This code allows to play with the process mentioned in the paper Long time asymptotic behavior of a self-similar fragmentation equation.

Installation

A script is available for an easy creation of the conda environment and compilation of auxiliary functions:

$ source install.bash

How to use ?

A toy example can be ran with:

$ python main.py

Fragmention process simulation

We consider a process in which a particle of mass $M$ lives a random time (depending on $M$) and split in $\lambda$ particle of mass $M/\lambda$ when dying. The process is then repeated. It is straightforward to simulate such a process.

Theoretical behaviour - concentration function

If we assume the random life times to follow an exponential distribution of parameter $\alpha$, the equation of a such process is given by:

$$\begin{cases} \partial_t c(t,x) = \lambda^{2+\alpha}x^\alpha c(t,\lambda x) - x^\alpha c(t,x) \\\ % c(0,x) = \delta_M(x) c(0,x) = u_0(x) \end{cases}$$

where $c(t,x)$ denotes the number of particle of mass $x$ at time $t$. We prove that the solution can be obtained with a series expansion of operators applied to the initial condition. More precisely:

$$\forall (t,x)\in \mathbb{R}_+^2,\quad c(t,x) = \left[\sum_{k=0}^\infty\frac{t^k}{k!}\mathscr{L}_k\mathcal{F}_\alpha^k\right]u_0(x)$$

with:

$$\mathscr{L}_k := \sum_{i=0}^k (-1)^{k-i} \binom{k}{k-i}_{\lambda^{-\alpha}} \lambda^{2i+\alpha i (i-1)/2 - \alpha} \mathscr{D}_\lambda^i$$

and

$$\mathscr{D}_\lambda f(x) :=f(\lambda x) \quad\mathrm{and}\quad \mathscr{F}_\alpha f(x) :=x^\alpha f(x).$$

Theoretical behaviour - moment evolution

We also focus our attention on the the Mellin transform:

$$C(t,\sigma) = \int_0^\infty c(t,x)x^{\sigma-1} \mathrm{d}x.$$

We prove in the article that this exact expansion holds:

$$C(t,\sigma) = \sum_{k=0}^\infty \frac{(\lambda^{-\alpha},\lambda^{2-\sigma})_k}{k!} (-t)^k$$

where the Pochhamer symbol is defined as:

$$\forall n\geq 0,\quad (a,q)_n := \prod_{i=0}^{n-1}(1-aq^i).$$

This code simulates the process and compute the exact series expansion for the population (case $\sigma=1$). Here are some comparisons between experimental results and our series expansion for the population evolution ($\sigma=1$).

Estimation of $\lambda$

We are also interested in computing the inverse problem. From an observation (or several), can we recover the parameter $\lambda$ ? Here are the estimations depending on the order of summation.

Interesting ?

If you have any questions, feel free to contact us. We will be more than happy to answer ! 😀

If you use it, a reference to the paper would be highly appreciated.

@article{agazzotti2024long,
  title={Long time asymptotic behavior of a self-similar fragmentation equation},
  author={Agazzotti, Gaetano and Deaconu, Madalina and Lejay, Antoine},
  year={2024}
}

Tested on