Diffusion of interacting Brownian particles: Jamming and anomalous diffusion

The free self-diffusion of an assembly of interacting particles confined on a quasi-one-dimensional ring is investigated both numerically and analytically. The interparticle pairwise interaction can be either attractive or repulsive and the energy barrier opposing thermal hopping of two particles one past the other is finite. Thus, for sufficiently long times, self-diffusion becomes normal or conventional diffusion. However, depending on the particle density, subdiffusive transients with exponent $1∕2$ and suppression of normal diffusion are observed. Above a certain density threshold, a sudden drop to zero of the diffusion coefficient for attractive particles signals the transition to a jammed phase. Furthermore, a Gaussian cluster of attractive particles condenses, by shrinking in size, for densities larger than such density threshold; lower density clusters spread out, regardless of the interaction sign, through a diffusion mechanism that is anomalous at short times, and normal for sufficiently long times. These effects could be observed in systems with colloidal particles, vortices, electrons, among other interacting particle systems.

Figure 1

(Color online) Mean square displacement $⟨\delta x^2\left(t\right)⟩$, defined in Eq. (4), versus $t$ for an assembly of $N$ particles on a ring of length $l$ interacting via the pair potential (3). (a) Molecular dynamics simulations for $N={10}^3$, $l=400$, $kT=0.07$, $\lambda =0.05$, and $g=0.1$. Inset: as in the main panel but for $N=2\times {10}^3$ and $g=0.06$. (b) Monte Carlo simulations (model 1) for $N=3.5\times {10}^3$, $l=3.5\times {10}^4$, $kT=0.2$, and different values of $p$. Inset: the corresponding fitting diffusion constant $D$ versus $p$. (c) Simulations of model 2 for the same assembly parameters as in (b) and different values of $r$. Inset: the diffusion exponent $\alpha$ versus $r$.

Figure 2

(Color online) Self-diffusion constant $D$ versus density $n$ for an assembly of attractive (open blue circles, $g=-0.02$, $\lambda =0.5$) and repulsive particles (solid red squares, $g=0.08$, $\lambda =0.05$). The constant $D$ is given in units of $D_0=kT$ and the density in units of ${\lambda }^{-1}$ (see text). The pair interaction potential is defined in Eq. (3). Other simulation parameters $l=2$ and $kT=0.5$. The solid line represents our “one-particle” approximation (6) for the repulsive case.

Figure 3

(Color online) (a) The potential $V\left({\sigma }^2\right)$ of Eq. (16) in units of $2kT{\sigma }_c^2$ for attractive (main panel) and repulsive (inset) particles. (b) The corresponding time dependence of ${\sigma }^2\left(t\right)$. Top inset: $g>0$, ${\sigma }^2\left(0\right)∕{\sigma }_c^2=0.1$; main panel: $g<0$, ${\sigma }^2\left(0\right)∕{\sigma }_c^2=0.9$; and bottom inset: $g<0$, ${\sigma }^2\left(0\right)∕{\sigma }_c^2=1.1$. The curves ${\sigma }^2\left(t\right)$ have been obtained by integrating numerically Eq. (15); $\sigma$ is expressed in units of ${\sigma }_c$ and $t$ in units of ${\sigma }_c^2∕kT$.

Figure 4

(Color online) Dispersion $\sigma \left(t\right)$ versus time $t$ showing the decay of a Gaussian cluster of interacting particles. This was obtained via molecular dynamics simulations of $N_1=50$ particles, initially forming a Gaussian packet with $\sigma \left(0\right)=3.3$ and interacting via the pair potential (3) with $\lambda =1$ and different $g$. Other simulation parameters are $l=400$ and $kT=1$: (a) cluster surrounded by a uniform particle distribution with $n=0.125$; (b) cluster expanding in vacuo with $n=0$. Insets: Logarithmic plot of ${\sigma }^2\left(t\right)$ for $g=10$; the slopes $t^{2∕3}$ and $t$ are drawn for the reader’s convenience.

Figure 5

(Color online) Decay of a Gaussian cluster of interacting particles, ${\sigma }^2\left(t\right)$ vs $t$. Monte Carlo simulations of a Gaussian cluster of $N_1=600$ particles diffusing against a uniform particle distribution with density $n=0.1$ (curve set A) and in vacuo with $n=0$ (curve set B). Particle kinetics of Eq. (5): (a) model 1; (b) and model 2. Other simulation parameters: $l=1.8\times {10}^4$ and $kT=1$; the curves weakly depend on the relevant kinetic parameter $p$ or $r$.