Custom flow in overdamped Brownian dynamics

Abstract

When an external field drives a colloidal system out of equilibrium, the ensuing colloidal response can be very complex and obtaining a detailed physical understanding often requires case-by-case considerations. In order to facilitate systematic analysis, here we present a general iterative scheme for the determination of the unique external force field that yields a prescribed inhomogeneous stationary or time-dependent flow in an overdamped Brownian many-body system. The computer simulation method is based on the exact one-body force balance equation and allows to specifically tailor both gradient and rotational velocity contributions, as well as to freely control the one-body density distribution. Hence compressibility of the flow field can be fully adjusted. The practical convergence to a unique external force field demonstrates the existence of a functional map from both velocity and density to external force field, as predicted by the power functional variational framework. In equilibrium, the method allows to find the conservative force field that generates a prescribed target density profile, and hence implements the Mermin-Evans classical density functional map from density distribution to external potential. The conceptual tools developed here enable one to gain detailed physical insight into complex flow behaviour, as we demonstrate in prototypical situations.

Phase coexistence of active Brownian particles

“…with ∂ρ/∂t = 0 in steady state. The internal force density field, as occurring in (3), was proven to be a "kinematic" functional of the density and the current [35,36], i.e. F int (r, ω, t) = F int ([ρ, J, J ω ], r, ω, t).…”

“…Once the ρ n are known, one can (trivially) determine the J x n and J y n via (36) and (37). As an aside, note that the sum rule 2v f ρ 0 = J x 1 + J y 1 , which can be derived from inserting (27) and (28) into (8), is satisfied by (36) and (37). Note also that the coupling of the orientational and the translational motion occurs now (only) via the shifted indices n ± 1 in (36) and (37).…”

“…Some more details, also about power functional theory for active Brownian particles, and more generally orientation-dependent models can be found in [38]; an overview of different methods to sample the one-body current in BD simulations is given in [36].…”

Active interface polarization as a state function

“…All occurring terms in (3) can be sampled in computer simulations; see, e.g., Ref. [22]. We first consider the total polarization for systems with vanishing total flux through the boundaries of volume V at all times t, i.e., ∂V ds · J(r, ω, t) = 0, where ds denotes the vectorial surface element and ∂V indicates the surface of volume V. Here V is arbitrary and can be chosen to be either the system volume, an enclosing larger volume that contains the system, or a subvolume of the system.…”

Superadiabatic demixing in nonequilibrium colloids

“…where the angles denote an average both over the noise as before, but also over the set of initial states; the velocity of particle i is given by a symmetric time derivative, v i (t) = (r i (t + Δt) − r i (t − Δt)) ∕ (2Δt) (see e.g., appendix A of ref. 40 for the derivation). We obtain the microscopically resolved velocity profile of species α as the ratio v α ðr; tÞ ¼ J α ðr; tÞ ρ α ðr; tÞ : ð5Þ…”