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Problem statement

I am trying to determine the mean free path $\lambda$ of a so-called Lévy dust, i.e. $M$ points in a square $L\times L$ environment with distances between subsequent points distributed according to a Lévy-stable distribution with stable parameter $\alpha$. It is critical to note that the environment has periodic boundary conditions. I thought $\lambda$ would be simple to determine, by simply letting a walker start on such a point, and execute ballistic motion with a fixed randomly sampled travel angle $\theta \in [0,2\pi)$ until detection of a point. Then, you can simply compute $\lambda$ as the average point-to-point distance. Averaging over different environments and a significant number of point-to-point distances, $\lambda$ can be easily computed.

However, due to the Lévy dust being scale-free (as long as $M$ is sufficiently small, see also Ferreira, et al. (2012)), the mean free path does not only represent the sparsity of the points within the box, but also reflects the spatial heterogeneity of the Lévy dust. As Ferreira and his colleagues argue, they state that (I added the parts between brackets):

It is known that for a uniform distribution, $\lambda$ is inversely proportional to the global (point) density. In patchy (fractal or non-fractal, i.e. Lévy dusts) distributions with fixed global density, the length of the mean free path also reflects the level of (point) aggregation.

They further state that as $\alpha\rightarrow 1$, the widely spread point distribution leads to large values of $\lambda$. As $\alpha \rightarrow 3$, a small number of dense clusters appears, surrounded by extensive empty spaces, also leading to large values of $\lambda$. This implies a minimization of the mean free path when $\alpha$ is 'just right', i.e. somewhere for $1 < \alpha < 3$.

However, what I find when running the above described experiment is none of this behaviour. I actually see a steady increase in the mean free path as $\alpha$ grows. Most surprisingly, I expected that the mean free path as $\alpha \rightarrow 1$ would resemble the value of the mean free path of a uniform distribution: $$ \lambda = \frac{L^2}{2RM}, $$ where $R$ is the radius of the points. However, I do not find this result in my own Monte Carlo experiments. Moreover, the results presented by Ferreira et al., also do not show this mapping towards the uniform limit of $\lambda$ as $\alpha\rightarrow 1$: they are consistently below the value of the above equation. Their results do display the aforementioned minimisation of the mean free path, which leaves me to believe my current implementation is wrong.

Questions

Therefore, my question is two-fold. First, is my assumption that the mean free path for a uniform distribution should be recovered as $\alpha\rightarrow 1$ correct? Or is the 'scale-free-ness' of the distribution still apparent?

Second, is my Monte Carlo approach of determining the mean free path correct? For this, I will put the pseudo-code below.

# PSEUDO-CODE FOR MONTE CARLO APPROACH TO DETERMINING MEAN FREE PATH
# Variables
# M: number of points
# L: environment size (L x L)
# alpha: Levy stable parameter
# distance: distance travelled between successive point encounters

levy_dust = distribute(M, L, alpha)   # Generate points as a Levy dust
walker = random(levy_dust)            # Initialize the walker on a random point
angle = random(0,2*pi)                # Random walking angle, i.e. ballistic motion
k = 0
while k < K:
    walker.walk(angle)
    if resource encountered:
        mean_free_path += travelled_distance
        travelled_distance = 0
            

Note that the above pseudo-code does provide me with the correct answer if points are uniformly distribution, therefore leading me to believe the code I implemented is correct.

Additionally, I could not find any sources on the interpoint distance distribution of such Lévy dusts, so any additional sources would be a tremendous help. Thanks a lot for any help anyone is able to offer.

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  • $\begingroup$ Why not try your code on a uniform distribution and see if you get the expected result? $\endgroup$
    – jklebes
    Commented Mar 3, 2021 at 9:58
  • $\begingroup$ Indeed this was something I did, and as expected I get the expected result for a uniform distribution. This makes me believe my implementation is correct, however that further raises doubt on the results presented in the paper that I mentioned. $\endgroup$ Commented Mar 3, 2021 at 10:27
  • $\begingroup$ Interesting, a final possibility is that while the algorithm is ok you have an effect of periodic boundary conditions and insufficient system size, which shows up in the Levy case and not the uniformly distributed case. $\endgroup$
    – jklebes
    Commented Mar 3, 2021 at 10:37
  • $\begingroup$ Very true, I believe I did some experiments with different systems sizes, but all of them amounted to the same results, however I can experiment a bit more thoroughly with this. Currently, my system size is set to L=1000, and this indeed might influence the Levy statistics to some degree. Is there something to identify if the underlying Levy dust distribution is correct, other than recording the sampled point-to-point distances when generating the point distribution? $\endgroup$ Commented Mar 3, 2021 at 11:11
  • $\begingroup$ Not familiar with the distribution, just guessing around at your simulation problems $\endgroup$
    – jklebes
    Commented Mar 3, 2021 at 11:35

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