Dynamical Spreading and Memory Retention Under Power Law Potential

Abstract

Power law potentials dictate interactions across scales and matter, controlling the structure and dynamics of inanimate, and living systems. Though the equilibrium distributions of particles with a power law repulsion were extensively studied, their unconfined dynamical evolution gained far less attention -- Yet, living matter is inherently out of equilibrium and is seldom static. Here, we investigate the overdamped dynamic spreading of a dense suspension of particles under repulsive pair-potential of the form $1/r^k$. Coarse graining the pair interactions, we predict that the suspension spreads in a self-similar form, with its radius growing in time as $t^{1/(k+2)}$, independent of the spatial dimension ($d$). We confirm this prediction experimentally in quasi-two dimensions using perpendicularly magnetized colloids with dipolar repulsion ($k=3$). Numerical simulations corroborate the experiments for the $k=3$ case and further predict a categorically different behavior at a critical power law: when $k<d-2$, the initial distribution is no longer concentrated at the origin. Instead, particles accumulate at the perimeter and retain a long-lived memory of their original pattern. We demonstrate that below this threshold, the initial distribution seeds the resulting pattern, encoding the future structure of an unconfined, and dynamically evolving system.

Figures (27)

• Figure 1: Experiments and simulations of particles spreading under a $1/r^3$ pair-potential. (A) Schematics of the experimental setup. Inset shows the sample with a $\times40$ magnification. Triangular organization of the particles can be seen. Blue trajectories of five particles are shown over 60 seconds. (B-C) Snapshots at three different times with $\times 4$ magnification: (B) Initially, (C) after 16 minutes, and (D) after about one hour. (E) The standard deviation as a function of time showing $R \sim t^{1/5}$ (Eq. 1). (F) Density as a function of distance at four different times ($t = 1500,3000,4500,5500$s). (G) Rescaled-density with respect to $t^{2/5}$ as a function of the self-similarity parameter $r/t^{1/5}$ showing the four curves collapse to a single one. The inset plots $(\rho t^{2/5})^{3/2}$ versus $\eta^2$, which, is linearly decreasing (see Eq. \ref{['f in thesis']}). (H-J) results from molecular dynamics simulations of 3000 particles with a $1/r^3$ potential showing the same spreading as in the experiment. (H) Snapshots of the simulation at three different times ($t = 0.002, 10, 4800$). (I) The same snapshots from (H) but rescaled according to Eq. 1. (J) Rescaled average density showing a collapse to a single curve and a fit to the theoretical prediction given by Eq. \ref{['f in thesis']}. The inset plots $(\rho t^{2/5})^{3/2}$ versus $\eta^2$ showing the linear predicted trend.
• Figure 2: Simulations in the long-range limit $k< d$. (A) Radius versus time for $k = (-1,0,1)$ in a simulation of 10000 particles, along with the theoretical scaling, $R(t) \propto t^{\frac{1}{k+2}}$, shown as a dashed line. (B) Snapshots of the distribution at steady-state, with color coding according to the density for k = 1 (top left), k = 0 (bottom), and k = -1 (top right). (C--E) Rescaled density according to $\rho t^{2\beta}$ as a function of the self-similar variable $\eta = r/t^{\beta}$. Bright blue dots give the theoretical predictions given in Eq. \ref{['f for k<d']}. A single curve can be seen, but it is, in fact, ten plots at different times that collapse to a single curve. The inset shows the separate, unscaled densities as a function of radius, with color going from light blue at early times to dark blue at later ones.
• Figure 3: Collision of two suspensions of $N=2000$ particles each, for $k=3$ (in B) and $k=-1$ (in C). (A) The initial configuration. (B) and (C) show the configuration at ($t=2.5\cdot 10^5$). (D) and (E) plots of the corridor length that does not contain particles between the two suspensions. (D) For $k=3$, plotted on a semi-log scale, and (E) for $k=-1$, plotted on a log-log scale. Note how it shrinks exponentially for $k=3$ and grows as a power law for $k=-1$. The diagram above C shows two soap bubbles with a similar interface. (F) Time to reach an isotropic steady state (which is indistinguishable from a single population), as a function of the power law exponent $k$. (G) Four initial populations form an H-shaped particle-free zone for $k=-1$. On the right is the shape formed by four soap bubbles.
• Figure 4: $R$ as a function of time in a $1D$ suspension with $k=2$ : $\beta = \frac{1}{4}$.
• Figure 5: Density profile at different times in a $1D$ suspension with k=2.