Custom flow in overdamped Brownian dynamics

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. To facilitate systematic analysis, here we present a general iterative scheme for the determination of the unique external force field that yields 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.

Figure 1

One-body density (a) and current profiles (b), external force field (c), and external potential (d) as a function of the $x$-coordinate in a system with target density profile ${\rho }_{}\left(x\right)=c_1{sin}^2(\pi x/L)+c_2$ with $c_1{\sigma }^2=0.12$ and $c_2{\sigma }^2=0.24$ after $k=40$ iterations. The inset in (a) shows the difference between target and sampled density profiles. Results are shown for different values of the target current: $J_0\sigma \tau =1$ (blue dotted line), $J_0\sigma \tau =0.5$ (orange dashed line), $J_0\sigma \tau =0.1$ (violet solid line), and $J_0\sigma \tau =0$ (green dot-dashed line), which is in equilibrium. The inset in (b) is a close-view for $J_0\sigma \tau =0.5$. The system is two-dimensional and homogeneous in the $y$ direction with $N=30$ particles in a square periodic box of side $L/\sigma =10$ at $k_{B}T/\epsilon =1$. The data is obtained by averaging over 25 BD realizations (MC realizations in equilibrium).

Figure 2

$x$ component (a1) and $y$ component (a2) of the velocity profile sampled after $k=40$ iterations using BD simulations. (b1) Sampled density profile for the steady state with constant density profile. $x$ (b2) and $y$ (b3) components of the external force field that produces the steady state with constant density. (c1) Sampled density profile for the steady state with inhomogeneous density profile. $x$ (c2) and $y$ (c3) components of the external force field that generates the steady state with inhomogeneous density. In both steady states we set $N=30,L/\sigma =10$, and $k_{B}T/\epsilon =0.5$. The bin size is set to 0.05 $\sigma$ in both directions and the origin or coordinates is located in the middle of the box. Results are averages over 100 BD realizations.

Figure 3

Inversion in full nonequilibrium as demonstrated by dynamics of confinement. Density profile (left column), current profile (middle column), and external force field (right column) as a function of $x$ for three different situations. (a) Time evolution of the system, density (a1) and current (a2), after switching on a static external field (a3). At $t=0$ the system is in equilibrium with vanishing external field. Results at four times are shown: ${\tau }_1/\tau =0.08,{\tau }_2/\tau =0.48,{\tau }_3/\tau =1.0$, and ${\tau }_4/\tau =2.2$. At ${\tau }_4$ the system is near equilibrium with the applied external field (a3). Profiles are obtained by averaging over $\sim {10}^8$ different trajectories. In panels (b) we show a system that evolves following the same dynamics as in (a) but two times faster. The current (b2) is therefore twice as large as the current in the original system, and the required external field (b3) is time-dependent. A movie showing a system that evolves three times faster is presented in the Supplemental Material [33]. Panels (c) show the time evolution in a system that reproduces the same dynamics as (a) but with a global motion toward the right (note that the spatial average of the current (c2) at any time is $J_0\sigma \tau =0.5$). The required external field (c3) is time-dependent. The horizontal and vertical dashed lines in the plots of the external field are drawn to help the comparison between systems. In all cases we set $N=30,L/\sigma =10$, and $k_{B}T/\epsilon =1.0$.

Figure 4

One-body density profile (a), one-body current (b), and external force field (c) as a function of the $x$-coordinate for different number of iterations, $k$, as indicate in the legend of (a). The target current is set to $J_0\sigma \tau =0.1$. The target density profile is ${\rho }_{}\left(x\right)=c_1{sin}^2(\pi x/L)+c_2$ with $c_1{\sigma }^2=0.12$ and $c_2{\sigma }^2=0.24$, which is practically identical to the sampled density profile after 40 iterations (violet solid line). Two-dimensional system with $N=30$ particles in a periodic box of side $L/\sigma =10$ at temperature $k_{B}T/\epsilon =1$. The data has been obtained by averaging over 25 BD realizations. The total simulation time of iteration $k$ of one BD realization is set to ${\tau }_k/\tau ={\tau }_0{2}^{(k-1)/3}$, with ${\tau }_0/\tau =100$. The bin size is $\mathrm{\Delta }x/\sigma =0.05$.

Figure 5

(a) Scaled error of the density profile $\mathrm{\Delta }{\rho }^{*}=\mathrm{\Delta }\rho L^2{\sigma }^2$ (green circles) and of the current profile $\mathrm{\Delta }{J}^{*}=\mathrm{\Delta }J{L}^2{\tau }^2$ (yellow squares) as a function of the iteration number $k$ (bottom axis) and the scaled simulation time of the iteration ${\tau }_k/\tau$ (upper axis) in nonequilibrium steady state using BD simulations. (b) Scaled error of the density profile $\mathrm{\Delta }{\rho }^{*}=\mathrm{\Delta }\rho L^2{\sigma }^2$ as a function of the iteration number $k$ (bottom axis) and the number of Monte Carlo steps at iteration ${\tau }_k/\tau$ (upper axis) in equilibrium using MC simulations. In both panels, the data has been obtained by averaging over 25 realizations. Note the logarithmic scale of the vertical axis and of the upper horizontal axis.