Excluded-volume effects in the diffusion of hard spheres

Excluded-volume effects can play an important role in determining transport properties in diffusion of particles. Here, the diffusion of finite-sized hard-core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions. The result is a nonlinear diffusion equation for the one-particle distribution function, with excluded-volume effects enhancing the overall collective diffusion rate. An expression for the effective (collective) diffusion coefficient is obtained. Stochastic simulations of the full particle system are shown to compare well with the solution of this equation for two examples.

Figure 1

Excluded [red (dark gray) and black] and available [blue (light gray)] area in a solution of black particles for the placement of an additional test particle. (a) The area available with point particles is the whole domain. (b) The area available (to the center of the test particle) with finite-size particles is reduced. Modified from Minton [14].

Figure 2

Marginal distribution function $p({\mathbf{x}}_1,t)$ at time $t=0.05$ with normally distributed initial data and $N=400$. (a) Solution $p({\mathbf{x}}_1,t)$ of Eq. (4a) for point particles ($\epsilon =0$). (b) Histogram for $\epsilon =0$. (c) Solution $p({\mathbf{x}}_1,t)$ of Eq. (11) for finite-sized particles ($\epsilon =0.01$). (d) Histogram for $\epsilon =0.01$. Histograms computed from ${10}^4$ realizations of Eq. (1) with $\Delta t={10}^{-5}$. All four plots have the same color bar.

Figure 3

Stationary marginal distribution function $p_s\left({\mathbf{x}}_1\right)$ under the external potential $V$ for point particles and finite-size particles, with $N=1000$. (a) Point particles, $p_s\propto e^{-V}$. (b) Histogram for $\epsilon =0$. (c) Finite-size particles $p_s$ from Eq. (15) ($\epsilon =0.01$). (d) Histogram for $\epsilon =0.01$. Histograms computed with ${10}^9$ steps of the MH algorithm. All four plots have the same color bar.