Single-File Diffusion in a Box

We study diffusion of (fluorescently) tagged hard-core interacting particles of finite size in a finite one-dimensional system. We find an exact analytical expression for the tagged particle probability density function using a Bethe ansatz, from which the mean square displacement is calculated. The analysis shows the existence of three regimes of drastically different behavior for short, intermediate, and large times. The results are in excellent agreement with stochastic simulations (Gillespie algorithm).

Figure 1

Diffusing particles where mutual passage is excluded, i.e., $y_j\le y_{j+1}-\Delta$ for $j=1,\dots ,N-1$.

Figure 2

Particle trajectories, generated by the Gillespie algorithm, in a system where $N=11$.

Figure 3

(a) Tagged PDF [Eq. (11)] for the middle (red online) and the rightmost (blue online) particle, where $N\Delta /L=0.25$, at three instances of time, with $y_{5,0}/L=-0.15$ and $y_{11,0}/L=0.3$. (b) Equilibrium density [Eq. (16)] compared to the point-particles case and stochastic simulations ($\circ$) ($t/{\tau }_{\mathrm{eq}}=2$, 500 lattice points and ${10}^5$ ensembles).

Figure 4

Mean square displacement for a tagged particle placed in the middle of the system where $N=\{3,21,141\}$. Solid curves (blue online) show numerical calculations of $\mathcal{S}(t)$ based on Eq. (11). Straight lines are approximated analytic results for regimes (i)–(iii), see text.