3.10. Lab 9: Self-organized Criticality

Due Date: 11:59pm March 5, 2024

Labs should be submitted as a single Jupyter notebook to CourseLink Dropbox

This lab counts for 4% of the final grade

To do in advance of this lab:

  • Make sure you have run the code in Allen Downey’s Chapter 8 notebook.

3.10.1. Exercises

Complete the exercises corresponding to Exercise 8.1 – 8.3 in Chapter 8 of Think Complexity (version 2):

  1. Launch chap08.ipynb and run the code. There are three short exercises embedded in the notebook to complete.

  2. To test whether the distributions of T and S are heavy-tailed, we plotted their Pmf on a log-log scale, which is what Bak, Tang and Wiesenfeld show in their paper. But as we saw in Chapter 4, this visualization can obscure the shape of the distribution. Using the same data, make a plot that shows the cumulative distributions (CDFs) of S and T. What can you say about the shape of these distributions? Do they follow a power law? Are they heavy tailed?

    You might find it helpful to plot the CDFs on a log-x scale and on a log-log scale.

  3. In Section 8.5 we showed that the initial equilibrium of the sand pile model produces fractal patterns. But after we drop a large number of random grains, the patterns look more random.

    Starting with the example in Section 8.5, run the sand pile model for a while and then compute fractal dimensions for each of the 4 levels. Is the sand pile model fractal in steady state?

  4. Another version of the sand pile model, called the “single source” model, starts from a different initial condition: instead of all cells at the same level, all cells are set to 0 except the center cell, which is set to a large value.

    Write a function that creates a SandPile object, sets up the single source initial condition, and runs until the pile reaches equilibrium. Does the result appear to be fractal?

    You can read more about this version of the sand pile model at http://math.cmu.edu/~wes/sandgallery.html.

3.10.2. Submission

Please submit a single Jupyter notebook which completes the above exercises, filling in Allen Downey’s Chapter 8 notebook. There are already placeholders in the notebook for your work.