Transcript Diapositive 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
Summary of Lecture 1 …
Boltzmann
The problem
to predict the behaviour of non-equilibrium
systems with many degrees of freedom
Gibbs
The proposed solution
Shannon
Jaynes
MaxEnt: a general information-theoretical
algorithm for predicting reproducible
behaviour under given constraints
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
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
•
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
1. biodiversity vs. resource supply
laboratory scale
continental scale (104 km2)
(Kassen et al 2000)
(O’Brien et al 1993)
bacteria
woody plants
Ln (nutrient concentration)
unimodal
monotonic
Barthlott et al (1999)
2. Species-energy power law
angiosperms
24 islands world-wide
# species
(S)
SE
0.62
Total Evapotranspiration, E (km3 / yr)
Wright (1983) Oikos 41:496-506
3. Species relative abundances
xn
s ( n) 
nc
Mean # species with
population n
xn

n
x 1
c 1
for large n
(Fisher log-series)
Many rare species
Few common species
Volkov et al (2005) Nature 438:658-661
6 tropical forests
ns (n)
 xn
log2 n
4. Self-thinning power law
Enquist, Brown & West (1998) Nature 395:163-165
m N
4 / 3
Lots of small plants
A few large plants
Can these different ecological patterns
(i.e. reproducible behaviours)
be explained by a single theory ?
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
Predicting reproducible behaviour ….
Constraints C
(e.g. energy
input, space)
System with
many degrees of
freedom (e.g.
ecosystem)
Reproducible
behaviour
(e.g. species
abundance
distribution)
C is all we need to predict reproducible behaviour
pi = probability that system is in microstate i
Macroscopic prediction: Q   pi Qi
i
Incorporate into pi only the information C
MaxEnt
… more generally we use Maximum
Relative Entropy (MaxREnt) …
 pi 
H  p q    pi log 
i
 qi 
 H  p q
= information gained about i
when using pi instead of qi
qi = distribution describing total ignorance about i
 
Maximize H p q w.r.t. pi subject to constraints C
 pi contains only the information C
… ensures baseline info = total ignorance
pi
contains only the info. C
Minimize:
Constraints
C
qi
 pi 
 H  p q    pi log 
i
 qi 
= information gained about i
when using pi instead of qi
total ignorance about i
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
Application to ecological communities
j = species label
rj = per capita resource use
nj = population
rS
nS
p(n1…nS) = ?
Maximize
 p
H  p q     p log 
n j 0
q

subject to constraints (C)
S
S
R   n j rj
N  nj
j 1
j 1
r2
n2
r1
n1
where (Rissanen 1983)
microstate
1
qn1...nS   
j 1 n j  1
S
The ignorance prior
1
qn1...nS   
j 1 n j  1
S
For a continuous variable x  (0,), total ignorance  no scale
Under a change of scale …
x  x  λx
qλxd λx  qxdx
… we are just as ignorant as before (q is invariant)
 λqλx   qx 
1
 qx  
x
qλx   qλx 
the Jeffreys prior
Solution by Lagrange multipliers
(tutorial exercise)
where
probability that species j has abundance n:
mean abundance of species j:
B-E
mean number of species with abundance n:
modified
Bose-Einstein
distribution
Example 1: N-limited grassland community
(Harpole & Tilman 2006)
rj
S = 26 species
(j = 1 …. 26)
N  62 m
2
R  5.9 g N m-2 yr1
Predicted relative abundances
R  5.9 g N m-2 yr1
Shannon diversity index
exp(Hn)
Ropt( pred.)  9.3
1
R  5.9 g N m yr
-2
R
( obs.)
opt
+8
9

+6
+2
+4
rj (N use per plant)
Community nitrogen use,
R (g N m-2 yr-1)
Example 2: Allometric scaling model for rj
rm
per capita
resource use
metabolic
scaling
exponent
α
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
 r j   j  1
α
On longer timescales, S =  and N variable

μ  0
S* = # species with
α = 2/3
nj 1
S=
S*  R
N variable
1 / 1α 
MaxREnt predicts a monotonic
species-energy power law
S*  R
1 / 1α 
α  2 / 3  1 / 1  α  0.60
α  3 / 4  1 / 1  α  0.57
Wright (1983) :
SE
0.62
s (n)  mean # species with population n vs. log2n
n n
ns ( n ) 
x
n 1
For R large, R is partitioned equally among
the different species
n r r  C
cf. Energy Equipartition
of a classical gas
1
n r  
r
rm
α
m  n 
1/ α
α  3 / 4  1 / α  4 / 3
m N
4 / 3
Summary of Lecture 2 …
Boltzmann
ecological patterns =
maximum entropy behaviour
Gibbs
Shannon
Jaynes
the explanation of ecological
patterns is not unique to
ecology