I am working on the hamiltonian of a system related to the extension of the Potts Model which is Cellular Potts Model. The total hamiltonian of the system is: $$ H = H_1 + H_2 $$
$$ H_1 = - J \sum_{\text{i,j neighbours}} (\tau(\sigma_i),\tau(\sigma_i))(1-\delta(\sigma_i,\sigma_j))+\lambda \sum_{\sigma_i} \big(v(\sigma_i)-V(\sigma_i)\big)^2 $$
$$ H_2 = - \mu(C_i - C_j) $$ where i, j are lattice sites, $σ_i$ is the cell at site i, $\tau(\sigma)$ is the cell type of cell $\sigma$, J is the coefficient determining the adhesion between two cells of types $\tau(\sigma)$, $\tau(\sigma')$, $\delta$ is the Kronecker delta, $v(\sigma)$ is the volume of cell $\sigma$, $V(\sigma)$ is the target volume, $\lambda$ is a Lagrange multiplier determining the strength of the volume constraint, $\mu$ is the strength of chemotactic movement and $C_{i}$ and $C_{j}$ are the concentration of the chemokine at site i and j, respectively (Wikipedia).
From $H_2$ above, C is defined by:
$$ \dfrac{\partial C}{\partial t} = \alpha \nabla^{2}C - \beta C + S_{i} $$ $\alpha$ is the diffusion constant, $\beta$ is decay constant and $S_{i}$ is the Secretion rate.
I have 3 layers of cells in 3D, let's say cell A, cell B and cell C placed on top themselves. I am applying a secretion field on cell C which will diffuse at every Monte Carlo Step (MCS). If I take any value, say 30 to be my secretion rate, $S_{i}$, I would like to get an insight on how I can intuitively think about what happens to my cell and how correct parameters should be selected for the other constants in the diffusion field and also $\mu$ in $H_2$ such that the total hamiltonian of the system is still minimized. Or is there a way to calculate this using forward time-centered space (FTCS) scheme to calculate the values/ other parameters.