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:

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

,

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:

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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 so we need to be accurate in the derivatives:

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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)):

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