Maximum Entropy, Maximum Entropy Production and their Application to Physics and Biology

1. Maximum Entropy, Maximum Entropy Production and their Application to Physics and Biology Roderick C. Dewar Research School of Biological Sciences The Australian National University

2. Summary of Lecture 1 … Boltzmann The problem to predict the behaviour of non-equilibrium systems with many degrees of freedom The proposed solution MaxEnt: a general information-theoretical algorithm for predicting reproducible behaviour under given constraints Gibbs Shannon Jaynes

3. Part 1: Maximum Entropy (MaxEnt) – an overview • Part 2: Applying MaxEnt to ecology Part 3: Maximum Entropy Production (MEP) • Part 4: Applying MEP to physics & biology Dewar & Porté (2008) J Theor Biol 251: 389-403

4. Part 2: Applying MaxEnt to ecology • The problem: explaining various ecological patterns • - biodiversity vs. resource supply (laboratory-scale) • - biodiversity vs. resource supply (continental-scale) • - the “species-energy power law” • - species relative abundances • - the “self-thinning power law” • The solution: Maximum (Relative) Entropy • Application to ecological communities • - modified Bose-Einstein distribution • - explanation of ecological patterns is not unique to ecology

5. Part 2: Applying MaxEnt to ecology • The problem: explaining various ecological patterns • - biodiversity vs. resource supply (laboratory-scale) • - biodiversity vs. resource supply (continental-scale) • - the “species-energy power law” • - species relative abundances • - the “self-thinning power law” • The solution: Maximum (Relative) Entropy • Application to ecological communities • - modified Bose-Einstein distribution • - explanation of ecological patterns is not unique to ecology

6. Ln (nutrient concentration) 1. biodiversity vs. resource supply laboratory scale (Kassen et al 2000) continental scale (104 km2) (O’Brien et al 1993) woody plants bacteria unimodal monotonic

7. Barthlott et al (1999)

8. 2. Species-energy power law angiosperms 24 islands world-wide # species (S) Total Evapotranspiration, E (km3 / yr) Wright (1983) Oikos 41:496-506

9. 3. Species relative abundances Mean # species with population n for large n (Fisher log-series) Many rare species Few common species

10. Volkov et al (2005) Nature 438:658-661 6 tropical forests

11. 4. Self-thinning power law Enquist, Brown & West (1998) Nature 395:163-165 Lots of small plants A few large plants

12. Can these different ecological patterns (i.e. reproducible behaviours) be explained by a single theory ?

13. Part 2: Applying MaxEnt to ecology • The problem: explaining various ecological patterns • - biodiversity vs. resource supply (laboratory-scale) • - biodiversity vs. resource supply (continental-scale) • - the “species-energy power law” • - species relative abundances • - the “self-thinning power law” • The solution: Maximum (Relative) Entropy • Application to ecological communities • - modified Bose-Einstein distribution • - explanation of ecological patterns is not unique to ecology

14. pi = probability that system is in microstate i Macroscopic prediction: Incorporate into pionly the information C Predicting reproducible behaviour …. System with many degrees of freedom (e.g. ecosystem) Constraints C (e.g. energy input, space) Reproducible behaviour (e.g. species abundance distribution) C is all we need to predict reproducible behaviour MaxEnt

15. Maximize w.r.t. pi subject to constraints C pi contains only the information C … more generally we use Maximum Relative Entropy (MaxREnt) … = information gained about i when using pi instead of qi qi = distribution describing total ignorance about i

16. contains only the info. C total ignorance about i … ensures baseline info = total ignorance pi Minimize: Constraints C = information gained about i when using pi instead of qi qi

17. Part 2: Applying MaxEnt to ecology • The problem: explaining various ecological patterns • - biodiversity vs. resource supply (laboratory-scale) • - biodiversity vs. resource supply (continental-scale) • - the “species-energy power law” • - species relative abundances • - the “self-thinning power law” • The solution: Maximum (Relative) Entropy • Application to ecological communities • - modified Bose-Einstein distribution • - explanation of ecological patterns is not unique to ecology

18. Application to ecological communities p(n1…nS) = ? j = species label rj = per capita resource use nj = population Maximize rS nS subject to constraints (C) r2 n2 where (Rissanen 1983) r1 n1 microstate

19. The ignorance prior For a continuous variable x (0,), total ignorance  no scale Under a change of scale … … we are just as ignorant as before (q is invariant) the Jeffreys prior

20. Solution by Lagrange multipliers (tutorial exercise) B-E where probability that species j has abundance n: mean abundance of species j: modified Bose-Einstein distribution mean number of species with abundance n:

21. Example 1: N-limited grassland community (Harpole & Tilman 2006) rj S = 26 species (j = 1 …. 26)

22. Predicted relative abundances Shannon diversity index exp(Hn) +8 +6 Community nitrogen use, (g N m-2 yr-1) +2 +4 rj (N use per plant)

23. Example 2: Allometric scaling model for rj metabolic scaling exponent per capita resource use adult mass West et al. (1997) : α = 3/4 Demetrius (2006) : α = 2/3 Let’s distinguish species according to their adult mass per individual

24. On longer timescales, S =  and S* = # species with α = 2/3 S = 

25. MaxREnt predicts a monotonic species-energy power law Wright (1983) :

26. mean # species with population n vs. log2n

27. For large, is partitioned equally among the different species cf. Energy Equipartition of a classical gas

28. Summary of Lecture 2 … Boltzmann • ecological patterns = maximum entropy behaviour • the explanation of ecological patterns is not unique to ecology Gibbs Shannon Jaynes