Short description of the model

[for further details see: Erdélyi Z, Pasichnyy M, Bezpalchuk V, Tomán J, B. Gajdics, Gusak A, Stochastic Kinetic Mean Field Model, Computer Physics Communications 204: pp. 31-37. (2016) (available online: http://dx.doi.org/10.1016/j.cpc.2016.03.003)]

Rate of change of concentration in each site $i$ of a three-dimensional grid is defined according to conservation of matter and the corresponding local flux balance at each site:

\begin{equation}
\frac{dc_i}{dt}=-\sum_{j=1}^Z\left[c_i(1-c_j)\left(\Gamma_{i,j}^{mean-field} + \delta\Gamma_{i,j}^{Lang}\right) \left. \\
\right. - c_j(1-c_i)\left(\Gamma_{j,i}^{mean-field} + \delta\Gamma_{j,i}^{Lang}\right)\right]. \label{eq:master}
\end{equation}

where $c_i$ is the atomic fraction of A atoms at site $i$, $c_j$ is the atomic fraction on a neighbouring site $j$, and the total number of nearest neighbours is $Z$. $c_i(1-c_j)$ is in fact the probability that the site $i$ is occupied by an A atom and a neighbouring $j$ site by a B atom; i.e. an A-B exchange is possible. $\Gamma_{i,j}^{mean-field}$ is the probability of such an exchange per unit time in mean-field approximation, i.e. the jump rate of A atoms from site $i$ to a neighbouring site $j$ and backward jumps of B atoms ($\Gamma_{j,i}^{mean-field}$ is for an exchange of an A and a B atoms being on site $j$ and $i$, respectively):

\begin{equation}
\Gamma_{i,j}^{mean-field}= \Gamma_{0} \exp\left(-\frac{\hat{E}_{i,j}}{kT}\right) \label{eq:Gamma_mean_field}
\end{equation}

with

\begin{equation}
\hat{E}_{i,j} = \left(M-V\right) \sum_{l=1}^Z c_l + \left(M+V\right) \sum_{n=1}^Z c_n \label{eq:E_ij}
\end{equation}

where $V_{\alpha,\beta}$ ($\alpha,\beta =$ A, B) are the pair interaction energies, $M= \frac{V_{AA} - V_{BB}}{2}$, $V = V_{AB} - \frac{V_{AA} + V_{BB}}{2}$ and $\Gamma_0 = \nu\exp\left(\frac{-E_0 + Z(V_{AB}+V_{BB})}{kT}\right)$ ($\nu$ is the attempt frequency and $E_0$ is the saddle point energy).

Note that $V$ is the regular solid solution parameter – proportional to the heat of mixing – and $M$ determines the strength of the composition dependence of the tracer diffusion coefficient (diffusion asymmetry). Last but not least, $\delta\Gamma^{Lang}$ in eq. \ref{eq:master} are the noise terms, which are random additions to the mean-field exchange rates:

\begin{equation}
\delta\Gamma_{i,j}^{Lang} = \frac{A_n}{\sqrt{dt}}\sqrt{3}\left(2 random - 1\right) \label{eq:Gamma_Lang}
\end{equation}

where $random$ is a uniform random number between $0$ and $1$. It is easy to check that the random expression $\sqrt{3}\left(2 random - 1\right)$ has the mean squared value equal to $1$. Here $A_n$ is the noise of the amplitude and $dt$ is the time time step.

Actually, eqs. (\ref{eq:master}-\ref{eq:Gamma_Lang}) have to be used to calculate the time evolution of the composition at each site of a 3D lattice. (see open source code: Downloads) With $A_n = 0$, we perform a fully mean-field calculation, whereas with increasing $A_n$ the calculation becomes more and more stochastic, that is the dispersion of composition becomes higher.