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

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

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

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

Origin of quantum mechanics: motivation – interpretation – derivation (?) Is there any? (Holland, 1993) –optical analogy –quantized solutions –standard (probability) – de Broglie – Bohm – stochastic –hydrodynamic –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) –stochastic –de Broglie-Bohm

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

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)?

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

Landau fluid

Alternate fluid Korteweg fluids:

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

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

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

Constraints: isotropic, second order Liu equations

Solution: Simplification:

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

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

N S t s unstable stable 2 Coulomb-Mohr

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

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