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Self-organized criticality - student project post-mortem

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We just got finished with our student team project, which you can find here, and I thought I'd do a little post mortem on it.

It's a neat little project that implements various simulations of self organized criticality on 2D grids. What is self organized criticality, you might ask? Dunno, I can't tell you.

All right, I do know a little. Imagine the ising system I've written about before. In the version without an external magnetic field, it has a single important parameter that we can set - the temperature. If you sweep through the values of temperature, you can find a single point where the behavior of the system changes qualitatively - order wins over disorder at temperatures below roughly 2.72 in reasonable [set everything to 1] units. Near that value - at criticality - you get large scale behavior, huge fluctuations, exponential slowdowns, etc.

Self organized criticality, as I currently understand it, is basically that, except that as you run your simulation, you realize that it displays criticality for a wide range of parameters (here - temperature). To quote Wikipedia (emphasis mine):

the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, self-organized criticality.

Here's an example, a simulation of forest fire (yellow being fire, green being trees, and purple being ash, from which trees can regrow):

In [30]:
import SOC
model = SOC.Forest(L=100, f=0.0001)  # f being probability of lightning strike
model.run(500, wait_for_n_iters=500)
model.animate_states(notebook=True)

Waiting for wait_for_n_iters=500 iterations before collecting data. This should let the system thermalize.

HBox(children=(FloatProgress(value=0.0, max=1000.0), HTML(value='')))









    




























What we did look for from a practical standpoint in our simulations was power law scaling of the number of iterations for each avalanche in a system, of avalanche size... and we did find that!


Here's another model we implemented, called the Manna model:













In [19]:



    


manna = SOC.Manna(L=30)
manna.run(500, wait_for_n_iters = 500)
manna.animate_states(notebook=True)



    















    






Waiting for wait_for_n_iters=500 iterations before collecting data. This should let the system thermalize.










    








HBox(children=(FloatProgress(value=0.0, max=1000.0), HTML(value='')))










    















    




























This deserves a bit of a longer runtime:













In [23]:



    


manna_long = SOC.Manna(L=30, save_every = 100)
manna_long.run(50000, wait_for_n_iters = 20000)



    















    






Waiting for wait_for_n_iters=20000 iterations before collecting data. This should let the system thermalize.










    








HBox(children=(FloatProgress(value=0.0, max=70000.0), HTML(value='')))










    
























In [26]:



    


manna_long.get_exponent(low = 10, high=100)



    















    

















    






y = 23317.107 exp(-1.3904 x)










    
Out[26]:








{'exponent': -1.3903903386497791, 'intercept': 4.367674673827714}

























You can find the project on GitHub here.


Apparently, SOC has applications basically everywhere, but I haven't dug into those yet. There's are two articles on my to-read list that relate SOC to ELMs in fusion reactors and turbulent transport. I might do writeups on those in a later post.













    
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