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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 …. Boltzmann. The problem - PowerPoint PPT Presentation
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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: 389-403
• 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 (104 km2) (O’Brien et al 1993)
monotonic
woody plants
Barthlott et al (1999)
Wright (1983) Oikos 41:496-506
2. Species-energy power law
62.0ES
angiosperms
24 islands world-wide
# species (S)
Total Evapotranspiration, E (km3 / yr)
3. Species relative abundances
cn
xns
n
)( 1c
Mean # species with population n
1xMany rare species
Few common species
n
xn for large n
(Fisher log-series)
Volkov et al (2005) Nature 438:658-661
)(nsn
n2log
6 tropical forests
nx
Enquist, Brown & West (1998) Nature 395:163-165
4. Self-thinning power law
3/4 Nm
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)
pi = probability that system is in microstate i Macroscopic prediction:
Incorporate into pi only the information C
i
iiQpQ
MaxEnt
… more generally we use Maximum Relative Entropy (MaxREnt) …
i i
ii q
ppqpH log
qpH = information gained about i when using pi instead of qi
qi = distribution describing total ignorance about i
qpHMaximize w.r.t. pi subject to constraints C
pi contains only the information C
qi
pi
i i
ii q
ppqpH log
= information gained about i when using pi instead of qi
total ignorance about i
contains 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
r1
r2
rS
j = species labelrj = per capita resource use nj = population
n1
n2
nS
S
jjjrnR
1
S
jjnN
1
subject to constraints (C)
Maximize
q
ppqpH
jnlog
0
Application to ecological communities
p(n1…nS) = ?
where (Rissanen 1983)
S
j jS nnnq
11 1
1 ...microstate
The ignorance prior
S
j jS nnnq
11 1
1 ...
dxxqxdxq λλxxx λ
xqxq λλ
xqxq λλ x
xq1
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)
2m 62 N
S = 26 species (j = 1 …. 26)
1-2 yrm N g 5.9 R
rj
1-2 yrm N g 5.9 R
+2+4
+6
+81-2 yrm N g 5.9 R
rj (N use per plant)
Community nitrogen use, (g N m-2 yr-1)R
9)obs.(opt R
3.9)pred.(opt R
Predicted relative abundances
Shannon diversity index exp(Hn)
Example 2: Allometric scaling model for rj
Demetrius (2006) : α = 2/3
αmr
West et al. (1997) : α = 3/4
per capita resource use
adult mass
metabolic scaling exponent
α1 jrj
Let’s distinguish species according to their adult mass per individual
variable N
S =
α = 2/3
On longer timescales, S = and
variable N
S* = # species with 1jn 0μ
α1/1 * RS
MaxREnt predicts a monotonic species-energy power law
α1/1 * RS
60.0α1/13/2α
62.0ES
Wright (1983) :
57.0α1/14/3α
mean # species with population n vs. log2n
nxn
nnsn
1)(
)(ns
3/4 Nm
For large, is partitioned equally among the different species
Crrn cf. Energy Equipartition of a classical gas
r
rn1
αmr
α/1 nm
3/4α/14/3α
R R
Summary of Lecture 2 …
Boltzmann
Gibbs
Shannon
Jaynes
ecological patterns = maximum entropy behaviour
the explanation of ecological patterns is not unique to ecology