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Stochastic growth of a stable phase from a metastable phase

When a stable phase is adjacent to a metastable phase with a planar interface, the stable phase grows. We propose a stochastic model describing the phase growth accompanying latent heat. The model is based on an energy-conserving Potts model with a kinetic energy term defined on a sparsely random lattice only in one direction. For this model, we calculate the stable and metastable phases exactly using statistical mechanics. Performing numerical simulations, we find that the growth law is qualitatively different from that of the deterministic phase-field model.


I. INTRODUCTION
When a stable phase contacts a metastable phase, the stable phase grows, and eventually, the metastable phase vanishes. This phenomenon is ubiquitously observed in nature [1]. The basic understanding of phase growth is that the propagation velocity of the interface between the two phases is proportional to the difference in free energy densities, where the proportional constant is the mobility [2]. There are cases where propagating interfaces exhibit instabilities leading to a wide variety of patterns [3][4][5]. Even without the instabilities of interfaces, the phenomenon could be qualitatively affected in realistic situations. For example, latent heat is generated at the interface in growth processes, which diffuses into the bulk regions. The effect may modify the law of the propagation of interfaces.
Among several models describing phase growth, the phase-field model may be the most standard description that takes the effect of latent heat into account [6][7][8][9][10][11]. The model is given as a set of coupled partial differential equations of the order parameter field and the temperature field. It has been known that the displacement of the planar interface is proportional to the square root of the time interval when the extent of the metastability ∆ is less than unity [12][13][14]. Here, ∆ is defined as where T c is the equilibrium transition temperature, T ms is the temperature of the heat bath in contact with the metastable phase, δs is the entropy jump per unit volume, and c p is the specific heat capacity per unit volume under constant pressure. The √ t-dependence was numerically observed [12] and recently it has been derived through a systematic perturbation method [13]. Now, we address whether the phase-field model is appropriate for describing phase growth. First, we briefly review the fundamental assumptions of the model. The starting point is the entropy functional consisting of the spatial integration of the local entropy density and the gradient term [9,10,15,16], where the entropy density is a function of the internal energy density and the number density. Then, following the Onsager principle, one can determine the evolution equation of these densities so that the irreversible currents are given as linear combinations of thermodynamic forces. The obtained equation is equivalent to the phase-field model [9,10]. This form of the equation was also introduced in the context of the dynamical behavior near the critical point [15,17]. Because such models derived from the Onsager principle are defined in a mesoscopic regime, thermal noises are inevitable in the description, where the noise intensity is determined by the fluctuation-dissipation relation of the second kind. Thus, it is not evident whether noise effects are irrelevant to the phase growth, which is assumed in the phase-field model.
Even worse, a typical interface width is on the order of 10 −7 cm [25]. Thus, precisely speaking, the interface may be out of the mesoscopic description. The interface is interpreted as a unique region where unexpected physical processes occur. In the phase-field model, all such properties are universally represented by only the gradient term. This is another issue concerning the validity of the phase-field model.
Predictions of the phase-field model should be checked by well-controlled experiments or numerical simulations of statistical mechanics models. However, to our knowledge, no experimental report has shown the √ t dependence. Although many statistical mechanics models have been studied in the context of phase growth [18][19][20][21][22][23][24][25][26][27], the generation of latent heat is not taken into account. Considering these situations, this paper proposes a statistical mechanics model describing the phase growth accompanying the latent heat.
The model we study is the q-state Potts model [28] with an additional variable representing the kinetic energy at each site, whose stochastic time evolution satisfies the detailed balance condition at equilibrium [29][30][31]. Thus, in equilibrium statistical mechanics, we can determine the phase diagram at equilibrium. The model exhibits the order-disorder transition as the temperature decreases from a sufficiently high level. Furthermore, owing to the presence of kinetic energy, the model can describe the conversion from potential energy to kinetic energy, which corresponds to the generation of latent heat. We study the model defined on a two-dimensional lattice, where each site is sparse-randomly connected in one direction and local in the other direction. From this setting, we can precisely identify the metastable phase in addition to the equilibrium properties. The proposal of the model is one of the claims in this paper.
By numerically simulating the model, we measure the displacement of the interface R(t). Let T s be the temperature of the heat bath in contact with the stable phase. We find the scaling relation R(t) = L xR (Dt/L 2 x ), where L x is the system size between the two heat baths. The scaling functionR(z) showsR(z) z 0.5 for z z c and R(z) z α for z z c , where the cross-over value z c and the exponent α, 0.5 ≤ α ≤ 1, depend on T s . The scaling relation in the late stage is not observed in the phase-field model. This result indicates that the stochastic phase growth involves a different universality class from that described by the phase-field model. This paper is organized as follows. In Sec. II, we introduce the model. In Sec. III, we analyze the model via equilibrium statistical mechanics. We identify the metastable phase in addition to equilibrium properties. In Sec. IV, we report the results of numerical simulations. In Sec. V, we make some concluding remarks. The technical details of the theoretical calculation are summarized in Appendix A, and the value of ∆ is estimated in Appendix B. Throughout the paper, the Boltzmann constant is set to unity, and β is always connected to the temperature T via β = 1/T .

A. Hamiltonian
be a two-dimensional lattice. For any site i ∈ Λ, a collection of sites connected to site i is denoted as B i . We assume that set B i is decomposed as where Fig. 1a. In this paper, we use a sparse-randomly connected lattice defined by is a one-to-one random map from {1, · · · , L y } to {1, · · · , L y } that satisfies σ + (i x , i y ) = i y , and b − (i x , ) is defined by the inverse of the map. This is a special case of the random graphs introduced in [32]. See Fig. 1b for the illustration.
On each site i ∈ Λ, the q-state variable σ i ∈ {1, 2, · · · , q} and the positive valued kinetic energy p i ∈ R are defined. The collections of variables, [σ i ] i∈Λ and [p i ] i∈Λ , are simply denoted by σ and p. The Hamiltonian we study is where δ(·, ·) represents the Kronecker delta.

Fully Coupled System
(a) Square lattice.

Fully Coupled System
(b) Sparse-randomly connected lattice.

B. Model with energy conservation
Before presenting our model, we first describe a model for a thermally isolated system. We describe the stochastic time evolution in which the stationary distribution is given by the microcanonical ensemble associated with the Hamiltonian H(σ, p). The stochastic process satisfies the detailed balance condition with respect to the uniform distribution on the energy surface H(σ, p) = E, where E is the total energy, which is invariant under time evolution.
We perform the following five procedures at one step: 1. A site i ∈ Λ is chosen randomly with equal probability.
2. The potential energy difference dE is calculated for a transition σ i → σ i chosen randomly with equal probability.
3. The transition σ i → σ i is accepted if p i − dE ≥ 0, and then the kinetic energy is changed to p i = p i − dE.

4.
A site k and another site j ∈ B k are chosen randomly.
5. The transition (p k , p j ) → (p k − dp, p j + dp) is accepted if p k − dp ≥ 0, where dp is a numerical parameter.
Let (p 0 i ) i∈Λ be a collection of the initial value of kinetic energy for each site. From the evolution rule, p i is written The set of all possible values of p i is denoted by P i . The phase space of the model Σ as a function of E is then expressed by the discrete set The transition probability determined by the procedures is expressed as W (σ , p |σ, p). Let t ∈ Z be the discrete time and P t (σ, p) be the probability of taking the state (σ, p) at t-step. Then, we have Because W (σ , p |σ, p) = W (σ, p|σ , p ) holds, the stationary distribution is given by the microcanonical form where |Σ(E)| denotes the number of elements of set Σ(E).

C. Model with two heat baths
In the model, we attach two heat baths, one on the left side i x = 1 and one on the right side i x = L x of the system introduced in the previous subsection. The temperatures of the left and right heat baths are denoted by T L and T R , respectively. The stochastic time evolution of the model is given by imposing an additional rule at the boundaries. We perform the following procedures when a site i with i x = 1 or i x = L x is chosen in procedure 1 of the time evolution described in the previous subsection: 1. The potential energy difference dE is calculated for a transition σ i → σ i chosen randomly.

The transition
3. The value of p i is replaced with a new one sampled from the distribution Note that (7) satisfies the detailed balance condition The stochastic rule in the previous subsection is used except at the boundaries i x = 1 and i x = L x . The transition probability determined by the procedures is expressed asW (σ , p |σ, p). Let t ∈ Z be the discrete time andP t (σ, p) be the probability of taking state (σ, p) at each t-step. Then, we havẽ When T L = T R = T , the detailed balance conditioñ holds. Thus, the stationary distribution is given by the canonical form where Z tot (β) = σ,p e −βH(σ,p) .

III. EQUILIBRIUM STATISTICAL MECHANICS
In this section, for reference in studying dynamical processes, we confirm some results of equilibrium statistical mechanics using the canonical ensemble for the configuration space of σ, where P can (σ) = p P can (σ, p). We study with an infinitely small external potential − i h ex δ(σ i , 1) in the Hamiltonian, and the free energy density defined as It should be noted that the free energy density f and the partition function Z are defined in the configuration space of σ.
In the calculation of m and f , we conjecture that the contribution from loops in the lattice can be ignored in the large-size limit [32][33][34]. On the basis of this conjecture, the thermodynamic phase can be determined using the model on a Cayley tree with three branches corresponding to coordination number 4. Concretely, it has been known that m and f are calculated from the probability of the state σ ∈ {1, · · · , q} at a site connected with a cavity site, which is denoted as u(σ). As shown in Sec. A, we first have the self-consistent equation for u(σ), with γ = e β − 1. Using the solutions of (16), we express the order parameter m(β) and the free energy density f (β) as Here we notice that (16) has the trivial solution u 0 (σ) ≡ 1/q for any β. There exists a temperature β sp beyond which (16) has another solution denoted as u * (σ), where with c * = 1/q. The temperature β sp is called the spinodal point. Using these two solutions u 0 and u * , we have (m 0 (β), m * (β)) and (f 0 (β), f * (β)). We display (m 0 (β), m * (β)) in Fig. 2a, and (f 0 (β), f * (β)) in Fig. 2b. The equilibrium transition temperature β c is identified as f 0 (β c ) = f * (β c ). 4 we have (m 0 (β), m * (β)) and (f 0 (β), f * (β)). We display (m 0 (β), m * (β)) in Fig. 2a, and (f 0 (β), f * (β)) in Fig. 2b. The equilibrium transition temperature β c is identified as f 0 (β c ) = f * (β c ). To study the free energy landscape, we definẽ whereũ Setting c 0 = 1/q, we have f 0 (β) =f (β, c 0 ) and f * (β) = f (β, c * ). Furthermore, by straightforward calculation, we confirm ∂ ∂c e −βf (β,c) = 0 (22) at c = c 0 and c = c * . See Appendix A 4 for the derivation. Therefore, displayingf (β, c) in Fig. 3, we see that the solution u * appears at the spinodal point β = β sp , and the equilibrium transition occurs at β = β c . We also find that ∂ 2f /∂c 2 = 0 at c = c 0 and β = β un , which means that the trivial solution u 0 loses stability. Letting F (u) be the right-hand side of (16), the solutions of the self-consistent equation are given by the fixed points u n+1 = F (u n ). The temperature β un is also characterized by the onset of the instability of the trivial solution for this iteration equation. For the case q = 10, we find that β sp 1.34, β c 1.39, and β un 1.78. FIG. 3. Free energy landscapes for (a) β < βsp, β = βsp, β = βc, and βc < β < βun from below, and (b) βc < β < βun, β = βun, and β > βun from below.

IV. NUMERICAL SIMULATION
Throughout this section, we simulate the model with q = 10 and choose the numerical parameter dp = 0.01T c . Before presenting our results, in Fig. 4, we show the statistical average of i δ(σ i , 1)/|Λ| for the model given in Sec. II, where T L = T R = β −1 and L x = L y = 100. The numerical result agrees with the theoretical calculation in Sec. III. To study the free energy landscape, we definẽ whereũ at c = c 0 and c = c * . See Appendix A 4 for the derivation. Therefore, displayingf (β, c) in Fig. 3, we see that the solution u * appears at the spinodal point β = β sp , and the equilibrium transition occurs at β = β c . We also find that ∂ 2f /∂c 2 = 0 at c = c 0 and β = β un , which means that the trivial solution u 0 loses stability. Letting F (u) be the right-hand side of (16), the solutions of the self-consistent equation are given by the fixed points u n+1 = F (u n ). The temperature β un is also characterized by the onset of the instability of the trivial solution for this iteration equation. For the case q = 10, we find that β sp 1.34, β c 1.39, and β un 1.78. FIG. 3. Free energy landscapes for (a) β < βsp, β = βsp, β = βc, and βc < β < βun from below, and (b) βc < β < βun, β = βun, and β > βun from below.

IV. NUMERICAL SIMULATION
Throughout this section, we simulate the model with q = 10 and choose the numerical parameter dp = 0.01T c . Before presenting our results, in Fig. 4, we show the statistical average of i δ(σ i , 1)/|Λ| for the model given in Sec. II, where T L = T R = β −1 and L x = L y = 100. The numerical result agrees with the theoretical calculation in Sec. III. Then, we prepare a metastable ordered phase of the temperature Here, we fix T L = 1.01T c so that the ordered phase is metastable, and T R is assumed to be greater than T c so that the disordered phase is stable. We prepare configurations at t = 0 via the following steps. First, we remove the coupling between Second, we prepare a configuration where σ i = 1 and p i = T L for any i satisfying i x ≤ L x /2, while each σ i is randomly chosen with equal probability and p i = T R for any i satisfying i x > L x /2 + 1. Third, we make the system evolve up to 50 MCS. The reached configuration is set to be an initial state at t = 0 in the following analysis. Note that s MCS corresponds to the t = sL x L y step in the procedure described in Sec. II.
The stable disordered phase grows from the initial configuration. To describe the phase growth, we observe an order parameter profile defined as Fig. 5a. We identify the interface position does not move anymore, all the curves become flat in the long-time limit. The guideline represents R(t) = At 0.5 with A 0.18. We also confirmed that the L y dependence is hardly visible for this system size. From these results, we reasonably conjecture that the behavior in the large-L x limit is described by the phase-field model because the extent of the metastability ∆ is evaluated as 0.05, as shown in Appendix B. Then, we prepare a metastable ordered phase of the temperature Here, we fix T L = 1.01T c so that the ordered phase is metastable, and T R is assumed to be greater than T c so that the disordered phase is stable. We prepare configurations at t = 0 via the following steps. First, we remove the coupling between Second, we prepare a configuration where σ i = 1 and p i = T L for any i satisfying i x ≤ L x /2, while each σ i is randomly chosen with equal probability and p i = T R for any i satisfying i x > L x /2 + 1. Third, we make the system evolve up to 50 MCS. The reached configuration is set to be an initial state at t = 0 in the following analysis. Note that s MCS corresponds to the t = sL x L y step in the procedure described in Sec. II.
The stable disordered phase grows from the initial configuration. To describe the phase growth, we observe an order parameter profile defined as We define the value φ(x, t) for any real x via linear interpolation. An example of φ(x, t) is displayed in Fig. 5a. We identify the interface position R(t) from φ(x, t) such that φ(R(t), t) = 0.5. R(t) is uniquely determined because φ crosses 0.5 only once as far as we observe. In Fig. 5b, we show an example of configuration σ with the interface position. In Fig. 6, we show the statistical average of R(t) for the case T R = 3T c > T sp . The five graphs correspond to results for different systems sizes (L x , L y ) = (100, 100), Then, we prepare a metastable ordered phase of the temperature Here, we fix T L = 1.01T c so that the ordered phase is metastable, and T R is assumed to be greater than T c so that the disordered phase is stable. We prepare configurations at t = 0 via the following steps. First, we remove the coupling between i Second, we prepare a configuration where σ i = 1 and p i = T L for any i satisfying i x ≤ L x /2, while each σ i is randomly chosen with equal probability and p i = T R for any i satisfying i x > L x /2 + 1. Third, we make the system evolve up to 50 MCS. The reached configuration is set to be an initial state at t = 0 in the following analysis. Note that s MCS corresponds to the t = sL x L y step in the procedure described in Sec. II.
The stable disordered phase grows from the initial configuration. To describe the phase growth, we observe an order parameter profile defined as for 1 ≤ x ≤ L x . We define the value φ(x, t) for any real x via linear interpolation. An example of φ(x, t) is displayed in Fig. 5a. We identify the interface position R(t) from φ(x, t) such that φ(R(t), t) = 0.5. R(t) is uniquely determined because φ crosses 0.5 only once as far as we observe. In Fig. 5b, we show an example of configuration σ with the interface position. In Fig. 6, we show the statistical average of R(t) for the case T R = 3T c > T sp . The five graphs correspond to results for different systems sizes (L x , L y ) = (100, 100), does not move anymore, all the curves become flat in the long-time limit. The guideline represents R(t) = At 0.5 with A 0.18. We also confirmed that the L y dependence is hardly visible for this system size. From these results, we reasonably conjecture that the behavior in the large-L x limit is described by the phase-field model because the extent of the metastability ∆ is evaluated as 0.05, as shown in Appendix B. Then, we prepare a metastable ordered phase of the temperature T L in 1 ≤ i x ≤ L x /2 and a stable disordered phase of the temperature Here, we fix T L = 1.01T c so that the ordered phase is metastable, and T R is assumed to be greater than T c so that the disordered phase is stable. We prepare configurations at t = 0 via the following steps. First, we remove the coupling between i x = L x /2 and i x = L x /2 + 1 by setting B + i = ∅ for i x = L x /2. Second, we prepare a configuration where σ i = 1 and p i = T L for any i satisfying i x ≤ L x /2, while each σ i is randomly chosen with equal probability and p i = T R for any i satisfying i x > L x /2 + 1. Third, we make the system evolve up to 50 MCS. The reached configuration is set to be an initial state at t = 0 in the following analysis. Note that s MCS corresponds to the t = sL x L y step in the procedure described in Sec. II.
The stable disordered phase grows from the initial configuration. To describe the phase growth, we observe an order parameter profile defined as for 1 ≤ x ≤ L x . We define the value φ(x, t) for any real x via linear interpolation. An example of φ(x, t) is displayed in Fig. 5a. We identify the interface position R(t) from φ(x, t) such that φ(R(t), t) = 0.5. R(t) is uniquely determined because φ crosses 0.5 only once as far as we observe. In Fig. 5b, we show an example of configuration σ with the interface position. In Fig. 6, we show the statistical average of R(t) for the case T R = 3T c > T sp . The five graphs correspond to results for different systems sizes (L x , L y ) = (100, 100), does not move anymore, all the curves become flat in the long-time limit. The guideline represents R(t) = At 0.5 with A 0.18. We also confirmed that the L y dependence is hardly visible for this system size. From these results, we reasonably conjecture that the behavior in the large-L x limit is described by the phase-field model because the extent of the metastability ∆ is evaluated as 0.05, as shown in Appendix B. does not move anymore, all the curves become flat in the long-time limit. The guideline represents R(t) = At 0.5 with A 0.18. We also confirmed that the L y dependence is hardly visible for this system size. From these results, we reasonably conjecture that the behavior in the large-L x limit is described by the phase-field model because the extent of the metastability ∆ is evaluated as 0.05, as shown in Appendix B. Then, we prepare a metastable ordered phase of the temperature T L in 1 ≤ i x ≤ L x /2 and a stable disordered phase of the temperature Here, we fix T L = 1.01T c so that the ordered phase is metastable, and T R is assumed to be greater than T c so that the disordered phase is stable. We prepare configurations at t = 0 via the following steps. First, we remove the coupling between i x = L x /2 and i x = L x /2 + 1 by setting B + i = ∅ for i x = L x /2. Second, we prepare a configuration where σ i = 1 and p i = T L for any i satisfying i x ≤ L x /2, while each σ i is randomly chosen with equal probability and p i = T R for any i satisfying i x > L x /2 + 1. Third, we make the system evolve up to 50 MCS. The reached configuration is set to be an initial state at t = 0 in the following analysis. Note that s MCS corresponds to the t = sL x L y step in the procedure described in Sec. II.
The stable disordered phase grows from the initial configuration. To describe the phase growth, we observe an order parameter profile defined as for 1 ≤ x ≤ L x . We define the value φ(x, t) for any real x via linear interpolation. An example of φ(x, t) is displayed in Fig. 5a. We identify the interface position R(t) from φ(x, t) such that φ(R(t), t) = 0.5. R(t) is uniquely determined because φ crosses 0.5 only once as far as we observe. In Fig. 5b, we show an example of configuration σ with the interface position. In Fig. 6, we show the statistical average of R(t) for the case T R = 3T c > T sp . The five graphs correspond to results for different systems sizes (L x , L y ) = (100, 100), does not move anymore, all the curves become flat in the long-time limit. The guideline represents R(t) = At 0.5 with A 0.18. We also confirmed that the L y dependence is hardly visible for this system size. From these results, we reasonably conjecture that the behavior in the large-L x limit is described by the phase-field model because the extent of the metastability ∆ is evaluated as 0.05, as shown in Appendix B. An unexpected behavior of R(t) is observed for the case T R = 1.2T c > T sp .
As shown in Fig. 7a, four graphs for different system sizes, (L x , L y ) = (100, 100), (200, 100), (300, 100), and (400, 100) do not overlap, while the L y dependence is hardly visible. We then consider a finite size scaling to plot R/L x as a function of t/L 2 x . The result is displayed in Fig. 7b. We find the scaling relation works well, whereR is a scaling function. The scaling function indicates the cross-over fromR(z) z 0.5 tō R(z) z α with 0.5 ≤ α ≤ 1. It should be noted that the scaling relation in the late stage is not observed in the phase-field model. 6 An unexpected behavior of R(t) is observed for the case T R = 1.2T c > T sp .
As shown in Fig. 7a, four graphs for different system sizes, (L x , L y ) = (100, 100), (200, 100), (300, 100), and (400, 100) do not overlap, while the L y dependence is hardly visible. We then consider a finite size scaling to plot R/L x as a function of t/L 2 x . The result is displayed in Fig. 7b. We find the scaling relation works well, whereR is a scaling function. The scaling function indicates the cross-over fromR(z) z 0.5 tō R(z) z α with 0.5 ≤ α ≤ 1. It should be noted that the scaling relation in the late stage is not observed in the phase-field model. The exponent α does not seem universal but depends on the value of T R . Decreasing T R further, we find that the late stage becomes dominant and α increases. For the case T R = T L = 1.01T c < T sp , α turns out to be close to unity, as shown in Fig. 8.

V. CONCLUDING REMARKS
We have proposed a stochastic model that describes phase growth. We propose a q-state Potts model with an additional variable representing the kinetic energy at each site and define a stochastic process satisfying the detailed balance condition. By designing a sparse-randomly connected lattice in one direction, we have explicitly calculated the thermodynamic properties via equilibrium statistical mechanics. Using this model, we have numerically observed that the interface between the stable and metastable phases moves following the scaling relation (25) with the scaling functionR(z). The scaling function shows a cross-over fromR(z) z 0.5 toR(z) z α , where the cross-over value of z and the exponent α depends on the temperature of the heat bath attached to the stable phase.
These results suggest that the interface motion qualitatively changes when stochastic processes are considered. When the driving force originating from the difference in free energy densities is the main factor in the interface motion, the result is consistent with the solution of the phase-field model. Contrary to this case, when the driving force is weak, the behavior cannot be described by the phase-field model, presumably because of strong fluctuation effects. We do not yet have a theoretical understanding of the observed behavior. Nevertheless, it is reasonable to interpret the observation as renormalization of the interface motion even apart from the critical point. To our knowledge, this aspect has never been studied. Therefore, it is a challenging problem to theoretically derive our observation.
It should be noted that the results are not specific to the model on the tailored lattice. For example, in Figs. 9a, 9b, and 9c, we show the results for the q-state Potts model on the square lattice whose parameter is the same as that of the model on the sparsely random lattice. The observed behavior is qualitatively similar to that reported in the previous section. Therefore, we ex- An unexpected behavior of R(t) is observed for the case T R = 1.2T c > T sp .
As shown in Fig. 7a, four graphs for different system sizes, (L x , L y ) = (100, 100), (200, 100), (300, 100), and (400, 100) do not overlap, while the L y dependence is hardly visible. We then consider a finite size scaling to plot R/L x as a function of t/L 2 x . The result is displayed in Fig. 7b. We find the scaling relation works well, whereR is a scaling function. The scaling function indicates the cross-over fromR(z) z 0.5 tō R(z) z α with 0.5 ≤ α ≤ 1. It should be noted that the scaling relation in the late stage is not observed in the phase-field model. The exponent α does not seem universal but depends on the value of T R . Decreasing T R further, we find that the late stage becomes dominant and α increases. For the case T R = T L = 1.01T c < T sp , α turns out to be close to unity, as shown in Fig. 8.

V. CONCLUDING REMARKS
We have proposed a stochastic model that describes phase growth. We propose a q-state Potts model with an additional variable representing the kinetic energy at each site and define a stochastic process satisfying the detailed balance condition. By designing a sparse-randomly connected lattice in one direction, we have explicitly calculated the thermodynamic properties via equilibrium statistical mechanics. Using this model, we have numerically observed that the interface between the stable and metastable phases moves following the scaling relation (25) with the scaling functionR(z). The scaling function shows a cross-over fromR(z) z 0.5 toR(z) z α , where the cross-over value of z and the exponent α depends on the temperature of the heat bath attached to the stable phase.
These results suggest that the interface motion qualitatively changes when stochastic processes are considered. When the driving force originating from the difference in free energy densities is the main factor in the interface motion, the result is consistent with the solution of the phase-field model. Contrary to this case, when the driving force is weak, the behavior cannot be described by the phase-field model, presumably because of strong fluctuation effects. We do not yet have a theoretical understanding of the observed behavior. Nevertheless, it is reasonable to interpret the observation as renormalization of the interface motion even apart from the critical point. To our knowledge, this aspect has never been studied. Therefore, it is a challenging problem to theoretically derive our observation.
It should be noted that the results are not specific to the model on the tailored lattice. For example, in Figs. 9a, 9b, and 9c, we show the results for the q-state Potts model on the square lattice whose parameter is the same as that of the model on the sparsely random lattice. The observed behavior is qualitatively similar to that reported in the previous section. Therefore, we ex- The exponent α does not seem universal but depends on the value of T R . Decreasing T R further, we find that the late stage becomes dominant and α increases. For the case T R = T L = 1.01T c < T sp , α turns out to be close to unity, as shown in Fig. 8.

6
An unexpected behavior of R(t) is observed for the case T R = 1.2T c > T sp .
As shown in Fig. 7a, four graphs for different system sizes, (L x , L y ) = (100, 100), (200, 100), (300, 100), and (400, 100) do not overlap, while the L y dependence is hardly visible. We then consider a finite size scaling to plot R/L x as a function of t/L 2 x . The result is displayed in Fig. 7b. We find the scaling relation works well, whereR is a scaling function. The scaling function indicates the cross-over fromR(z) z 0.5 tō R(z) z α with 0.5 ≤ α ≤ 1. It should be noted that the scaling relation in the late stage is not observed in the phase-field model. The exponent α does not seem universal but depends on the value of T R . Decreasing T R further, we find that the late stage becomes dominant and α increases. For the case T R = T L = 1.01T c < T sp , α turns out to be close to unity, as shown in Fig. 8.

V. CONCLUDING REMARKS
We have proposed a stochastic model that describes phase growth. We propose a q-state Potts model with an additional variable representing the kinetic energy at each site and define a stochastic process satisfying the detailed balance condition. By designing a sparse-randomly connected lattice in one direction, we have explicitly calculated the thermodynamic properties via equilibrium statistical mechanics. Using this model, we have numerically observed that the interface between the stable and metastable phases moves following the scaling relation (25) with the scaling functionR(z). The scaling function shows a cross-over fromR(z) z 0.5 toR(z) z α , where the cross-over value of z and the exponent α depends on the temperature of the heat bath attached to the stable phase.
These results suggest that the interface motion qualitatively changes when stochastic processes are considered. When the driving force originating from the difference in free energy densities is the main factor in the interface motion, the result is consistent with the solution of the phase-field model. Contrary to this case, when the driving force is weak, the behavior cannot be described by the phase-field model, presumably because of strong fluctuation effects. We do not yet have a theoretical understanding of the observed behavior. Nevertheless, it is reasonable to interpret the observation as renormalization of the interface motion even apart from the critical point. To our knowledge, this aspect has never been studied. Therefore, it is a challenging problem to theoretically derive our observation.
It should be noted that the results are not specific to the model on the tailored lattice. For example, in Figs. 9a, 9b, and 9c, we show the results for the q-state Potts model on the square lattice whose parameter is the same as that of the model on the sparsely random lattice. The observed behavior is qualitatively similar to that reported in the previous section. Therefore, we ex-

V. CONCLUDING REMARKS
We have proposed a stochastic model that describes phase growth. We propose a q-state Potts model with an additional variable representing the kinetic energy at each site and define a stochastic process satisfying the detailed balance condition. By designing a sparse-randomly connected lattice in one direction, we have explicitly calculated the thermodynamic properties via equilibrium statistical mechanics. Using this model, we have numerically observed that the interface between the stable and metastable phases moves following the scaling relation (25) with the scaling functionR(z). The scaling function shows a cross-over fromR(z) z 0.5 toR(z) z α , where the cross-over value of z and the exponent α depends on the temperature of the heat bath attached to the stable phase.
These results suggest that the interface motion qualitatively changes when stochastic processes are considered. When the driving force originating from the difference in free energy densities is the main factor in the interface motion, the result is consistent with the solution of the phase-field model. Contrary to this case, when the driving force is weak, the behavior cannot be described by the phase-field model, presumably because of strong fluctuation effects. We do not yet have a theoretical understanding of the observed behavior. Nevertheless, it is reasonable to interpret the observation as renormalization of the interface motion even apart from the critical point. To our knowledge, this aspect has never been studied. Therefore, it is a challenging problem to theoretically derive our observation.
It should be noted that the results are not specific to the model on the tailored lattice. For example, in Figs. 9a, 9b, and 9c, we show the results for the q-state Potts model on the square lattice whose parameter is the same as that of the model on the sparsely random lattice. The observed behavior is qualitatively similar to that reported in the previous section. Therefore, we expect that our discovery will be observed in experiments. It is interesting to observe such new fluctuation effects in macroscopic dynamics. 7 pect that our discovery will be observed in experiments. It is interesting to observe such new fluctuation effects in macroscopic dynamics. In this section, we derive formulas (16), (17), (18), and (22) in Sec. III.

Derivation of (16)
We study a Cayley tree with a root site connected with four sites in the first generation. Each site in the n-th generation (n ≥ 1) is connected with three sites in the n + 1-th generation. See Fig. 10 for the illustration of the Cayley tree. Let Z be the partition function of the Potts model on the lattice. We consider the partition function of a system in which a root site is replaced by the cavity and the state of a site in the first generation is fixed as σ ∈ {1, · · · , q}, which is denoted byZ 1 (σ ). Z is then the partition function of the model expressed as A graphical representation is displayed in Fig. 11. By setting In this section, we derive formulas (16), (17), (18), and (22) in Sec. III.

Derivation of (16)
We study a Cayley tree with a root site connected with four sites in the first generation. Each site in the n-th generation (n ≥ 1) is connected with three sites in the n + 1-th generation. See Fig. 10 for the illustration of the Cayley tree. Let Z be the partition function of the Potts model on the lattice. We consider the partition function of a system in which a root site is replaced by the cavity and the state of a site in the first generation is fixed as σ ∈ {1, · · · , q}, which is denoted byZ 1 (σ ). Z is then the partition function of the model expressed as A graphical representation is displayed in Fig. 11. By setting (b) 7 pect that our discovery will be observed in experiments. It is interesting to observe such new fluctuation effects in macroscopic dynamics. In this section, we derive formulas (16), (17), (18), and (22) in Sec. III.

Derivation of (16)
We study a Cayley tree with a root site connected with four sites in the first generation. Each site in the n-th generation (n ≥ 1) is connected with three sites in the n + 1-th generation. See Fig. 10 for the illustration of the Cayley tree. Let Z be the partition function of the Potts model on the lattice. We consider the partition function of a system in which a root site is replaced by the cavity and the state of a site in the first generation is fixed as σ ∈ {1, · · · , q}, which is denoted byZ 1 (σ ). Z is then the partition function of the model expressed as A graphical representation is displayed in Fig. 11. By setting In this section, we derive formulas (16), (17), (18), and (22) in Sec. III.
1. Derivation of (16) We study a Cayley tree with a root site connected with four sites in the first generation. Each site in the n-th generation (n ≥ 1) is connected with three sites in the n + 1-th generation. See Fig. 10 for the illustration of the Cayley tree. Let Z be the partition function of the Potts model on the lattice. We consider the partition function of a system in which a root site is replaced by the cavity and the state of a site in the first generation is fixed as σ ∈ {1, · · · , q}, which is denoted byZ 1 (σ ). Z is then the partition function of the model expressed as A graphical representation is displayed in Fig. 11. By setting we rewrite (A1) as DefiningZ n (σ) and G n similarly, we have the iterative equationZ whose graphical representation is shown in Fig. 12. We define u n (σ) as which corresponds to the probability of the state σ of the cavity-connected site in the n-th generation. By substituting (A6) into (A5), we obtain We also have using σ u n (σ) = 1. From (A7) and (A8), we derive the iterative equation for u n (σ), Assuming homogeneity in the equilibrium state, u n (σ) is independent of n in the large-size limit. This provides (16).

Derivation of (17)
The order parameter m for the model is calculated by the expectation value of δ(σ, 1) at the root site. That is, (A11) By replacing u 1 with the solution of the self-consistent equation (16), we obtain (17).
3. Derivation of (18) To derive the free energy density, we use a tactical method manipulating graphs. We first remove one edge connected to the root site. The partition function of this system with σ at the root site and σ at the other site connected by the removed edge isZ 0 (σ)Z 1 (σ ). See Fig. 13. We thus express the partition function Z as Z = σ,σ e βδ(σ,σ )Z 0 (σ)Z 1 (σ ) (A12) where G 0 ≡ σZ 0 (σ) and u 0 (σ) ≡Z 0 (σ)/G 0 . Note that u 0 (σ) also satisfies (A9). Next, we prepare four independent systems. The partition function of the total system is Z 4 . We remove one edge connected to the root site for each graph. Then, we combine four graphs with the root site by adding one site. See Fig. 14. The partition function of this system, Z 0 , is expressed aš Similarly, another Cayley tree is obtained by combining the other graphs with another added site, and the partition function is written aš The free energies of the original system and the new system are −T log Z 4 and −T logŽ 0Ž1 , respectively. The difference in free energy is equal to 2f , where f is the free energy density, because the two systems have the same free energy density in the large-size limit and the new system is the original system with two sites added. That is,  By replacing u 0 and u 1 with the solution of the selfconsistent equation (16), we obtain (18).
where P can (σ, p) is given in (12). Using the free energy density f calculated in Sec. III, we express the energy density h as where g is the potential energy density given by g(β) ≡ ∂ ∂β (βf (β)) .
As with the free energy density, g 0 (β) and g * (β) denote the potential energy densities corresponding to the trivial solution u 0 and the nontrivial solution u * of (16), respectively. In Fig. 15a, g 0 (β) and g * (β) are displayed. Then, the latent heat per unit volume T c δs at the equilibrium transition temperature is determined by the entropy jump defined as δs ≡ β c (g 0 (β c ) − g * (β c )).
For the model with q = 10, we obtain T c δs 1.07. Next, we consider the heat capacity per unit volume C expressed as Using similar notations, we obtain C 0 (β) and C * (β) from g 0 (β) and g * (β). These are shown at the bottom of Fig. 15b. For the cases q = 10 and T L = 1.01T c , we obtain C * (T L ) 6.95. Therefore, for the model we numerically study, we have ∆ 0.05, which is less than unity. Note that in the stochastic model studied in this paper, C corresponds to c p in the phase-field model.