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

2. 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 & Maritan (in preparation)

3. The problem: to predict non-equilibrium fluxes from given constraints The solution: apply MaxEnt to microscopic paths (Jaynes) - irreversibility in information-theoretical terms - MaxEnt implies maximum irreversibility Max irreversibility  MEP (steady states) Max irreversibility  Fokker-Planck equation (dynamics) Part 3: Maximum Entropy Production (MEP)

4. The problem: to predict non-equilibrium fluxes from given constraints The solution: apply MaxEnt to microscopic paths (Jaynes) - irreversibility in information-theoretical terms - MaxEnt implies maximum irreversibility Max irreversibility  MEP (steady states) Max irreversibility  Fokker-Planck equation (dynamics) Part 3: Maximum Entropy Production (MEP)

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

6. Cold plate, T c Hot plate, T h Ra < 1760 conduction TT Cold plate, T c Hot plate, T h Ra > 1760 convection H = ? Turbulent heat flow (Raleigh-Bénard convection)

7. F sw F lw + H + E C, H 2 0, O 2, N  T,  Ecosystem energy & mass fluxes

8. From among all those possible flux patterns compatible with the constraints, which one is reproducibly selected?

9. The problem: to predict non-equilibrium fluxes from given constraints The solution: apply MaxEnt to microscopic paths (Jaynes) - irreversibility in information-theoretical terms - MaxEnt implies maximum irreversibility Max irreversibility  MEP (steady states) Max irreversibility  Fokker-Planck equation (dynamics) Part 3: Maximum Entropy Production (MEP)

10. t = 0 t = τ Γ = microscopic path of system + environment Path entropy = path Γ followed in reverse

11. t = 0 t = τ Γ = microscopic path of system + environment Path entropy Irreversibility = path Γ followed in reverse (equilibrium)

12. I(p) is finite, t = 0 t = τ Γ = microscopic path of system + environment Path entropy Irreversibility Physical constraints C = path Γ followed in reverse (equilibrium) i.e. macroscopic fluxes can run in reverse

13.  possible flux 1. Maximise subject to  normalisation  physical constraints C 2. Maximise w.r.t. F  reproducible flux w.r.t. Step 1  To select F, maximise H in two steps …

14. Step 1

15.  possible flux 1. Maximise subject to  normalisation  physical constraints C w.r.t. if then if then The Kuhn-Tucker optimisation conditions:

16. Step 2

17. 2. Maximise w.r.t. F  reproducible F λ = F = 0 i.e. equilibrium When μ = 0 we get Therefore non-equilibrium systems (F  0) must satisfy μ > 0 i.e. maximum irreversibility F  0 max S max I (cf. Boltzmann)

18. The problem: to predict non-equilibrium fluxes from given constraints The solution: apply MaxEnt to microscopic paths (Jaynes) - irreversibility in information-theoretical terms - MaxEnt implies maximum irreversibility Max irreversibility  MEP (steady states) Max irreversibility  Fokker-Planck equation (dynamics) Part 3: Maximum Entropy Production (MEP)

19. SW LW Latitudinal heat transport F = ? 170 W m -2 300 W m -2 TT F  0 t = 0t = τ micropath Γ

20.  Γ p Γ = 1 normalisation  Γ p Γ f Γ = F flux  Ω  Γ p Γ u Γ = u density  V  u Γ /  t = –  f Γ conservation law Max H : Subject to : (Dewar 2003) FΩFΩ uVuV Max I : F  0 max H max I = entropy production MEP

21. heat flow   (1/T) stress  strain mass flow   (-μ/T) reaction rate  affinity EP Γ =  V Σ flux Γ  force Thermodynamic EP emerges from constraint of energy & mass balance.... from energy balance from mass balance.... and this is why MEP is so general Dewar (2003)

22. The problem: to predict non-equilibrium fluxes from given constraints The solution: apply MaxEnt to microscopic paths (Jaynes) - irreversibility in information-theoretical terms - MaxEnt implies maximum irreversibility Max irreversibility  MEP (steady states) Max irreversibility  Fokker-Planck equation (dynamics) Part 3: Maximum Entropy Production (MEP)

23. ‘Bubble dynamics’ (cf. Jaynes 1996) Macroscopic state of the system : ‘bubble’ of probability at time t in the space of macrostates X

24. Maximum irreversibility  macroscopic dynamics System statewith probability can be expressed in terms of p(X,t) and v(X,t) Max I w.r.t. v(X,t) subject to constraints  reproducible v(X,t)  temporal evolution of p(X,t) under C Probability current: Conservation of probability : Conditional state velocity : at time t Path probability:

25. Maximise I w.r.t. v(X, t) subject to Onsager driftdiffusion Example: Gaussian fluctuations (near equilibrium) t t +dt Non-eq. forcing  Fokker-Planck equation:  p/  t = -  (pv)

26. How does the MEP bubble evolve? Gaussian ‘Entropy hill function’ : Fluctuation- driven climb up entropy hill Bubble size adjusts to local entropy curvature

27. ‘Entropy hill function’ : Fluctuation- driven climb up entropy hill Bubble size adjusts to local entropy curvature Example : How does the MEP bubble evolve?

28. Summary of Lecture 3 … Boltzmann Gibbs Shannon Jaynes MaxEnt (reproducible behaviour)  systems arbitrarily far from equilibrium obey maximum irreversibility (Max I) Max I governs selection of non- equilibrium steady states (MEP) and macroscopic dynamics (e.g. Fokker-Planck)