Unique Additive Information Measures: Boltzmann-Gibbs-Shannon, Fisher and Beyond

1. Unique additive information measures – Boltzmann-Gibbs-Shannon, Fisher and beyond Peter Ván BME, Department of Chemical Physics Thermodynamic Research Group Hungary • Introduction – the observation of Jaynes • Weakly nonlocal additive information measures • Dynamics – quantum mechanics • Power law tails • Microcanonical and canonical equilibrium distributions • Conclusions

2. 1 Entropy in local statistical physics (information theoretical, predictive, bayesian) (Jaynes, 1957): The measure of information is unique under general physical conditions. (Shannon, 1948; Rényi, 1963) • Extensivity (density) • Additivity (unique solution)

3. The importance of being unique: Robustness and stability: link to thermodynamics. • Why thermodynamics? • thermodynamics is not the theory of temperature • thermodynamics is not a generalized energetics • thermodynamics is a theory of STABILITY • (in contemporary nonequilibrium thermodynamics) • The most general well-posedness theorems use explicite thermodynamic methods • (generalizations of Lax idea) • The best numerical FEM programs offer an explicite thermodynamic structure • (balances, etc…) Link to statistical physics: universality (e.g. fluid dynamics)

5. 2 Entropy in weakly nonlocal statistical physics (estimation theoretical, Fisher based) (Fisher, …., Frieden, Plastino, Hall, Reginatto, Garbaczewski, …): • Extensivity • Additivity • Isotropy

6. (unique solution) Boltzmann-Gibbs-Shannon Fisher There is no estimation theory behind!

7. What about higher order derivatives? • What is the physics behind the second term? What could be the value of k1? • What is the physics behind the combination of the two terms? Dimension dependence Dynamics - quantum systems Equilibrium distributions with power law tails

8. Second order weakly nonlocal information measures • Extensivity • Additivity • Isotropy in 3D

9. (unique solution) Depends on the dimension of the phase space.

10. 3 Quantum systems – the meaning of k1 probability distribution Fisher Boltzmann-Gibbs-Shannon • Mass-scale invariance (particle interpretation)

11. Why quantum? A Time independent Schrödinger equation B Probability conservation Momentum conservation + Second Law (entropy inequality) Equations of Korteweg fluids with a potential

12. (Fisher entropy) Schrödinger-Madelung fluid Bernoulli equation Schrödinger equation

13. Logarithmic and Fisher together? Nonlinear Schrödinger eqaution of Bialynicki-Birula and Mycielski (1976) (additivity is preserved): Stationary equation for the wave function: Existence of non dispersive free solutions – solutions with finite support - Gaussons

14. 4 MaxEnt calculations – the interaction of the two terms (1D ideal gas)

15. symmetry There is no analitic solution in general There is no partition function formalism Numerical normalization One free parameter: Js

17. Conclusions • Corrections to equilibrium (?) distributions • Corrections to equations of motion • Unique = universal • Independence of micro details • General principles in background Thermodynamics Statistical physics Information theory?

18. Concavity properties and stability: Positive definite If k is zero (pure Fisher): positive semidefinite!

19. One component weakly nonlocal fluid basic state constitutive state constitutive functions Liu procedure (Farkas’s lemma):

20. reversible pressure Potential form: Euler-Lagrange form Variational origin

21. (Fisher entropy) Schrödinger-Madelung fluid Bernoulli equation Schrödinger equation

22. Alternate fluid Korteweg fluids: