Aggregation-fragmentation-diffusion model for trail dynamics

We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one, and also one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail $P\left(w\right)\sim w^{-\gamma }$ for small weight $w$. We obtain the exponent $\gamma$ analytically and find that it varies continuously with the two model parameters. The exponent $\gamma$ can be positive or negative, so that in one range of parameters small-weight trails are abundant and in the complementary range they are rare.

Figure 1

Illustration of (a) fragmentation with probability $p$, with a fixed ratio between the two fragments, characterized by $r$, and (b) random hopping with probability $1-p$.

Figure 2

A numerical realization of the aggregation-fragmentation-diffusion process with $r=0.01$ and $p=0.05$. The color coding illustrates weights from low (blue) to high (yellow) values, and the color bar is in natural logarithmic scale. Unoccupied sites are dark purple.

Figure 3

Numerically computed trail concentration $c\left(p\right)$ versus the fragmentation probability $p$. The dashed line corresponds to the analytic expression (5), and the circles to Monte Carlo simulations on a lattice of $N={10}^6$ sites.

Figure 4

The normalized second and third moments $m_2\left(n\right)=M_2\left(n\right)/2$ and $m_3\left(n\right)=M_3\left(n\right)/6$, versus (a) $n^{-1/2}$ and (b) $n$. Shown are results of Monte Carlo simulations on a ring with ${10}^8$ sites for the case $r=p=1/2$. A fourth-order polynomial fit to $m_k\left(n\right)$ versus $n^{-1/2}$ yields the asymptotic values 1.0001 and 1.0002 for $m_2\left(\infty \right)$ and $m_3\left(\infty \right)$, respectively.

Figure 5

The power-law tail of the weight density $P\left(w\right)$. The curves shown here correspond to different values of $p$, indicated in the legend, and are obtained from Monte Carlo simulations on a lattice of $N={10}^6$ sites. Dashed lines correspond to the theoretical predictions of Eqs. (1) and (2). The value of $r$ was chosen to be $r=0.01$. The inset displays the case $r=0.25$ and $p=0.7$ for which $\gamma =-0.32$.

Figure 6

(a) The exponent $\gamma$ for different values of $p$ and $r$. The lines correspond to the theoretical predictions, given by Eq. (2), for different values of $r$; see the legend. The inset shows the region of negative $\gamma$ as shaded. (b) When data are plotted as $r^{\gamma -1}+{(1-r)}^{\gamma -1}$ they collapse into the curve $(3-p)/(1-p)$ (solid line) in accord with (2).

Figure 7

The exponent $\gamma$ versus the parameter $r$ for small values of $p$.

Figure 8

(a) Voids are defined as closed regions of unoccupied sites, and they are defined by an area $S$ (blue region) and duration $T$. (b),(c) Probability distribution function of the void durations and sizes, respectively, for two different values of $p$ and $r=0.01$. Solid red lines correspond to a power law with exponents $-3/2$ (b) and $-4/3$ (c).