## Active model B+

Here we will consider the active model B+, which is a model for phase-separation in active matter (see previous post). We consider a scalar order parameter ϕ(r,t). The free energy is given by:

$F[\phi]=\int dV\left\{ -\frac{A}{2}\phi^{2}+\frac{A}{4}\phi^{4}+\frac{K}{2}|\nabla\phi|^{2}\right\}$

The dynamics for active model B+, is given by (see Tjhung, Nardini, Cates, PRX, (2018)):

$\frac{\partial\phi}{\partial t}=\underbrace{\nabla^{2}}_{(3)}(-A\phi+A\phi^{3}-K\underbrace{\nabla^{2}\phi}_{(2)}+\lambda|\underbrace{\nabla}_{(1)}\phi|^{2})$ $-\underbrace{\nabla\cdot}_{(1)}(\zeta\underbrace{(\nabla^{2}\phi)}_{(2)}\underbrace{\nabla}_{(1)}\phi)+\sqrt{2D}\underbrace{\nabla\cdot}_{(1)}\boldsymbol{\Lambda}$,

where λ and ζ are the activity parameters. (λ = ζ = 0 corresponds to the passive/equilibrium limit.) Λ is Gaussian white noise with zero mean and delta-function correlation:

$\left<\Lambda_\alpha(\mathbf{r},t)\Lambda_\beta(\mathbf{r}',t')\right>=\delta_{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')$.

Numerically, we have to discretize the Laplacian and gradient operator in the ϕ-dynamics. First, for derivatives in (1), we must use higher order derivatives. This is because the noise is of order $\sqrt{\Delta t/\Delta x\Delta y}$ so we need to be accurate in the derivatives:

$\partial_{x}\phi=\frac{\frac{1}{280}\phi_{i-4,j}-\frac{4}{105}\phi_{i-3,j}+\frac{1}{5}\phi_{i-2,j}-\frac{4}{5}\phi_{i-1,j}+\frac{4}{5}\phi_{i+1,j}-\frac{1}{5}\phi_{i+2,j}+\frac{4}{105}\phi_{i+3,j}-\frac{1}{280}\phi_{i+4,j}}{\Delta x}$.

For Laplacian in (2), we use the isotropic form of the numerical Laplacian because we nucleate small bubbles, which is bad for 2ϕ (see Pooley, Furtado, PRE, (2007)):

$\nabla^{2}\phi=\frac{1}{\Delta x\Delta y}\left[\begin{array}{ccc} -\frac{1}{2} & 2 & -\frac{1}{2}\\ 2 & -6 & 2\\ -\frac{1}{2} & 2 & -\frac{1}{2} \end{array}\right]\phi_{ij}$.

For derivative in (3), we apply (1) twice to ensure detailed balance exactly on the lattice.