## Abstract

In standard descriptions, the master equation can be obtained by coarse-graining with the application of the hypothesis of full local thermalization that is equivalent to the local thermodynamic equilibrium. By contrast, fast transformations proceed in the absence of local equilibrium and the master equation must be obtained with the absence of thermalization. In the present work, a non-Markovian master equation leading, in specific cases of relaxation to local thermodynamic equilibrium, to hyperbolic evolution equations for a binary alloy, is derived for a system with two order parameters. One of them is a conserved order parameter related to the atomistic composition, and the other one is a non-conserved order parameter, which is related to phase field. A microscopic basis for phenomenological phase-field models of fast phase transitions, when the transition is so fast that there is not sufficient time to achieve local thermalization between two successive elementary processes in the system, is provided. In a particular case, when the relaxation to local thermalization proceeds by the exponential law, the obtained coarse-grained equations are related to the hyperbolic phase-field model. The solution of the model equations is obtained to demonstrate non-equilibrium phenomenon of solute trapping which appears in rapid growth of dendritic crystals.

This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

## 1. Introduction

Coarse-graining is a fundamental procedure to transfer from a microscopic level of analysis to a mesoscopic level of description. Evolution of many systems and transformations between various phase states is governed by the driving force derived from a coarse-grained free energy functional of the Ginzburg–Landau type [1–8]. In this way, several useful advances of the coarse-graining procedure were made for the phase-field equations which describe the dynamics of local equilibrium systems [9]. For instance, a complete coarse-grained derivation of a phase-field model for precipitation and phase separation was presented by Bronchart *et al*. [10]. Their derivation leads to a mesoscopic nonlinear Fokker–Planck equation finally equivalent to a Cahn–Hilliard equation with noise and with definite expressions for the mobilities and the moments of the noise. Using the coarse-graining procedure in the phase-field theory allows for the obtention of a mesoscopic free energy to model properties of materials [11].

In the description of strongly non-equilibrium processes, special attention is paid to the analysis of rapid transport processes, wave propagation and kinetic phenomena [12–14]. In particular, the description of fast phase transitions in a macroscopic continuous framework often requires hyperbolic evolution equations taking into account a small but non-vanishing relaxation time related to local internal thermalization of the system [15–18]. This is in contrast with usual parabolic evolution equations, which are valid in the limit of vanishing relaxation time, i.e. when the transitions between coarse-grained states are slow enough so that their characteristic evolution time is much longer than the microscopic relaxation time. Such hyperbolic equations yield results in much better agreement to the observed results (for instance, for the rapid dendritic solidification [19]), than the usual parabolic equations, but often they are phenomenological, based on macroscopic grounds.

The lack of local equilibrium is not the only possible phenomenon leading to hyperbolic evolution equation. For instance, purely dynamical features, relating to the length of the jumps of the particles or to the waiting time between two successive jumps of the particles may also contribute to generalized master equations with memory (see Mendez *et al*. [14] for an overview). Here, instead, we focus our attention on the lack of time for local thermal equilibration. In this case, non-Markovian effects arise even in the case that the length of jumps or the waiting times have a single well-specified value.

In Jou & Galenko [20], we proposed as a microscopic basis for such equations a non-Markovian master equation for the separation process of a binary alloy. In that paper, we considered only a single internal variable, which was considered to be, in one case, the chemical identity of the atoms (A or B) and, in another case, the momentum of the atoms. We illustrated how the usual master equation describing the evolution of the system became a non-Markovian master equation when the microscopic relaxation time was not negligible in comparison with the time between two successive elementary steps of the ongoing separation process at the coarse-grained level.

In the present paper, we generalize the results of Jou & Galenko [20] for two different internal variables, one of them conserved (related to the chemical identity of the atoms) and the other one non-conserved, related to the phase field which is often used as a tool to describe fast phase transitions [15,16], by differentiating between the ordered phase (solid one) and the disordered phase (liquid one, for instance). This brings our analysis closer to the previous works on phase-field models for fast phase transitions, where the coarse-grained description using both the concentration and the phase field are used [21].

## 2. Definition of the system

The binary system being considered in this paper is an alloy composed of *A* and *B* atoms at sites *i*=1,2,…,*N*−1,*N*, each of which can be characterized by two parameters: (i) their chemical identity (*A* or *B*), denoted by *s* (*s*=1 being *A* and *s*=−1 being *B*), and (ii) their order parameter *η* with the meaning of the phase-field variable with the value of *η*=1 in an ordered phase and *η*=0 in a disordered phase. Thus, the order parameters are characterized by
2.1Note that, alternatively, the order parameter *η* in equation (2.1) could indicate the spatial direction of atoms magnetization, in which case *η*=±1 would be preferable to *η*=0;1.

For further definition, the alloy’s system is considered as a discrete medium consisting of atoms every one of which has its *i*th site. Each site may be in one of the four different states, {*A*_{1},*A*_{0},*B*_{1},*B*_{0}}, with the probability *p*_{i}(*s*,*η*) of the mentioned state defined by parameters *s*=(*A*,*B*) and *η*=(1,0). If both features are independent of each other we will have *p*_{i}(*s*,*η*)=*p*_{si}*p*_{ηi} with *p*_{si} the probability of finding atom *A* in the *i*th site (and correspondingly 1−*p*_{si} the probability of finding an atom *B*) and *p*_{ηi} the probability of finding *η*=1 at site *i* (and 1−*p*_{ηi} the probability of finding *η*=0 at the site *i*).

An instantaneous configuration of the alloy at an atomistic level is given by , with *p*_{i} the above-mentioned probability. The configuration of the alloy evolves by the following master equation for the probability distribution of finding configuration at time *t*:
2.2Here, is the configuration identical to except that atoms at sites at *i* and *j* have been mutually exchanged (but not their respective spins) and atom at site *k* has changed the *η*-parameter from state 1 to state 0 or *vice versa*. Figure 1 shows configurations of an alloy’s system, for example, in which the order parameter changes from *η*=0 to *η*=1 according to definition (2.1). The notation *k*=0 indicates that *η* does not change in any site, i.e. means that the particles have been mutually changed only at *i* and *j* but label *η* remains the same as before in any site. stands for the transition rate from configuration to and *vice versa* at the transition rate .

In the case when labels *s*_{i}(*A* or *B*) or labels *η*_{i}(1 or 0) may be considered as independent of each other, equation (2.2) includes the evolution equations for concentrations *s*_{i} (*i*=1,…,*N*) or for configurations *η*_{i}. Thus, we may concentrate our attention independently on each one. Indeed, in Jou & Galenko [20], we considered the evolution of configurations *s*_{i}, and we did not consider *η*_{i} at all. Now, we can decouple master equation (2.2) for concentration and the alloy’s state and analyse them independently. In more general situations, the dynamics of *s*_{i} may depend on the state of *η*_{i} and the description will be more cumbersome, leading to coupled equations for the evolution of *s*_{i} and *c*_{i}.

### (a) Conserved order parameter

We consider the concentration as conserved order parameter. Then, equation (2.2) describes mixtures composed of *A* and *B* atoms which will be called for one-symmetry phases [5,10] by taking a constant value for *η* according to equation (2.1) at every site. In this case, equation (2.2) reduces to [5,10]
2.3Here, is the probability distribution corresponding to the configuration of *s*_{i}, denoted as , is the transition rate to change configuration to , where is a configuration identical to except that atoms at sites *i* and *j* have been mutually exchanged. The stars in the sums mean that equation (2.3) is restricted to first-neighbour exchanges.

If the system has time enough to thermalize in the time between two consecutive changes of the mentioned state, the energy probability distribution *f*_{eq} at sites *i* and *j* is defined by the canonical distribution, namely
2.4where *β*=(*k*_{B}*T*)^{−1} Boltzmann factor, with the temperature *T* and the Boltzmann constant *k*_{B}, is the interaction energy between site *i* and the rest of the system in configuration when site *i* is occupied by an atom *A*. Obviously, an analogous definition is true for . The hypothesis about full thermalization has been used in Bronchart *et al*. [10] to set the expression for the transition rates as
2.5with , *θ* being a characteristic attempt frequency, and *E*_{s} the energy of the barrier that particles must surpass for the jump exchange of atoms between sites *i* and *j* to take place, and *δ*(*p*_{i}) are the Kronecker deltas related to the presence of *A* in *i* and *B* in *j*. Equation (2.5) has been used to describe precipitation in a binary mixture, where the process from a well-mixed initial state to a two-phase final state with components *A* and *B* forming two different regions takes place [10].

### (b) Non-conserved order parameter

For the non-conserved order parameter *η*, equation (2.2) describes inhomogeneous phase state or different spin’s direction in a chemically one-component system [22]. The *η*-configuration is labelled by and, the general equation (2.2) can be decomposed such that the corresponding master equation
2.6describes the evolution of the probability distribution with the rate characterizing the transition in phase state parameter, i.e. from 0 to 1 (or from 1 to 0) at site *i* by definition (2.1). If the time between two consecutive changes of *η* is long enough for local thermalization, the transition rates have the form [10]
2.7with and *θ* being a characteristic attempt frequency, *E*_{b} is the energy barrier which must be surpassed for the phase change from *η*_{i}=0 to *η*_{i}=1 (or *vice versa*), and is the equilibrium probability distribution function between the states *η*_{i}=1,0.

In what follows, we concentrate on advancing the Master equation (2.6) for fast phase transformation and on coarse-grained procedure for the single-order parameter *η*.

## 3. Non-Markovian master equations

In usual circumstances of transitive processes, one can assume conditions of local energy thermalization. This means that the time scale between successive measurements of the elementary state, let us say time *t*_{0}, is long enough for thermalization of the system after the exchange of atom *A* at site *i* and atom *B* at site *j*, or after the change of *η* from 0-state to 1-state at site *k*. In this case, master equations (2.3) and (2.6) describe instant exchanges of atoms and of *η*-states, respectively.

In many specific situations and with modern experimental techniques, one can reach very short time periods for transitive processes. For example, in rapid dendritic crystallization of binary alloys [19] one can reach a growth velocity up to 50 (m s^{−1}) for the undercooling up to 350 (K). To estimate the characteristic temporal and spatial scales, we assume the diffusion coefficient *D*≈10^{−9}(m^{2} s^{−1}) at the dendrite growth velocity *V* ≈10(m s^{−1}). In this case, a thickness of solute diffusion boundary layer ahead of the solid–liquid interface is estimated as ℓ_{D}=*D*/*V* ≈10^{−10}(m), which has nano-length scale. The time for crystallizing of local volume by diffusion mechanism has the order of *t*_{S}=*D*/*V* ^{2}≈10^{−11}(s), which is comparable with the time for atomic jumps by diffusion mechanism. This particular example shows that solidification may proceed so fast that atoms may have no time to exchange their positions and conditions of local thermalization may not exist. To generalize this situation to other processes with high driving forces [19], one must assume that the phase transformation proceeds very rapidly compared to motion of local chemical or structural disturbance, which does not have enough time to relax to its local equilibrium. In these specific cases, change of microstates should be described by a non-Markovian master equation taking into account this lack of thermalization.

### (a) Change of the phase state

In Jou & Galenko [20] we considered how equation (2.3) should be modified when the time scale between successive measurements of the atomic state, *t*_{0}, is not long enough for internal local thermalization of the system after the atoms are changed. This allowed us to generalize the idea about coarse-grained derivation presented in Bronchart *et al*. [10] to fast phase transitions [15]. The main feature characterizing a fast transition is that time *t*_{0} elapsed between successive elementary steps of the physical process is not long enough to allow for a true thermalization of the system. Therefore, we considered the lack of energy thermalization for the elementary steps characterizing particle exchanges between sites *i* and *j*. Here, we consider the analogous problem with the change of *η*_{i} (for instance, to change the phase state from 0 to 1 or from 1 to 0).

Consider that the thermalization of *η*_{i} takes some finite time *τ*_{ηi} which in the simplest relaxation time approximation may be described as
3.1*f*_{ηi} being the probability distribution of *η*_{i} between its two states 0 and 1. Thus, one has two microscopic characteristic times: the time *t*_{0} between successive changes of *η* and the local thermalization time *τ*_{ηi}. A slow transition occurs for *t*_{0}≫*τ*_{ηi} and fast transitions correspond to *t*_{0}≈*τ*_{ηi} or *t*_{0}<*τ*_{ηi}. For the diffusion process this situation is considered in details in Jou & Galenko [23].

If *t*_{0} is not much longer than *τ*_{ηi}, the factor *f*^{(eq)}_{ηi}(*t*) in equation (2.7) for the transition rate should be replaced by the factor (corresponding to the solution of equation (4.7))
3.2i.e. the process becomes non-Markovian as the transition probabilities at *t* do not depend only on *t* but also on *t*−*t*_{0}. This will imply that the transition rate will become dependent also on the time interval *t*_{0}, i.e. . If one considers not only a single time interval *t*_{0} but a number of time intervals, *t*′=*nt*_{0}, one should write the transition rate in more general terms. Formally, equation (2.6) should then be replaced by
3.3In equations (3.3), we have taken the integration limits from 0 to *t*, because we assume that the process begins at *t*=0. An alternative extreme possibility which is also often adopted would have been to assume that the process began at . Since, in fact, the relaxation times are relatively short compared to macroscopic observation times, these different expressions are essentially equivalent.

The corresponding transition kernels and may have different forms, depending on the various pathways for relaxation to local thermodynamic equilibrium [16,20]. The simplest exponential form in *t*−*t*′,
leads from the master equation (3.3) to the second-order non-Markovian master equation
3.4Equation (3.4) is the hyperbolic master equation describing change of phase state with exponential relaxation to local thermalization. In the case that equation (3.2) should be generalized taking into account not only *f*^{(eq)}_{ηi}(*t*) and *f*^{(eq)}_{ηi}(*t*−*t*_{0}) but also *f*^{(eq)}_{ηi}(*t*−2*t*_{0}), for instance, an evolution equation of third order in time derivatives would be needed instead of equation (3.4), and so on.

### (b) Evolution of the alloy’s configuration

Analogously to non-conserved order parameter, §3a, equation (2.3) can be transformed to the equation for the evolution of the alloy’s configuration as
3.5The non-Markovian master equation (3.5) is analogous to well-known equations with memory as, for instance, the Mori functional equation (see Mori [24,25] and ch. 7 in Jou *et al*. [12]). This equation is known as the ‘generalized master equation’ in the context of continuous-time random walks and it is discussed in length in the book by Mendez *et al.* [14].

The memory kernels and in equation (3.5) govern the current evolution of the alloy’s configuration through its past relaxation of local states. When the kernels have a simplest exponential form,
equation (3.5) becomes
3.6where *τ*_{D} is the relaxation time for the rate of probability distribution change, , and (and ) is the transition rate from configurations to (and *vice* *versa*) at a final moment *t*=*t*′ of relaxation to locally thermalized state (with a configuration identical to , except that atoms at sites *i* and *j* have been mutually exchanged).

Equation (3.6) presents the hyperbolic master equation describing the evolution of the alloy’s configuration with exponential relaxation to local thermalization. This equation can be derived using a continuous-time random walk framework [14] and it has been investigated in Jou & Galenko [20], where the coarse-grained partial differential equation
3.7was obtained^{1} for the solute concentration *c* in a binary alloy under the stochastic influence of noise described by the function *ξ**_{n}(*t*). With the mobility *M*_{c} of solutes and the chemical potential
3.8the hyperbolic equation (3.7) can model the phase segregation by the spinodal mechanism (*ϵ*_{cn}(*T*,*c*_{n};*d*)∇^{2}*c*≠0) or by usual diffusion (*ϵ*_{cn}(*T*,*c*_{n};*d*)∇^{2}*c*=0), where *ϵ*_{cn}(*T*,*c*_{n};*d*) is the gradient factor which, in general, may depend on the size *d* of the coarse-graining cell (see discussions about such dependence in [10,20] and in §6).

## 4. Coarse-graining in the dynamics of non-conserved order parameter

We now derive the phase-field models for the system described by equation (3.3). In particular, we analyse the dynamics described by the equation known as the time-dependent Ginzburg–Landau equation [26–28] or, in particular, Allen–Cahn equation [29]) with a single non-conserved order parameter. Such dynamics is known as ‘the Model-*A*’ by classification of Halperin *et al.* [30].

Let us split the system into cells of linear size *d* and assume that *N*_{d}=(*d*/*a*)^{3} is the number of sites in each cell (in the three-dimensional space), where *a* is the characteristic distance between microscopic sites (figure 2). Further, we define mesoscopic configuration at cell *n* by *c*_{n} and defined as
4.1The coarse-grained description of the conserved order parameter was described in [20]. Here, we focus our attention on the evolution of describing the coarse-grained state of the cell of the system with reference to the non-conserved variable *η*.

The relation between the probability distribution function of the mesoscopic coarse-grained configuration and that of the microscopic configuration is given by
where the sum refers to all the particles *i* belonging to the coarse-grained box *n*. Thus, the evolution equation for may be obtained from that for (see equation (2.6) or, in more concrete terms, equation (3.3)) as
4.2where the sum on *m* refers to all the coarse-grained boxes and the coarse-grained transition rates are given by
4.3with 〈⋯ 〉 standing for the equilibrium average.

Because of (2.6) and taking into account the canonical form for the equilibrium distribution, equation (4.3) may be approximated as
4.4with given by
4.5with 1 and 0 referring to the values *η*_{i}=1 and *η*_{i}=0, respectively (recall that refers to the Dirac delta-function). In equation (4.4), the dependence in *t*−*t*′ comes from the fact that the average is not done over but over an expression like equation (3.2), with *t*−*t*_{0} replaced by *t*′.

By following an analogous procedure to that shown in Bronchart *et al*. [10], the linearized evolution equation for the coarse-grained values may be written as^{2}
4.6where is the chemical potential difference within the cell *n* described by
4.7as defined in equation (4.5). The transport kernel generalizing usual transport coefficients is given by
4.8Eventually, the noise term *ξ*_{n}(*t*) in equation (4.6) is described by
4.9which exhibits memory effects (see explanations in Jou & Galenko [20] and references therein). Equation (4.6) is in fact a generalized Langevin equation with coloured noise [14]. As said in the third paragraph of the present introductory section, equations of this kind may arise from several effects, but here we have considered that the dominant effect is the lack of internal local equilibrium.

In the particular case of an exponential memory kernel [16,20],
(with the relaxation time *τ*_{η} of *η*), we may rewrite equation (4.6) together with equation (4.9) as
4.10where the noise term is characterized by
4.11In the case that interactions between *η*-variable of different cells are also considered, the term would be generalized to , with equation (4.8) incorporating the values of and , respectively.

Equation (4.10) describes the evolution of the coarse-grained order parameter which is non-conserved and characterizes the dynamics of spin orientation or *η* the phase change. Introducing the memory effects and taking into account exponential relaxation function (4.11), the dynamics is described by the hyperbolic equation (4.10), which includes inertia by the term ∝∂^{2}*η*_{n}(*t*)/∂*t*^{2}. This, in particular, describes finite rotational speed and time for spins orientation change from +1 to −1 or *vice versa* that is the result of introducing rotational kinetic energy into energetic equation for magnets dipole dynamics [31].

An explicit form of the chemical potential from equation (4.10) can be written with the definition of the free energy. Following previous studies [5,10,32], we consider the free energy in the form of the discrete Ginzburg–Landau functional
4.12which is true for spatially inhomogeneous system having gradient pre-factor *ϵ*_{ηn}. In contrast to Jou & Galenko [20], where *p*_{i} was referred only to the probability to find atom *A* in site *i*, we assume in equation (4.12) that *p*_{i}(*s*,*η*) refers to the probability to find *A* with *η*=1 in site *i*, i.e. it takes into account the two variables *s* and *η* defined by equation (2.1).

The expression for the effective macroscopic (*d*-independent, see §6) free energy will depend on the system and the situation at hand. In particular, one could use an effective free energy density incorporating the neighbour interactions
4.13which might be reduced to the particular form
4.14where *z* is the coordination number of the lattice, and summation over *m* means a sum over nearest-neighbour sites (*m*=0,1,2 or, alternatively, *m*=−1,0,+1). Equation (4.14) is suitable in the limit *d*→*a*, i.e. when *d* has the minimum possible value, which is not strictly zero but is equal to the distance *a* between atomic sites, and it is well defined, as a given value of *d* is specified, see §6.

The explicit expression for the chemical potential is obtained from the derivative of the free energy (4.12) and (4.14) with respect to the local occupation probability as
4.15where
4.16is the contribution into the chemical potential from the particles (parameters + and −) and their mixing, is the discrete Laplacian for the lattice, and *μ*_{+}=*z*(*ϵ*_{‡}−*ϵ*_{±}) and *μ*_{−}=*z*(*ϵ*_{=}−*ϵ*_{±}) are the chemical potentials of the parameters + and −, respectively.

The continuum limit would correspond to both *d* and *a* going to zero. In this case, the several differences between values of chemical potentials at different neighbouring cells may be written in form of gradients. The physical problem, however, may be found in the *d*-dependence of the free-energy and the chemical potential, see §6. Therefore, we consider the continuous limit *d*→0 and take the local (near-neighbour) approximation, *βl*_{nm}(*μ*_{m}−*μ*_{n}), which tends to
with *M*_{η}(*T*,*η*) the mobility given by *βl*_{n n+1} with *η* the average non-conserved order parameter, i.e. the phase-field variable. As a result, equations (4.10)–(4.16) present the hyperbolic equation for the phase field
4.17with the noise described by the function *ξ**_{n}(*t*), the relaxation time *τ*_{η} of the gradient flow (which is the rate of change of the phase field ∂*η*/∂*t*) and the chemical potential
4.18

## 5. Predictions of the coarse-grained phase-field equations

The coupled system of equations (3.7) and (4.17) predicts phase change and chemical diffusion in a binary system. Within the classification of the work [30], this coupled system of equations is related to ‘Model C’, namely to the system of hyperbolic equations for the conserved order parameter *c* and non-conserved order parameter *η*. For the demonstration of the phase change and chemical diffusion described by the hyperbolic ‘Model C’, we consider the change of concentration with the increase of solidification velocity leading to non-equilibrium solute trapping [33]. This phenomenon has been evidenced experimentally in dendritic solidification of Ni–B alloy melts [34] and it plays one of the main roles in formation of metastable phases [19].

To analyse the solute trapping during rapid solidification, we solve equations (3.7) and (4.17) for rapid solidification of the Si-0.25 at.% As alloy with material parameters given in Galenko *et al*. [35]. The equations were solved with the absence of terms responsible for the noise, *ξ**_{n}(*t*)=0, and for the concentration separation by the spinodal or binodal mechanism, *ϵ*_{cn}(*T*,*c*_{n};*d*)∇^{2}*c*=0. We neglect the role of noise because the main focus of our interest lies in the present work on the effects of memory in the evolution of the average value of the concentration. The free energy in the chemical potentials, equations (3.8) and (4.10), has been taken by the coupling of functions *c* and *η*, i.e. *μ*_{(hom)}(*T*,*c*_{n};*η*_{n};*d*), as described in Echebarria *et al*. [36]. This allowed us to obtain a quasi-stationary regime of the diffuse interface moving with the velocity *V* at a given undercooling Δ*T*=*T*_{m}−*T*, where *T*_{m} is the melting temperature of Si and *T* the actual temperature in the Si-0.25 at.% As alloy’s sample (see details of our analytical treatments, numerical method and computations in Galenko *et al*. [35]).

One-dimensional solution of equations (3.7) and (4.17) for the average value of *c*(*x*) is shown in figure 3 for the range of high solidification velocity, 0.1<*V* (m s^{−1})<2.56. In this range, one can clearly see that the overall concentration *c*(*x*) drastically changes from a widely distributed profile, figure 3*a*, to a smooth and constant distribution in two segregated phases with a small spike within the diffuse interface, figure 3*d*. Note that in the presence of the noise term, *ξ**_{n}(*t*), the solution of equation (4.17) would have a family of fluctuating curves instead of the single smooth curve *c*(*x*) plotted in each panel of figure 3.

If the overall concentration *c*(*x*) can be decomposed into the solid concentration, *c*_{S}(*x*), and liquid concentration, *c*_{L}(*x*) (these are auxiliary concentrations, actually measuring the derivative of the grand potential of each phase at the local chemical potential, see details in [35]), the non-equilibrium segregation coefficient, *k*(*V*), is defined as the ratio between maximum concentrations of the solute As in the solid and liquid phases of the Si–As alloy
5.1With the increase of the velocity *V* the spike of overall concentration gradually decreases and, as figure 3*d* shows, at highest growth velocity one gets , i.e. the liquid and solid concentrations are equal to the nominal concentration with *k*(*V*)=1 by the definition (5.1). Such a regime without chemical segregation is known as the diffusionless (chemically partitionless) regime in which the complete solute trapping by rapid interface occurs. The transition from a chemically partition regime, *k*(*V*)≠1, to the diffusionless regime with the complete solute trapping, *k*(*V*)=1, has been obtained in molecular dynamics simulations of rapidly solidifying binary mixtures [37].

Finally, we have to note again that the solution of equations (3.7) and (4.17) has been obtained without noise contribution. We have demonstrated the transition to complete diffusionless transformation with the complete solute trapping, figure 3, in which the main contribution is obtained from the deterministic part of diffusion. How coloured noise may influence the diffusion and dynamics of an interface is a subject of additional work (see, for example, [38] in which a special study of the influence of coloured noise on spinodal decomposition is given).

## 6. Final remarks and conclusion

In this paper, we have generalized our previous coarse-grained derivation of a non-Markovian master equation for a conserved order parameter [20]. Here, we have incorporated an additional non-conserved order parameter representing the phase (or for the spin). The essential feature leading to memory effects (non-Markovian character) is that the process is fast enough that the system has no time to reach thermodynamic equilibrium in the measurable time interval between two consecutive steps of the coarse-grained dynamics. This lack of a well-defined separation in the microscopic time scale *τ*_{η} and coarse-grained time scale *t*_{0} becomes specially relevant in fast processes, namely when *t*_{0}≫*τ*_{ηi} is not applicable, and requires a modification of the master equation.

A further aspect of interest is how the present procedure depends not only on the time scale but also on the space scale, namely on the size *d* of the coarse-graining boxes. Bronchart *et al.* [10] find that the shape of the chemical potentials *μ*_{n} appearing in the evolution equation of the coarse-grained concentration depends significantly on the size of the coarse-graining cell *d*. Thus, in strict terms, there is not a well-defined univocal continuum limit to the coarse-graining procedure. In equation (4.12) the mesoscopic free energy density *f* and stiffness *ϵ* have been written in terms of *d* to show that they depend on the cell size. For example, the stiffness for the *s*-variable (*A* or *B*) from equation (3.8) is defined by
6.1as the *d*-dependent energetic parameter of the free-energy (4.12) with *ϵ*_{AA}, *ϵ*_{BB} and *ϵ*_{AB} being the interaction energies between pairs *A*−*A*, *B*−*B* and *A*−*B* (or *B*−*A*), respectively. It is assumed here that the interaction between *A* atoms (or *B* atoms) exists also for different cells, making some average on their separations, which change with the cell size *d*. In the case that the chemical potential refers to changes in *η* for a one-component system, an analogous expression for *ϵ* could be used in equation (4.12) with the form
6.2Here, *ϵ*_{‡} and *ϵ*_{=} are referring to the interaction energy of ++ or −− neighbours, respectively, and *ϵ*_{±} refers to the interaction of +− or −+ neighbours. As in equation (6.1), a dependence on *T*, *c*_{n} and *d* may be expected.

It was additionally observed by Bronchart *et al.* [10] that in spite of free energies, mobilities and intensity of noise depend on the coarse-graining size *d*; three simulations carried out by them on the problem of precipitation in binary mixtures (with values *d*/*a*=6,8,10) yield approximately the same qualitative results for the volume fraction of precipitate as a function of time. Thus, the overall method proposed in Bronchart *et al*. [10] leads from the practical point of view to a consistent macroscopic formalism roughly independent on *d*. This way may be understood if the interaction is restricted to microscopic close neighbours, because the interaction neighbouring coarse-grained cells will be restricted to their common boundaries in such a way that its influence will become smaller for higher size of the cell.

Using these aspects of interest and remarks, the main conclusions, finally, can be stated as follows:

— Definition of a sample with local configuration in which a conserved variable (concentration of atoms) is coupling with a non-conserved variable (spin or phase state) has been given. The dynamics of local configurations in the elementary steps characterizing particle exchange between sites has been analysed. The dynamics is Markovian in the case of local energy thermalization that assumes instant exchanges of atoms between states. The dynamics of conserved and non-conserved variables in the situations considered here is not Markovian due to the lack of thermalization during short change of micro-configurations.

— The suggested master equation for the non-Markovian dynamics describes a change of the alloy’s configuration during fast phase transitions. In the limiting cases, i.e. when

*t*_{0}≫*τ*_{ηi}, this master equation describes locally thermalized systems (due to the instant relaxation to local thermalization or to local equilibrium) whereas, when*t*_{0}≈*τ*_{ηi}or*t*_{0}<*τ*_{ηi}, it may follow a hyperbolic dynamics of locally non-equilibrium systems (due to the finite exponential relaxation to local thermalization).— The coarse-graining procedure leads to a coupled system of equations for the non-conserved variable (phase field) and the conserved variable (concentration of atoms). One of the solutions of the model equations allows us to show existence of the non-equilibrium solute trapping which plays one of the main roles in formation of dendritic microstructure and metastable phases of rapidly solidifying alloys. The obtained transition to the complete solute trapping at highest diffuse-interface velocity and to the diffusionless phase transition is in agreement with the results of molecular dynamics simulations of rapidly solidifying binary mixtures.

— Though we have implicitly considered that the relaxation times of the conserved and non-conserved microscopic variables are of the same order, the present method would allow for more general possibilities, namely, a Markovian character for one of variables and non-Markovian for the other one, if their respective relaxation times are sufficiently different.

## Data accessibility

This article has no additional data.

## Authors' contributions

All authors contributed equally to the present article.

## Competing interests

We declare we have no competing interests.

## Funding

D.J. acknowledges support by the Direccion General de Investigación of the Spanish Ministry of Economy and Competitiveness under grant no. TEC 2015-67462 and of the Generalitat of Catalonia under grant no. 2009-SGR-00164. P.K.G. acknowledges the support by the RSF [grant no. 16-11-10095] and from the Administration of the Physical Department during his stay in Universitat Autònoma de Barcelona.

## Footnotes

One contribution of 16 to a theme issue ‘From atomistic interfaces to dendritic patterns’.

↵1 The hyperbolic equation (3.7) is obtained by (i) dividing the system into cells, (ii) expressing the probabilities and rates through concentrations and chemical potentials in cells, respectively, and (c) using the free energy in the form of the discrete Ginzburg–Landau functional, the concrete form for the effective free-energy density and an explicit expression for the chemical potential [20].

↵2 The procedure in Bronchart

*et al*. [10] takes into account dividing the system into cells as well as expressing the probabilities and rates through concentrations and chemical potentials in cells, respectively.

- Accepted September 7, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.