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

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

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

Part 1: MaxEnt – an overview • The problem: to predict “complex system” behaviour • The solution: statistical mechanics • - Boltzmann microstate counting (maximum probability) • - Gibbs algorithm (MaxEnt) • Applications of MaxEnt to equilibrium systems • - micro-canonical, canonical, grand-canonical distributions • Physical interpretation of MaxEnt • - frequency interpretation • - information theory interpretation (Jaynes) • Extension to non-equilibrium systems (Jaynes) • General properties of MaxEnt distributions

What is the problem? - to predict the macroscopic behaviour of systems having many interacting degrees of freedom cells, plants, ecosystems, economies, climates … many interacting degrees of freedom energy out energy in matter in matter out system open non-equilibrium environment

Poleward heat transport 170 W m-2 300 W m-2 Latitudinal heat transport H = ? SW T LW

Cold plate, Tc convection conduction Hot plate, Th Turbulent heat flow (Raleigh-Bénard convection) Ra < 1760 Ra > 1760 Cold plate, Tc T H = ? Hot plate, Th

Flw + H + E C, H20, O2, N Fsw Ecosystem energy & mass fluxes T, 

Global Circulation Models, Dynamic Ecosystem Models …. What is the problem? - to predict the macroscopic behaviour of systems having many interacting degrees of freedom cells, plants, ecosystems, economies, climates … many interacting degrees of freedom energy out energy in matter in matter out system open non-equilibrium environment many degrees of freedom  statistical mechanics

Boltzmann microstate counting Microstate i1 Macrostate A = less detailed description Microstate i2 W(A) = number of microstates that give macrostate A The most probable macrostate A is the one with the largest W(A) (assume microstates are a priori equiprobable) Ludwig Boltzmann (1844 - 1906) SB(A) = kBlog W(A) = Boltzmann entropy of macrostate A The most probable macrostate is the one of maximum entropy

ε3 ε2 ε1 Example: N independent distinguishable particles with fixed total energy E Microstate i = {the mthparticle is in state jm} Macrostate A = {njparticles are in state j} : maximise S = kBlog W subject to (large N)

Boltzmann entropy  Clausius entropy Given E, Smax = kBlogWmax = kB(βE + NlogZ)  underδE = δQ, Smax changes by δSmax = kBβ(δQ) cf. Clausius thermodynamic entropyδSTD = δQ/T Smax STD β1/kBT BUT: microstate counting only works for non-interacting particles

The Gibbs algorithm (MaxEnt) Maximise H = -ipi log pi with respect to {pi} subject to the constraints (C) on the system Gibbs algorithm pi = probability that system is in microstate i Macroscopic predictions via But how do we constructpi ? J Willard Gibbs (1839 - 1903) ‘minimise the index of probability of phase’

Three applications of MaxEnt (equilibrium systems) Distribution (pi) System constraints (C) • Closed, isolated • Closed • Open • Microcanonical • Canonical • Grand-canonical

Example 1: closed, isolated system in equilibrium Microstate i = any N-particle state with total energy Eirestricted to E Precise description of i and Ei depends on microscopic physics (CAN include particle interactions) C: N and E fixed Maximise subject to basis for Boltzmann’s microstate counting

C: N and fixed Example 2: closed system in equilibrium E Microstate i = any N-particle state (no restriction on Ei) Maximise subject to Hmax STD β 1/kBT

C: and fixed Example 3: open system in equilibrium E N Microstate i = any physically allowed microscopic state (no restriction on Ei or Ni) Maximise subject to Hmax STD β 1/kBT γ -μ/kBT

Frequency interpretation (Venn, Pearson, Fisher …) pi describes a physical property of the real world (frequency) System has Ωa priori equiprobable microstates N independent identical systems, ni = no. of systems in state i pi =ni /N = frequency of microstate i W = no. of microstates giving {n1,n2 … nΩ} MaxEnt coincides with large-N limit of Maximum Probability for multinomial W

minimum uncertainty : pi = 0 (i = 1,2...5), p6 = 1 H = 0 maximum uncertainty : pi = 1/6 (i = 1…6) H = log 6 Information theory interpretation (Shannon 1948, Jaynes 1957 …) • pi represents our state of knowledge of the real world • basic axioms foruncertainty H associated with pi •  the unique uncertainty function is • Applies to any discrete set of outcomes i Claude Shannon (1916-2001)

Q (= ΣipiQi) reproducible under C it is sufficient to encode only the information Cinto pi … … but this is precisely what MaxEnt does! H = -ipi log pi = missing information about i MaxEnt = max H subject to C all information other than C is thrown away Jaynes (1957b, 1978) Behaviour that is experimentally reproducible under conditions C must be theoretically predictable from Calone Edwin Jaynes (1922-1998)

The prediction game PREDICTION ESSENTIAL PHYSICS Reproducible behaviour Q Max H subject to C  pi Assumed constraints C experimental conditions conservation laws microstates (e.g. QM) MaxEnt test C  C' OBSERVATION Observed behaviour Qobs QobsQ  missing constraint

Information theory interpretation of MaxEnt general algorithm for predicting reproducible behaviour under given constraints can be extended to non-equilibrium systems (same principle, different constraints) ‘Maximum caliber principle’ (Jaynes 1980, 1996) Edwin Jaynes (aged 14 months) cf. Feynman path integral formalism of QM!

The second law in a nutshell WB after Jaynes 1963, 1988 A B reproducible macroscopic change . . S = kBlog W microscopic path in phase-space WA' WA AB reproducible  WB WA' = WA  SB  SA Liouville Theorem (Hamiltonian dynamics)

Some general properties of MaxEnt distributions subject to m + 1 constraints C Partition function: Orthogonality: Constitutive relation: Response-fluctuation & reciprocity relations: Stability-convexity relation:

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

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