Unique additive information measures – Boltzmann-Gibbs-Shannon, Fisher and beyond Peter Ván BME, Department of Chemical Physics Thermodynamic Research Group Hungary 1.Introduction – the observation of Jaynes 2.Weakly nonlocal additive information measures 3.Dynamics – quantum mechanics 4.Power law tails Microcanonical and canonical equilibrium distributions 5.Conclusions

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)

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)

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

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

– What about higher order derivatives? – What is the physics behind the second term? What could be the value of k 1 ? – What is the physics behind the combination of the two terms? Equilibrium distributions with power law tails Dynamics - quantum systems Dimension dependence

Second order weakly nonlocal information measures –Isotropy in 3D –Extensivity –Additivity

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

Fisher Boltzmann-Gibbs-Shannon –Mass-scale invariance (particle interpretation) – probability distribution 3 Quantum systems – the meaning of k 1

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

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

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

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

There is no analitic solution in general There is no partition function formalism symmetry Numerical normalization One free parameter: J s

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?

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

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

reversible pressure Potential form: Euler-Lagrange form Variational origin

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

Alternate fluid Korteweg fluids: