of every lattice site $i$ to an Ising-like
scalar variable as Das and Puri (2002); Majumder et al. (2018)

$\psi_{i}^{n}=2\delta_{\sigma_{i},n}-1;~{}n=1,2,\dots q.$ (6)

This gives a lattice of spin $\pm 1$ for every value of the Potts spin $n$. By
scanning through these lattices in $x$, $y$, and $z$ (for $d=3$) directions,
we measure the total number of interfaces or defects $N_{\rm def}$ in each of
the constituent Ising chains of the lattice, of course, by taking into account
of the periodic boundary conditions. For each Ising chain of length $L$, the
total number of linear domains will be $N_{\rm ld}=N_{\rm def}/2$, except for
the special cases where all the spins in the chain are either $+1$ or $-1$. In
the former case, the number of linear domains will be $1$ and for the latter
there will be no domains of $+1$ spins. Finally, by using the total number
$N_{+}$ of $+1$ spins present in the lattice, we calculate the average linear
size of domains of the spin state or species $n$ as

$\ell_{n}=\langle N_{+}/N_{\rm ld}\rangle,$ (7)

where the bracket indicates averaging over all possible directions. Thus, by
using the corresponding Ising-like lattice for every spin state or species, we
calculate the average linear size of domains of every spin state or species.
Finally, we do a change of identity of the domain lengths by
$\ell_{n}\rightarrow\ell_{\Re_{n}}$ using the previously introduced ranking of
the spin states or species. The average of domain lengths obtained from
different trajectories are then calculated in terms of $\ell_{\Re_{n}}$. Thus,
from now onward $\ell_{1}$ corresponds to the average linear domain size of
the winner and $\ell_{\Re_{n}}$ with $\Re_{n}>1$ correspond to losers. In the
traditional way, different spin states or species are not distinguished, and
average domain length $\ell$ is calculated by considering them alike, so that

$\ell=\frac{1}{q}\sum_{\Re_{n}=1}^{q}\ell_{\Re_{n}}.$ (8)

Figure 2:  Demonstration of fulfillment of the scaling criterion (1) of the
two-point equal-time correlation function during phase ordering in a square
lattice with $L=128$ for $q=4$, at $T=0.6T_{c}$. In (a) the average linear
domain length $\ell(t)$ is used as the scaling factor for the correlation
function $C_{\rm avg}(r,t)$, obtained from averaging over all the spin states
or species. The same for the correlation function of the winner species
$C_{1}(r,t)$, using the domain length of the winner $\ell_{1}(t)$ is shown in
(b). The insets show the corresponding unscaled data at different times.
Figure 3:  Time dependence of different domain lengths measured for a phase-
ordering system with $q=4$ using a square lattice having $L=128$, at
$T=0.6T_{c}$. Here, $\ell(t)$ correspond to the traditionally calculated
domain length obtained by averaging over all spin states or species. The other
lengths presented are for different spin states or species in terms of their
ranking using $\Re_{n}$ (see text for details). According to the ranking,
$\ell_{1}$ corresponds to the domain length of the winner species and the rest
are all losers. The dashed blue line represents the expected LCA growth $\sim
t^{1/2}$. The dashed black line represents a power-law decay $\sim t^{-2}$.
The shades in grey scale are introduced to roughly mark different regimes.

Before proceeding further with the domain length for the winner and loser, we
verify whether $\ell_{1}$ satisfies the criterion of a scaling phenomenon (1).
For that we calculate the two-point equal-time correlation function averaged
over all the spin states or species as

$\displaystyle C_{\rm
avg}(r,t)=\frac{1}{q}\sum_{n=1}^{q}\left[\langle\psi_{i}^{n}(t)\psi_{j}^{n}(t)\rangle-\langle\psi_{i}^{n}(t)\rangle\langle\psi_{j}^{n}(t)\rangle\right],$
(9)

where $r$ is the scalar distance between any two sites $i$ and $j$. The
corresponding scaling plots of $C_{\rm avg}(r,t)$ by using $\ell(t)$ is
presented in Fig. 2(a) for phase ordering in a square lattice system with
$q=4$ at $T=0.6T_{c}$. The collapse of data for different times implies the
fulfillment of the scaling criterion (1). Next, we calculate two-point equal-
time correlation function of the winner species

$\displaystyle
C_{1}(r,t)=\langle\phi_{i}^{1}(t)\phi_{j}^{1}(t)\rangle-\langle\phi_{i}^{1}(t)\rangle\langle\phi_{j}^{1}(t)\rangle,$
(10)

where $\phi_{i}^{1}$ is the Ising-like lattice variable obtained using the
winner spin state or species. The corresponding scaling plot of $C_{1}(r,t)$
using the domain length of the winner $\ell_{1}(t)$ as the scaling factor, is
shown in Fig. 2(b). Reasonably good collapse of data confirms the satisfaction
of the required scaling criterion (1). In fact at early times, for small $r$
the collapse of data in (b) is superior to the data presented in (a).

Figure 4:  Time dependence of the domain length of (a) the winner $\ell_{1}$
and (b) loser $\ell_{q}$ for different values of $q$ at $T=0.6T_{c}$ using a
square lattice system of size $L=128$. The blue dashed lines represent the LCA
growth law $\sim t^{1/2}$, and the black dashed lines represent the power-law
decay $\sim t^{-2}$.

Having the fulfillment of a scaling phenomenon established, we move on to deal
with time dependence of domain lengths in Fig. 3. There, the data for
$\ell(t)$, obtained by considering all $q$ spin states or species, show a much
slower growth compared to the expected LCA law $\sim t^{1/2}$. Such an
inconsistency with the expected growth law has always been a problem for
smaller system sizes, like $L=128$, presented here. This could be explained
via a combination of finite-time corrections and early advent of finite-size
effects. In such cases, there have been attempts to extract the asymptotic
growth by applying finite-size scaling via an ansatz considering an initial
domain length Majumder and Das (2010, 2011); Das et al. (2012). However, the
trend has always been to use larger system sizes to access larger length and
time scales. As mentioned, we extract the time dependence of $\ell_{\Re_{n}}$
by categorizing the spin states or species via a ranking. The different domain
lengths obtained this way are also plotted in Fig. 3. There, the data for
domain length $\ell_{1}(t)$ of the winner show remarkable consistency with the
expected $\sim t^{1/2}$ growth over an extended period starting from very
early time. Followed by that it enters a regime where it still grows, however,
with a smaller exponent. Finally, it hits the finite-size limit and remains
constant as long as we ran the simulations. The intermediate regime is
essentially an extended crossover regime from the scaling regime to the
finite-size limit. In contrast, the data for $\ell(t)$ show a much sharper
crossover to the finite-size limit.

Intriguing is the behavior of the domain length of the losers as shown in Fig.
3 by the data for $\ell_{\Re_{n}}$ with $\Re_{n}=2,3,\dots q$. In the scaling
regime (lightest grey zone) they show a much slower growth compared to the
winner or the expected LCA law. The slower growth in this regime is the root
cause of the slower growth for $\ell(t)$ in the scaling regime. After the
scaling regime, the growth of the losers gets seized for a very brief period,
and eventually it enters into a decay regime that coincides with intermediate
crossover regime for $\ell_{1}(t)$. For $\ell_{q}$, i.e., $\ell_{4}$, the
domain length of the last ranked spin state or species, the decay seems to be
consistent with $\sim t^{-2}$ indicating a $\sim t^{-z}$ behavior, of course,
subjected to further verification for other values of $q$, as will be done
subsequently.

Figure 5:  System size dependence of the kinetics of domain length of (a) the
winner and (b) loser for $q=4$ at $T=0.6T_{c}$. Square lattice system with
three different $L$, as mentioned, are used. The dashed blue lines in (a) and
(b) represent the LCA growth. The dashed black line in (b) represents a power-
law decay $\sim t^{-2}$.

Figure 6:  Representative snapshots illustrating time evolution during phase
ordering of the Potts model with (a) $q=2$ and (b) $q=4$ at $T=0$ for a square
lattice having $L=128$. Again, different colors correspond to different spin
states or species.

From now onward, we will consider $\ell_{1}$ as the growing length scale to
verify the expected scaling law for phase ordering in different cases.
Additionally, we will investigate the behavior of domain lengths of the last
ranked loser, i.e., $\ell_{q}$. In Fig. 4 (a), we compare the time dependence
of $\ell_{1}(t)$ for $2\leq q\leq 10$ using a square lattice of size $L=128$
at $T=0.6T_{c}$. Data for all $q$ follows the expected $\sim t^{1/2}$ behavior
for an extended period until they are affected by finite-size effects.
Importantly, none of them show any early-time corrections. The amplitude of
growth decreases as $q$ increases. Same power-law growth with different
amplitudes suggests that these growths can also be quantified via a universal
finite-size scaling function with a non-universal metric factor Majumder et
al. (2018); Janke et al. (2021). We have checked that an analogous plot using
the average domain length $\ell(t)$ cannot provide the same universal picture
for such a small system size.

The universality is not restricted to the behavior of $\ell_{1}(t)$ but also
gets reflected in the time dependence of $\ell_{q}(t)$ too, as shown in Fig.
4(b). There, $\ell_{q}(t)$ for all values of $q$ exhibits a non monotonic
behavior with first showing a growth followed by a decay. In the growth regime
data for all $q$ are almost parallel to each other. Albeit, the growth is much
slower than the LCA law. Then, for all $q$ there is a gradual crossover to the
decay regime. There, at long time, the data are again parallel to each other,
and are consistent with the $\sim t^{-2}$ behavior. This strongly suggests the
presence of a universal decay law

$\ell_{q}(t)\sim t^{-z},$ (11)

where $z=2$ is the dynamical exponent. The theoretical origin of the above
relation still needs to be investigated.

We also check the system size dependence of the behavior of the domain lengths
in Figs. 5(a) and (b), respectively, for the winner and loser. In (a) the data
for $\ell_{1}(t)$ from different $L$ follow each other until they enter the
finite-size affected regime, and are consistent with the expected $\sim
t^{1/2}$ behavior. In (b) too the data for smaller system sizes follow the
larger ones up to a certain time, and then gradually start decaying. In the
growth regime, data for all system sizes show a growth slower than the LCA
behavior $\sim t^{1/2}$. Importantly, in the decay regime all data are
consistent with $t^{-2}$ behavior, further numerically confirming the validity
of Eq. (11).

###  III.2 Phase ordering at $T=0$ in $d=2$ and $3$

In the previous subsection we have established the usefulness of disentangling
the growth of different spin states or species in terms of winner and losers.
Here, we explore the same in investigating the special case of phase ordering
in the $q$-state Potts model at $T=0$ in $d=2$ and also in $d=3$.

####  III.2.1 $d=2$

Phase ordering at $T=0$ in $d=2$ for $q=2$ Potts model that corresponds to the
Ising model, is the least challenging of all where one typically observes an
ideal representation of the associated scaling law. Of course, there could be
occasional cases of dynamic freezing, but overall a major fraction of
performed simulations still reach the ground state in a finite time for finite
lattices. A representative for time evolution of such a system using a square
lattice of size $L=128$ is shown in Fig. 6(a). The configuration at the latest
time ($t=3\times 10^{4}$ MCS) corresponds to a typical ground state at $T=0$,
where only one of the spin states or species survives.

Figure 7:  (a) Time dependence of measured domain lengths of the winner and
losers in $d=2$ for $q=4$ during phase ordering at $T=0$ using a square
lattice with $L=128$. The traditionally measured average domain length
$\ell(t)$ is also plotted. (b) Domain length $\ell_{1}$ of the winner species
as a function of time for different values of $q$ using the same square
lattice. (c) System size dependence of $\ell_{1}(t)$ at $T=0$ for a fixed
$q=4$. Dashed blue and black lines, respectively, represent the power-laws
$\sim t^{1/2}$ and $\sim t^{1/3}$.

For larger $q$, on the other hand, a significant fraction of the total
simulations get stuck. This observation is in accordance with the known
difficulty of reaching ground states for $q\geq d+1$ Lifshitz (1962); Safran
(1981). It was also argued that for $q\geq 3$ in $d=2$ due the emergence of
pinned configuration during the evolution, the system may get stuck in a
disordered state leading to glassy dynamics Petri (2003); de Oliveira et al.
(2004); De Oliveira et al. (2004). Relatively recently, from long time
simulations of the model it was further confirmed that for $q\geq 2$ in $d=2$
the probability of reaching ground states is non-zero Olejarz et al. (2013).
There, it was found that in the simulations where the system becomes static,
the configurations has perfectly flat interfaces. In Fig. 6(b) where we
present the evolution of the $q=4$ Potts model, the snapshot at the latest
time represent such a configuration with flat interfaces. Apart from these, we
too observed existence of late time configurations which were evolving ad
infinitum, as typically observed during phase ordering of the $d=3$ Ising
model at $T=0$ Olejarz et al. (2011a, b). We have also performed simulation
for other values of $q$ up to $q=10$. The overall picture remains the same
although the probability of reaching the ground state within our maximum
simulation time almost vanishes as $q$ increases, and episodes of observing
pinned configuration become almost certain.

Figure 8:  Time dependence of the domain length $\ell_{1}$ obtained from
simulations which did not freeze within the maximum simulation time, at $T=0$
in $d=2$ for different $q$ using a system with $L=128$. Dashed blue and black
lines represent possible power-law behaviors as indicated.

Having discussed the difficulties of reaching ground states, we move on to
study the domain-growth kinetics at $T=0$ by using our prescription of
categorizing the different spin states or species as the winner and losers.
Since for large $q$ a significant fraction of the simulations do not reach the
ground state, it is not straightforward to decide the winner. However, if one
looks at the time evolution as in Fig. 6(b) for $q=4$, then one can still
choose the purple spin state or species as the winner, as it is the one that
has been growing throughout, and has the largest size in the latest time. All
the other spin states or species are thus considered to be losers. Following
this we estimate all possible domain lengths, and present the corresponding
plots for $q=4$ in Fig. 7(a). There, we have also shown the data for the
average domain length $\ell(t)$ obtained by usual way of considering all spin
states or species alike. The data for the losers and $\ell(t)$ clearly show a
growth much slower than the LCA growth. The data for $\ell_{4}$ in fact hints
towards a power-law growth with an exponent $\alpha=1/3$. On the other hand,
the data for the winner again is fairly consistent with the $\sim t^{1/2}$
growth. Fitting with the form

$\ell_{1}(t)=At^{\alpha},$ (12)

in the ranges $[10:3\times 10^{3}]$, $[10^{2}:3\times 10^{3}]$, and
$[2^{2}:3\times 10^{3}]$, respectively, yield $\alpha=0.453(1)$, $0.474(1)$,
and $0.486(1)$. The slight deviation from $\alpha=1/2$ is due to the presence
of pinned configurations that do not allow the system to reach its ground
state within the time period up to which we run the simulations. Nevertheless,
our target here is not to obtain the ground state, rather to apply our
approach in quantifying the domain-growth kinetics with the data we obtain at
$T=0$ after running the simulations for a finite time. Hence, we abstain
ourselves in performing simulation for long times. Keeping in mind that the
ground state is not reached always and looking at the long-time flat behavior
of $\ell_{4}$ in Fig. 7(b), here, we do not explore the validation of Eq.
(11).

In Fig. 7(b) we present the domain-growth kinetics of the winner species at
$T=0$ for different values of $q$. For $q\leq 4$ the growth is almost
consistent with expected LCA growth with an exponent $\alpha=1/2$, until
finite-size effects creep in. The data for $q=6$ shows significant deviation
from the LCA right from early time, and at later times it is consistent with
the $\sim t^{1/3}$ behavior. This is a virtue of the slow dynamics for larger
$q$ due to pinning effects which will be overcome eventually in the long time
limit via thermally activated processes in the form of an avalanche Olejarz et
al. (2013). However, consistency of our data for $q=3$ and $4$ with the LCA
law, suggest that the bound $q\leq d+1$ is not a strict one. We present the
system size dependency of kinetics of $\ell_{1}$ at $T=0$ in Fig. 7 (c), which
show prefect consistent with the LCA growth for the smallest system size
$L=64$. For the largest system, i.e., $L=256$, the data seem to be weakly
deviating from the LCA growth, due to the slowly dynamics arising from pinning
effects. This suggests that pinning effects increase with system size.

It is evident from Fig. 7 that at $T=0$, the value where $\ell_{1}$ saturates
at long $t$, does not approach the finite-size limiting value $\approx L$.
This can be attributed to the same fact that at $T=0$ not all the simulations
could attain the ground state with all sites having the same spin states or
species. Since this effect increases as $q$ increases, one notices in Fig. 7
that the saturating value of $\ell_{1}$ decreases as a function of $q$. To
ignore the effect of the frozen dynamics, we now separate out the simulation
runs where the winner species could achieve a length $\ell_{1}\approx L$ until
$t=10^{5}$ MCS, which is significant for a system size as small as $L=128$.
The time dependence of $\ell_{1}$ obtained only from the non-freezed
simulations are shown in Fig. 8 for different $q$. Again data for $q\leq 4$
are perfectly consistent with the LCA growth for an extended time period
before encountering finite-size effects. In fact from a careful observation,
one can appreciate the fact that the data are more consistent with LCA growth,
in comparison with the data in Figs. 7. Although not apparent in the plot, the
increasing estimated error on $\ell_{1}$ as a function of $q$, is also a
consequence of the fact that the fraction of non-freezed simulations decreases
as $q$ increases.

In Fig. 8, one can also notice that for $q>4$ at long times the growth of
$\ell_{1}$ seems to be faster than the LCA growth. For the largest $q$, i.e.,
for $q=10$, apart from the long-time faster regime there exists an
intermediate time period when the growth of $\ell_{1}$ is extremely slow,
consistent with a $\sim t^{1/4}$ behavior. The slow phase correspond to the
presence of pinned configurations which eventually breaks via an avalanche
leading to the long-time faster growth regime. For $q=10$ the faster growth is
consistent with $\ell_{1}(t)\sim t^{3/2}$. Of course, since the pinning effect
is smaller for small $q$, the slope of the faster-growth regime appears to be
increasing as a function of $q$. A detail study of even larger $q$ values is
needed to extract any functional dependence and limiting value of the exponent
as a function of $q$ during the avalanche of pinned configurations.

####  III.2.2 $d=3$

After having an understanding of phase ordering in $d=2$ at zero temperature,
we now move on to explore the other special case of domain-growth kinetics in
$d=3$ at $T=0$. Here, we restrict ourselves to the unique case of the $q=2$
Potts model, i.e., the Ising model. For higher $q$ one needs to perform
rejection free MC simulations which we will present elsewhere.

Figure 9:  Domain-growth kinetics in the $q=2$ Potts model, i.e., the Ising
model, in terms of domain lengths $\ell$ and $\ell_{1}$, at $T=0$ in $d=3$.
Data from simulations that did not get stuck are used. System used is a cubic
lattice of length $L=128$. Dashed lines with different colors correspond to
different power-law behaviors as indicated. Different shades are guides to the
eyes to distinguish different regimes.

The zero temperature phase ordering of the Ising model poses challenges
because a major fraction of the simulations get stuck in local minima and thus
one experiences frequent dynamic freezing. Hence, the measured time dependence
of the average domain length deviates significantly from the expected LCA
growth. In this regard, a dynamic crossover in the time dependence of the
average domain length from a $\ell(t)\sim t^{1/3}$ at finite times to the
asymptotic $\ell(t)\sim t^{1/2}$ behavior has been argued Lipowski (1999),
which was later confirmed numerically by considering data from simulations
that did not get stuck Vadakkayil et al. (2019). This calls for the need of
simulations of larger system sizes to access large length scale and time
scales in order to realize the aforesaid crossover to the universal LCA
growth. In all these previous studies, however, the growth of the domains of
the winner and loser were not disentangled. In Fig. 9, we present the kinetics
of domain length of the winner for the $q=2$ Potts model using a cubic lattice
of $L=128$, at $T=0$ by considering only the simulations that evolved
completely in the given maximum run time of our simulation, i.e., until
$t=4\times 10^{4}$ MCS. There, we have also included the data for the average
domain length $\ell(t)$ that shows the presence of a single scaling law which
is slower than $\ell(t)\sim t^{1/2}$, albeit, faster than $\ell(t)\sim
t^{1/3}$. A fitting with the form (12) in the range $[10:200]$ yields
$\alpha=0.3703(3)$. On the other hand, $\ell_{1}(t)$ hints the presence of a
crossover from a slower growth at early times to $\ell_{1}(t)\sim t^{1/2}$
behavior at long $t$. The early-time growth seems to be significantly faster
than $\ell_{1}(t)\sim t^{1/3}$ behavior. Importantly, the crossover which was
previously thought to be observed only at large length scale and time limit,
is now realized much earlier in the behavior of $\ell_{1}$, and for a much
smaller system having linear dimension of $L=128$.

Figure 10:  System size dependence of domain-growth kinetics of the winner
spin state or species for the $q=2$ Potts model, i.e., the Ising model, in
$d=3$ at $T=0$. In (a) we show the data including the runs where the dynamics
got frozen, whereas (b) shows the data excluding them. In both figures dashed
lines with different colors correspond to different power-law behaviors, as
indicated.

In view of the results for $T=0$ in $d=2$, presented in the previous
subsection, for the $d=3$ case too, one can expect the effect of pinned
configurations. Thus, the crossover which is observed in Fig. 9 could
supposedly be attributed to the pinning effect that stalls the dynamics for
some time, and then overcome it via an avalanche. To further substantiate this
argument, in Fig. 10 we present the data of $\ell_{1}(t)$ for different system
for phase ordering systems having $q=2$ at $T=0$ in $d=3$. In Fig. 10(a) we
show the plot $\ell_{1}(t)$ obtained by considering all the simulations that
we ran, irrespective of the fact whether it got stuck within the maximum
allowed run time. The data for different system sizes nicely overlap with each
other, and are in agreement with the presence of a single scaling law which
seems to be slower than $\ell(t)\sim t^{1/2}$, but faster than $\ell(t)\sim
t^{1/3}$. A fitting of the data for $L=256$ using the form (12) in the range
$[10:2\times 10^{3}]$ provides $\alpha=0.365(5)$. The corresponding plots
using data considering only the simulations which could come out of the frozen
dynamics within the given simulation time is presented in Fig. 10(b). There,
the data for the smallest system size $L=64$ is perfectly consistent with
$\ell_{1}(t)\sim t^{1/2}$ growth, with no indication of a crossover. While the
absence of the crossover was expected as in previous studies using small
system sizes, the realization of the LCA growth is contrastingly surprising.
In Fig. 10(b), as the system size increases one can notice the emergence of a
crossover from an early-time slower growth to a faster one. However, the
crossover time shifts as a function of $L$ implying its system size
dependence. This cannot be the case for a true crossover in the scaling. Thus,
we infer that this is plausibly due to effect of pinned configurations that
stalls the dynamics at finite times, and becomes more prominent as system size
increases, similar to what we observed in the $d=2$ cases for $q\geq 4$. The
system comes out of this pinned dynamics in the form of an avalanche
exhibiting a growth even faster than the LCA law. The data for the largest
system size roughly indicate such a behavior in the post-crossover regime
before it hits the finite-size limit. However, the exponent of the power-law
seems to be smaller than $3/2$ which we obtained for the case of $q=10$ in
$d=2$ in Fig. 8.

##  IV Conclusion

In summary, we have presented results from phase-ordering dynamics of a multi-
species system modeled via the $q$-state Potts model in space dimension $d=2$
and $3$. We have focused on disentangling the dynamics of different species on
the basis of whether the species or spin state has survived as the dominating
species in the final state. If the species is the majority one then we call it
as the winner or else loser.

In $d=2$ at a finite temperature $T=0.6T_{c}$, below the critical temperature
$T_{c}$, we have observed that for $2\leq q\leq 10$ the domain length
$\ell_{1}$ of the winner shows nice agreement with the expected Lifshitz-Cahn-
Allen (LCA) growth law $\ell_{1}(t)\sim t^{1/2}$ without any finite-time
corrections for systems as small as a square lattice having a linear dimension
of $L=128$. This defies the usual tradition of using large system sizes. We
have shown that such an observation with smaller system sizes cannot be
realized if one uses the traditional average domain length $\ell(t)$ that is
calculated by considering all spin states or species alike. We have also shown
that the time dependence of the domain length of the loser exhibits a rich
dynamical behavior. At early times the domains of the loser show a growth that
is slower than the LCA growth. Followed by that it gradually starts decaying,
and in the long-time limit exhibits a $\ell_{q}\sim t^{-z}$ behavior where
$z=2$ is the dynamical exponent for nonconserved dynamics. To establish the
universality of the above relation its validity needs to be verified for
$d=3$. In the same spirit, it would be interesting to further verify it for
phase ordering in long-range interacting systems Christiansen et al. (2019);
Janke et al. (2019); Christiansen et al. (2021).

We have also explored the special case of phase ordering of the Potts model at
$T=0$ in $d=2$. There, we have shown for $2\leq q\leq 4$ that the time
dependence of the domain length of the winner is consistent the LCA growth,
provided one has separated the simulations which did not get stuck due to
frozen dynamics. The time dependence of $\ell_{1}(t)$ for large $q$ values
show that at intermediate times the dynamics is stalled due to pinning.
Subsequently, the system gets out of these pinned configuration which gets
manifested in the form of an avalanche exhibiting an ultra fast domain growth
$\ell_{1}\sim t^{3/2}$. In future, it would be worth to explore the scaling of
the domain length in the avalanche regime for larger $q$. For that one needs
to implement the rejection free $n$-fold algorithm for faster computation
Bortz et al. (1975).

In $d=3$, we have explored the special case of zero-temperature phase ordering
for $q=2$, i.e., for the Ising model. We have shown that if one examines the
time dependence of $\ell_{1}$, then the crossover in the growth from an early-
time slower growth to a faster $\sim t^{1/2}$ growth can be realized in
relatively smaller system size, e.g., in a cubic lattice of linear dimension
$L=128$. Using data from different system sizes we have shown that these
crossover in growth is system size dependent. That poses question about the
true nature of the crossover. Rather, it indicates the presence of pinned
dynamics which later is overcome in the form of an avalanche as observed for
higher $q$ in $d=2$. Nevertheless, in this regard, again a detailed study in
$d=3$ for larger $q$ is needed, which we take up as a future endeavor.

###### Acknowledgements.

The work was funded by the Science and Engineering Research Board (SERB),
Govt. of India in the form of Ramanujan Fellowship under file no. RJF/2021/
000044\.

## References

  * Moncton et al. (1977) D. Moncton, J. Axe, and F. DiSalvo, Phys. Rev. B 16, 801 (1977).
  * Berker et al. (1978) A. Berker, S. Ostlund, and F. Putnam, Phys. Rev. B 17, 3650 (1978).
  * Bak and Domany (1979) P. Bak and E. Domany, Phys. Rev. B 20, 2818 (1979).
  * Wu (1982) F.-Y. Wu, Rev. Mod. Phys. 54, 235 (1982).
  * Graner and Glazier (1992) F. Graner and J. Glazier, Phys. Rev. Lett. 69, 2013 (1992).
  * Chen et al. (2007) N. Chen, J. Glazier, J. Izaguirre, and M. Alber, Comp. Phys. Comm. 176, 670 (2007).
  * Szabó and Merks (2013) A. Szabó and R. Merks, Front. Onco. 3, 87 (2013).
  * Guisoni et al. (2018) N. Guisoni, K. Mazzitello, and L. Diambra, Front. Phys. 6, 61 (2018).
  * Chatterjee et al. (2020) S. Chatterjee, M. Mangeat, R. Paul, and H. Rieger, Europhys. Lett. 130, 66001 (2020).
  * Mangeat et al. (2020) M. Mangeat, S. Chatterjee, R. Paul, and H. Rieger, Phys. Rev. E 102, 042601 (2020).
  * Bray (2002) A. Bray, Adv. Phys. 51, 481 (2002).
  * Puri and Wadhawan (2009) S. Puri and V. Wadhawan, eds., _Kinetics of Phase Transitions_ (CRC Press, Boca Raton, 2009).
  * Lifshitz (1962) I. Lifshitz, Sov. Phys. JETP 15, 939 (1962).
  * Allen and Cahn (1979) S. Allen and J. Cahn, Acta Metall. 27, 1085 (1979).
  * Safran (1981) S. Safran, Phys. Rev. Lett. 46, 1581 (1981).
  * Sahni et al. (1983a) P. Sahni, G. Grest, M. Anderson, and D. Srolovitz, Phys. Rev. Lett. 50, 263 (1983a).
  * Sahni et al. (1983b) P. Sahni, D. Srolovitz, G. Grest, M. Anderson, and S. Safran, Phys. Rev. B 28, 2705 (1983b).
  * Kaski et al. (1985) K. Kaski, J. Nieminen, and J. Gunton, Phys. Rev. B 31, 2998 (1985).
  * Ferrero and Cannas (2007) E. Ferrero and S. Cannas, Phys. Rev. E 76, 031108 (2007).
  * Ibáñez De Berganza et al. (2007) M. Ibáñez De Berganza, V. Loreto, and A. Petri, Philos. Mag. 87, 779 (2007).
  * Loureiro et al. (2010) M. Loureiro, J. Arenzon, L. Cugliandolo, and A. Sicilia, Phys. Rev. E 81, 021129 (2010).
  * Olejarz et al. (2013) J. Olejarz, P. Krapivsky, and S. Redner, J. Stat. Mech.: Theo. Exp. 2013, P06018 (2013).
  * Lippiello et al. (2008) E. Lippiello, F. Corberi, A. Sarracino, and M. Zannetti, Phys. Rev. E 78, 041120 (2008).
  * Olejarz et al. (2011a) J. Olejarz, P. Krapivsky, and S. Redner, Phys. Rev. E 83, 030104 (2011a).
  * Olejarz et al. (2011b) J. Olejarz, P. Krapivsky, and S. Redner, Phys. Rev. E 83, 051104 (2011b).
  * Chakraborty and Das (2016) S. Chakraborty and S. Das, Phys. Rev. E 93, 032139 (2016).
  * Das and Chakraborty (2017) S. Das and S. Chakraborty, Euro. Phys. J. Spec. Top. 226, 765 (2017).
  * Vadakkayil et al. (2019) N. Vadakkayil, S. Chakraborty, and S. Das, J. Chem. Phys. 150, 054702 (2019).
  * Newman and Barkema (1999) M. Newman and G. Barkema, _Monte Carlo Methods in Statistical Physics_ (Oxford University Press, Oxford, 1999).
  * Landau and Binder (2014) D. Landau and K. Binder, _A Guide to Monte Carlo Simulations in Statistical Physics_ (Cambridge University Press, Cambridge, 2014).
  * Das and Puri (2002) S. Das and S. Puri, Phys. Rev. E 65, 026141 (2002).
  * Majumder et al. (2018) S. Majumder, S. Das, and W. Janke, Phys. Rev. E 98, 042142 (2018).
  * Majumder and Das (2010) S. Majumder and S. Das, Phys. Rev. E 81, 050102 (2010).
  * Majumder and Das (2011) S. Majumder and S. Das, Phys. Rev. E 84, 021110 (2011).
  * Das et al. (2012) S. Das, S. Roy, S. Majumder, and S. Ahmad, Europhys. Lett. 97, 66006 (2012).
  * Janke et al. (2021) W. Janke, S. Majumder, and S. Das, J. Phys.: Conf. Ser. 2122, 012009 (2021).
  * Petri (2003) A. Petri, Braz. J. Phys. 33, 521 (2003).
  * de Oliveira et al. (2004) M. de Oliveira, A. Petri, and T. Tomé, Physica A 342, 97 (2004).
  * De Oliveira et al. (2004) M. De Oliveira, A. Petri, and T. Tomé, Europhys. Lett. 65, 20 (2004).
  * Lipowski (1999) A. Lipowski, Physica A 268, 6 (1999).
  * Christiansen et al. (2019) H. Christiansen, S. Majumder, and W. Janke, Phys. Rev. E 99, 011301 (2019).
  * Janke et al. (2019) W. Janke, H. Christiansen, and S. Majumder, J. Phys. : Conf. Ser. 1163, 012002 (2019).
  * Christiansen et al. (2021) H. Christiansen, S. Majumder, and W. Janke, Phys. Rev. E 103, 052122 (2021).
  * Bortz et al. (1975) A. Bortz, M. Kalos, and J. Lebowitz, J. Comput. Phys. 17, 10 (1975).