First-passage distributions in a collective model of anomalous diffusion with tunable exponent

We consider a model system in which anomalous diffusion is generated by superposition of underlying linear modes with a broad range of relaxation times. In the language of Gaussian polymers, our model corresponds to Rouse (Fourier) modes whose friction coefficients scale as wave number to the power $2-z$. A single (tagged) monomer then executes subdiffusion over a broad range of time scales, and its mean square displacement increases as $t^{\alpha }$ with $\alpha =1/z$. To demonstrate nontrivial aspects of the model, we numerically study the absorption of the tagged particle in one dimension near an absorbing boundary or in the interval between two such boundaries. We obtain absorption probability densities as a function of time, as well as the position-dependent distribution for unabsorbed particles, at several values of $\alpha$. Each of these properties has features characterized by exponents that depend on $\alpha$. Characteristic distributions found for different values of $\alpha$ have similar qualitative features, but are not simply related quantitatively. Comparison of the motion of translocation coordinate of a polymer moving through a pore in a membrane with the diffusing tagged monomer with identical $\alpha$ also reveals quantitative differences.

Figure 1

(Color online) Mean squared position of the central monomer (in units of $b^2$) as a function of time (in units of ${\tau }_{\text{s}}$) for a polymer of length $N=257$. The curves correspond (left to right) to $z=1.25$, 1.5, 1.75, 2, and 2.25. Each curve is an average over $10000$ realizations.

Figure 2

(Color online) Logarithmic plot of absorption probability distribution as a function of time (in units of ${\tau }_{\text{s}}$) of the central monomer of a $N=257$ polymer, in the presence of an absorbing boundary at a distance $8b$ from the initial position of the monomer. The leftmost curved depicts the result for normal diffusion of a single particle. The rest of the curves correspond (left-to-right) to $z=1.25$, 1.5, 1.75, 2, and 2.25. For each $z$ value, $100000$ independent runs were performed. The inset depicts the value of $\theta$ obtained from the slopes of these graphs as a function of $\alpha$. The continuous line depicts the relation between these exponents proposed in Ref. [20] (see text).

Figure 3

(Color online) Semilogarithmic plot of the absorption probability density of a tagged monomer as a function of time (in units of ${\tau }_{\text{s}}$) for the central monomer in a $N=257$ polymer, in the interval between two absorbing boundaries at a distance $16b$. The monomer starts at the midpoint between the boundaries. Each curve is the result of $100000$ independent simulations. The different curves correspond (left to right) to $z=1.25$, 1.5, 1.75, 2, and 2.25.

Figure 4

(Color online) Probability density function of the central monomer $c=129$ in a polymer with $N=257$ monomers in the presence of one absorbing boundary. The monomer is initially located at a distance $8b$ from the absorbing boundary. The horizontal axis is in the scaled variable $\rho =x/\ell \left(t\right)$. The graphs correspond to different times (right to left) $t/{\tau }_{\text{s}}=2,32,128,1024,3500,6000$, and obtained from $100000$ independent runs.

Figure 5

(Color online) Probability density function of the central particle $\left(c=129\right)$ in chains of $N=257$ monomers in the presence of one absorbing boundary. The horizontal axis is in the scaled variable $\rho =x/\ell \left(t\right)$ (see text). The different curves correspond to $z=1.25$, 1.5, 1.73, 2, and 2.25 (bottom to top). The graphs are shifted vertically for clarity with increasing $z$ values by a factor of 5. The dashed lines have slopes $\varphi =1/\alpha$. Each curve was obtained from $100000$ independent runs.

Figure 6

(Color online) Probability density function of the central particle in a chain with $N=257$ monomers with absorbing boundaries at $X_{b1,2}=\pm 8b$ as a function of position measured from the center of the interval (in units of $b$) for $z=1.25$, 1.5, 1.75, 2, and 2.25 (broad to narrow). Each graph is the result of $100000$ independent runs.

Figure 7

(Color online) Probability density function of the central monomer in a chain with $N=257$ particles in the presence of absorbing boundaries at $X_{b1}=-8b$ and $X_{b2}=8b$, for $z=1.25$, 1.5, 1.75, 2, and 2.25 (bottom to top). Each curve is the result of $100000$ independent runs. The curves are shifted vertically for clarity, by a factor of 5 at each increasing $z$. The dashed lines have slopes $\varphi =1/\alpha$.

Figure 8

(Color online) The dots connected by a line depict the normalized PDF of the translocation coordinate, at times significantly exceeding the mean translocation time, for a two-dimensional self-avoiding polymer of length $N=128$ that did not yet translocate (from Ref. [34]). This is compared to the normalized PDF of a central particle (in a chain of $N=257$) performing anomalous diffusion controlled by exponent $z=1.25$ (i.e., $\alpha =0.8$), moving between two absorbing boundaries. For the purpose of comparison the range of the translocation coordinate and the range of monomer positions have been shifted and rescaled to the segment [0,1]. The inset shows the same quantities on the logarithmic scale; the dashed has slope 1.25.