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One component fluid mechanics - quantum (?) fluids Quantum potential Why Fisher information ?

Weakly nonlocal fluid mechanics Peter Ván Budapest University of Technology and Economics, Department of Chemical Physics. One component fluid mechanics - quantum (?) fluids Quantum potential Why Fisher information ? Two component fluid mechanics – sand (?) Conclusions.

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One component fluid mechanics - quantum (?) fluids Quantum potential Why Fisher information ?

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  1. Weakly nonlocal fluid mechanicsPeter VánBudapest University of Technology and Economics, Department of Chemical Physics • One component fluid mechanics - quantum (?) fluids • Quantum potential • Why Fisher information? • Two component fluid mechanics – sand (?) • Conclusions

  2. Why nonequilibrium thermodynamics? science of temperature Thermodynamics science of macroscopic energy changes Thermodynamics • general framework of any • Thermodynamics (?) macroscopic (?) • continuum (?) • theories • General framework: • fundamental balances • objectivity - frame indifference • Second Law reversibility – special limit

  3. Phenomenology– minimal or no microscopic information universality Second Law– “super-principle” – valid for all kind of dynamics – like symmetries Beyond local equilibrium – memory and inertia Beyond local state – nonlocality weak–short range - not gravity – higher order gradients

  4. weakly nonlocal Non-equilibrium thermodynamics basic balances • basic state: • constitutive state: • constitutive functions: Second law: (universality) Constitutive theory Method: Liu procedure, Lagrange-Farkas multipliers Special: irreversible thermodynamics

  5. Origin of quantum mechanics: motivation – interpretation – derivation (?) Is there any? (Holland, 1993) • optical analogy • quantized solutions • standard (probability) • de Broglie – Bohm • – stochastic • hydrodynamic • stochastic • de Broglie-Bohm • Kaniadakis • Frieden-Plastino • (Fisher based) • Hall-Reginatto Justified by the consequences. “The Theory of Everything.” (Laughlin-Pines, 2000) • Points of views • Equivalent • (for a single particle)

  6. Schrödinger equation: Madelung transformation: de Broglie-Bohm form: Hydrodynamic form:

  7. Fundamental questions in quantum mechanics: • Why we need variational principles? • (What is the physics behind?) • Why we need a wave function? • (What is the physics behind?) • Where is frame invariance(objectivity)?

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

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

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

  11. Landau fluid

  12. Alternate fluid Korteweg fluids:

  13. Origin of quantum potential – weakly nonlocal statistics: Unique under physically reasonable conditions. • Extensivity (mean, density) • Isotropy • Additivity

  14. Fisher Boltzmann-Gibbs-Shannon Extreme Physical Information (EPI) principle (Frieden, 1998) • Mass-scale invariance (particle interpretation)

  15. Two component weakly nonlocal fluid density of the solid component volume distribution function basic state constitutive state constitutive functions

  16. Constraints: isotropic, second order Liu equations

  17. Solution: Simplification:

  18. Entropy inequality: Pr Coulomb-Mohr isotropy: Navier-Stokes like + ...

  19. Properties 1 Other models: a) Goodman-Cowin configurational force balance b)Navier-Stokes type: somewhere

  20. S unstable stable s t N 2 Coulomb-Mohr

  21. Conclusions • Weakly nonlocal statistical physics • Universality (Second Law – super-principle) • independent of interpretation • independent of micro details phenomenological background behind any statistical-kinetic theory (Kaniadakis - kinetic, Frieden-Plastino - maxent) • Method - more theories/models • Material stability

  22. Thermodynamics = theory of material stability e.g. phase transitions (gradient systems?) • What about quantum mechanics? • There is a meaning of dissipation. • There is a family of equilibrium (stationary) solutions. • There is a thermodynamic Ljapunov function: semidefinite in a gradient (Soboljev ?) space

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