Aggregation-fragmentation-diffusion model for trail dynamics
Kyle Kawagoe, Greg Huber, Marc Pradas, Michael Wilkinson, Alain Pumir, and Eli Ben-Naim
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 $P(w) \sim w^{-\gamma}$ 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 tails are
abundant, and in the complementary range, they are rare.
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