Are Forest Fires HOT?

1. Are Forest Fires HOT? A mostly critical look at physics, phase transitions, and universality Jean Carlson, UCSB

2. Background • Much attention has been given to “complex adaptive systems” in the last decade. • Popularization of information, entropy, phase transitions, criticality, fractals, self-similarity, power laws, chaos, emergence, self-organization, etc. • Physicists emphasize emergent complexity via self-organization of a homogeneous substrate near a critical or bifurcation point (SOC/EOC)

3. Criticality and power laws • Tuning 1-2 parameters  critical point • In certain model systems (percolation, Ising, …) power laws and universality iff at criticality. • Physics: power laws are suggestive of criticality • Engineers/mathematicians have opposite interpretation: • Power laws arise from tuning and optimization. • Criticality is a very rare and extreme special case. • What if many parameters are optimized? • Are evolution and engineering design different? How? • Which perspective has greater explanatory power for power laws in natural and man-made systems?

4. Highly Optimized Tolerance (HOT) • Complex systems in biology, ecology, technology, sociology, economics, … • are driven by design or evolution to high-performance states which are also tolerant to uncertainty in the environment and components. • This leads to specialized, modular, hierarchical structures, often with enormous “hidden” complexity, • with new sensitivities to unknown or neglected perturbations and design flaws. • “Robust, yet fragile!”

5. “Robust, yet fragile” • Robust to uncertainties • that are common, • the system was designed for, or • has evolved to handle, • …yet fragile otherwise • This is the most important feature of complex systems (the essence of HOT).

6. Robustness of HOT systems Fragile Fragile (to unknown or rare perturbations) Robust (to known and designed-for uncertainties) Uncertainties Robust

7. Robustness Complexity Interconnection Aim: simplest possible story

8. The simplest possible spatial model of HOT. Square site percolation or simplified “forest fire” model. Carlson and Doyle, PRE, Aug. 1999

9. empty square lattice occupied sites

10. not connected connected clusters

11. connected clusters Draw lattices without lines.

12. 20x20 lattice

13. .4 .6 Density = fraction of occupied sites .8 .2

14. Assume one “spark” hits the lattice at a single site. A “spark” that hits an empty site does nothing.

15. A “spark” that hits a cluster causes loss of that cluster.

16. Think of (toy) forest fires.

17. yield density loss Think of (toy) forest fires.

18. yield density loss Yield = the density after one spark

19. no sparks 1 0.9 0.8 0.7 Y= yield 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1  = density

20. yield = Average over configurations. density=.5

21. no sparks sparks 1 0.9 “critical point” 0.8 Y= (avg.) yield 0.7 0.6 0.5 N=100 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1  = density

22. 1 0.9 “critical point” Y= (avg.) yield 0.8 limit N   0.7 0.6 0.5 0.4 0.3 0.2 c = .5927 0.1 0 0 0.2 0.4 0.6 0.8 1  = density

23. Y Fires don’t matter. Cold 

24. Everything burns. Y Burned 

25. Critical point Y 

26. This picture is very generic and “universal.” Y critical phase transition 

27. Statistical physics: Phase transitions, criticality, and power laws

28. Thermodynamics and statistical mechanics Mean field theory Renormalization group  Universality classes Power laws Fractals Self-similarity “hallmarks”or “signatures” of criticality

29. clusters   log(prob) log(size)  Statistical Mechanics Microscopic models Distributions and correlations Ensemble Averages (all configurations are equally likely)  = correlation length

30. criticality high density low density Renormalization group

31. Fractal and self-similar Criticality

34. Percolation   cluster

35. all sites occupied 1 critical point c 0 1 no  cluster Percolation P( ) = probability a site is on the  cluster P( ) 

36. miss  cluster full density hit  cluster lose  cluster Y  Y =( 1-P() ) + P() (  -P() ) =  - P()2 P() c

37. Tremendous attention has been given to this point. yield density

38. Power laws Criticality cumulative frequency cluster size

39. Average cumulative distributions fires clusters size

40. high density low density cumulative frequency Power laws: only at the critical point cluster size

41. Higher dimensions Other lattices Percolation has been studied in many settings. Connected clusters are abstractions of cascading events.

42. Edge-of-chaos, criticality, self-organized criticality (EOC/SOC) yield Claim: Life, networks, the brain, the universe and everything are at “criticality” or the “edge of chaos.” density

43. Self-organized criticality (SOC) yield Create a dynamical system around the critical point density

44. Self-organized criticality (SOC) Iterate on: • Pick n sites at random, and grow new trees on any which are empty. • Spark 1 site at random. If occupied, burn connected cluster.

45. lattice distribution fire density yield fires

46. 3 10 2 10 1 10 0 10 0 1 2 3 4 10 10 10 10 10 -.15

47. 18 Sep 1998 Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 4 data sets

48. 3 10 SOC FF 2 10 -1/2 1 10 0 10 -2 -1 0 1 2 3 4 10 10 10 10 10 10 10 Exponents are way off

49. yield density Edge-of-chaos, criticality, self-organized criticality (EOC/SOC) • Essential claims: • Nature is adequately described by generic configurations (with generic sensitivity). yield • Interesting phenomena is at criticality (or near a bifurcation).

50. Qualitatively appealing. • Power laws. • Yield/density curve. • “order for free” • “self-organization” • “emergence” • Lack of alternatives? • (Bak, Kauffman, SFI, …) • But... • This is a testable hypothesis (in biology and engineering). • In fact, SOC/EOC is very rare.