Evolution equation for tagged-particle density and correlations in single-file diffusion

We derive and study a theoretical description for single-file diffusion, i.e., diffusion in a one-dimensional lattice of particles with hard core interaction. It is well known that for this system a tagged particle has anomalous diffusion for long times. The novelty of the present approach is that it allows for the derivation of correlations between a tagged particle and other particles of the system at a given distance with empty sites in between. The behavior of the correlation gives deeper insights into the processes involved. The numerical integration of differential equations are in good agreement with Monte Carlo simulations.

Figure 1

Numerical evaluation of the approximation of Eq. (22). The quantity plotted is $\mathrm{Pr}(m_{i+j+1}^o/n_i^o\cdots m_{i+j}^o)+\mathrm{Pr}(m_{i-1}^o/n_i^o\cdots m_{i+j}^o)$ against position $i$ for different values of $j$: $j=1$ ($\circ$), $j=2$ ($\square$), $j=3$ ($\diamond$), and $j=4$ ($*$). Different values of concentration $c$ were used; from top to bottom: $c=0.7$, 0.5, and 0.3. The initial conditions are random. The sum of the conditional probabilities that individually depend on $i$ and $j$ coincides with the constant value $2p_1$. For $c=0.7$ (top curve), the range of the horizontal axis is larger than the range in which the tagged-particle density takes relevant values, between about $-10$ and 10 (see Fig. 2, $\diamond$). Lattice length $L=1000$, time $t={10}^4$, number of samples 500 000.

Figure 2

Tagged-particle density $n_i$, against position for different times in the anomalous regime: $t={10}^3$ ($\square$), $t={10}^4$ ($\diamond$), and $t={10}^5$ ($\circ$). Concentration $c=0.7$ was used. The initial condition is random. Dots correspond to Monte Carlo simulations in a lattice of length $L=1000$; number of samples: 100 000, and $\Gamma =1/2$. The curves are the results of numerical integration of Eq. (24) related to $n_i$ by Eq. (14) [27]. Each of the about 700 particles of each sample was considered as one tagged particle to reduce the statistical error.

Figure 3

Correlation $q_i=u_{-i-1,1}-u_{i,1}$ for different times. Same parameters as in Fig. 2.

Figure 4

Mean square displacement against time in log-log scale. Lower curve: random initial condition. Parameters as in Fig. 2. Circles correspond to Monte Carlo simulations and the curve to the numerical integration of Eq. (24) related to $n_i$ by Eq. (14). Dashed lines represent the asymptotic behaviors for short [$2\Gamma (1-c)t$] and long [Eq. (1), $b=2$] times, i.e., normal and anomalous regimes, respectively. For times ${10}^2\le t\le {10}^5$ the results obtained from numerical integration can be fitted by ${\Delta }^{2}x=A{t}^{1/2}$, where the value of $A$ is $0.30$; the corresponding theoretical value from Eq. (1) is $0.34$. Upper curve: initial condition with equidistant particles, concentration $c=0.5$ (lattice length 1000, number of samples 30 000 and $\Gamma =1/2$). The results were shifted one decade upwards for clarity. Squares correspond to Monte Carlo simulations and the curve to the numerical integration of Eq. (24). Dashed lines represent the asymptotic behavior for short times, $2\Gamma t$, and for large times [Eq. (1), $b=\sqrt{2}$] that hold for initially equidistant particles.

Figure 5

Integration results for the tagged-untagged particle correlation $u_{i,j}$ at $t={10}^3$, for equidistant particles at $t=0$ with concentration $c=0.5$. A logarithmic gray scale was used. The integration was performed in a lattice of size $-70\le i\le 70$ and $1\le j\le 80$. The method of integration is Runge-Kutta of order 5 with variable time interval. A small initial value of order ${10}^{-9}$ was used for $i\ne 0$ to avoid numerical instabilities.

Figure 6

Integration results for $n_i$ (dashed curve) and $u_{i,1}$ (solid curve) for $t={10}^3$. It can be seen that $u_{i,1}$ is slightly shifted to the right with respect to $i=0$. Same parameters as in Fig. 5.