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

Summary of Lecture 1 … 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 Boltzmann Gibbs Shannon Jaynes

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:

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 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 Part 2: Applying MaxEnt to ecology

Ln (nutrient concentration) unimodal 1. biodiversity vs. resource supply bacteria laboratory scale (Kassen et al 2000) continental scale (  10 4 km 2 ) (O’Brien et al 1993) monotonic woody plants

Barthlott et al (1999)

Wright (1983) Oikos 41: Species-energy power law angiosperms 24 islands world-wide # species (S) Total Evapotranspiration, E (km 3 / yr)

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

Volkov et al (2005) Nature 438: tropical forests

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

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

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 Part 2: Applying MaxEnt to ecology

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

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

qiqi pipi = information gained about i when using p i instead of q i total ignorance about icontains only the info. C … ensures baseline info = total ignorance Minimize: Constraints C

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 Part 2: Applying MaxEnt to ecology

r1r1 r2r2 rSrS j = species label r j = per capita resource use n j = population n1n1 n2n2 nSnS subject to constraints (C) Maximize Application to ecological communities p(n 1 …n S ) = ? where (Rissanen 1983) microstate

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

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

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

r j (N use per plant) Community nitrogen use, (g N m -2 yr -1 ) Predicted relative abundances Shannon diversity index exp(H n )

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

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

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

mean # species with population n vs. log 2 n

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

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