Maximum Entropy,
Maximum Entropy Production
and their
Application to Physics and Biology
Roderick C. Dewar
Research School of Biological Sciences
The Australian National University
Recall Lecture 3 …
Boltzmann
MaxEnt applied to non-equilibrium systems :
• maximum irreversibility
Gibbs
• steady-state flux selection by MEP
• macroscopic dynamics
Shannon
Jaynes
F0
max H
max I
FΩ
uV
Max H :
H  p    pΓ log pΓ
Subject to :
Γ pΓ = 1
Γ pΓ fΓ = F
Γ pΓuΓ = u
uΓ /t = –fΓ + QΓ
Γ
 τ
pΓ 
exp
Z path
 2k B
1
normalisation
flux  Ω
density  V
continuity equation

 1  Q Γ  
 f Γ    

V
T  T  
 pΓ  τ
I ( p)   p Γ log   
 p~  k
Γ
B
 Γ

(Dewar 2003)
 1  Q τ F
 
  dS
T  T  k B ΩT
 F  
V
= entropy production in V = entropy export across Ω
Max I:
MEP
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
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
Poleward heat transport
170 W m-2
Latitudinal
heat
transport
H=?
T
300 W m-2
SW
LW
cT24
Fsw cT14
Fsw=cT14+H
Equatorial
zone
Polar zone
H=?
T1
 1 1
EPmatter  H    
 T2 T1 
T2
H = DΔT
H=cT24
Trade-off between H and ΔT (Earth)
80
60
ΔT
EP
H
DMEP = 1.4
40
DEarth ≈ 1.7
20
0
0.01
0.1
1
10
100
Inter-zonal thermal diffusivity
D (W m-2 K-1)
MEP works elsewhere
T1
T0
T1
T0
EP
EP
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
Paltridge (1978) :
10-zone climate model
EPmatter 
10

zonei 1
 1
1
Fi 
 
 Ti1 Ti 
EPradiation ?
Planetary rotation rate ?
N pole
SWi
LWi
Equator
S pole
Fi
Ti
θi
Zonal temperature and cloud cover
Ti
θi
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
Raleigh-Bénard convection: MEP = max flux
(Ozawa et al 2001, after Malkus 1954)
1 1

F
 1  Q
EP   F         dS  F     F
V
T  T  ΩT
 Tc Th 
gαΔT d 3
Ra 
 Ra *  1708
κν
Cold plate, Tc
diffusion
d
F is maximum when the boundary
layer is marginally stable:
δ
convection
1
3
 ΔT   Ra
Nu  F /  k


d
Ra
*

 

F
diffusion
Hot plate, Th=Tc+ΔT
gαΔT 2δ 3
Ra b 
 Ra *
κν
ΔT / 2
F  Fb  k
δ
δ
M: slope = 1/3
(max flux)
1 1

F
 1  Q
EP   F         dS  F     F
V
T  T  ΩT
 Tc Th 
(max flux)
Ozawa et al. (2001) after Malkus, Busse

τΔU
 1  Q 1
EP   F        Q 
τ
T
T
T
T
V
V
 

M: slope = 1
M: slope = 1/3
(max flux)
1 1

F
 1  Q
EP   F         dS  F     F
V
T  T  ΩT
 Tc Th 
(max flux)
Ozawa et al. (2001) after Malkus, Busse

τΔU
 1  Q 1
EP   F        Q 
τ
T
T
T
T
V
V
 

M: slope = 1
Global entropy
production (mW m-2 s-1)
Tuning GCM parameters using MEP
(Kleidon et al. 2006)
total
vertical
horizontal
k = 0.4
von Karman parameter, k
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
Different growth morphologies are labelled by their
Miller indices <111>, <110> …...
….. the 3D orientations of the different crystal faces
that are growing :
<111>
<110>
Hill (1990) : crystallization of NH4Cl
F = L  (X - X0)
EP<110>
XMEP = 0.21
Xobs = 0.216
force X = liq - solid
EP = F  X
flux F
EP<111>
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
2-state chlorophyll model:
Juretić & Županović (2003)
optimal quantum yield = 97%
power transfer efficiency = 91%
High efficiency ( 90%) is due to
5-state chlorophyll model:
non-linear flux-force relation
cf. linear flux-force relations:
50% power transfer efficiency
optimal quantum yield = 94.6%
power transfer efficiency = 87.8%
MEP applications
• Horizontal heat flows on Earth, Mars and Titan (Lorenz et al. 2001)
• Horizontal heat flows and cloud cover on Earth (Paltridge 1975, 1978,
1981; O’Brien & Stephens 1993, 1995)
• Ocean thermohaline circulation (Shimokawa & Ozawa 2002)
• Rayleigh-Bénard convection (Malkus 1954)
• Shear turbulence (Malkus 1956, Busse 1970)
• Mantle convection (Lorenz 2002)
• Crystal growth morphology (Hill 1990)
• Energy dissipation in ecosystems (Schneider & Kay 1994)
• Photosynthetic free energy transduction (Juretić et al. 2003)
• Evolutionary optimisation of ATP synthase (Dewar et al. 2006)
F0F1-ATP synthase : Nature’s smallest rotary motor
pmf-driven H+ transport  γ stalk torsion  ATP synthesis
Key functional parameter :
κ = angular position of γ at which ADP+Pi  ATP (motor timing)
Transition rates between the 5 open (O) states of F1 were
calculated using the kinetic model of Panke & Rumberg (1999)
O:
O:ATP
JATP
O:ADP
O:ADP+Pi
O:Pi
Transition rates between the 5 open (O) states of F1 were
calculated using the kinetic model of Panke & Rumberg (1999)
O:
O:ATP
5
S state    pi log pi
i 1
JATP
O:ADP
EPATP
O:ADP+Pi
 J  ATP 

 RJ ATP log  

 J ATP 
O:Pi
MaxEnt predicts observed kinetic design of ATPase
Sstate and EPATP : simultaneous
maxima at κ = 0.598
XATP
(102 J mol-1)
JATP (s-1)
(κempirical fit  0.6)
Sstate (102)
EPATP
(10 J K-1 mol-1 s-1)
Relative angular position of γ at which ADP+Pi  ATP (κ)
MaxEnt and MEP …what next?
Theory
•
Boltzmann
MaxEnt basis of MEP :
info theory vs. max probability (N )
Applications in Global Change Science and beyond ….
Gibbs
Shannon
•
Climate and climate change :
EPwater, cloud & water vapour feedbacks
•
Plant and ecosystem responses to climate change :
MEP = plant optimisation (“survival of the likeliest”)
•
Climate-biosphere feedbacks :
MEP = Gaia for grown-ups
•
Other complex, non-equilibrium systems :
e.g. plasmas, economies, networks
Annual MEP Workshops
•
Jaynes
Bordeaux (2003-05), Split (2006), Jena (2007-09) ….