Weakly nonlocal fluid mechanicsPeter 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 anyThermodynamics (?) 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

aa ja basic balances ,...),( va

– basic state:– constitutive state:– constitutive functions:

a

)C(aj,...),,(C aaa

weakly nonlocalSecond law:

0)C()C(s ss j

Constitutive theory

Method: Liu procedure, Lagrange-Farkas multipliersSpecial: 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:

)(

2

2

xVmt

i

Madelung transformation:iSeR

Sm

:v2: R

de Broglie-Bohm form:

)( VUQM v

R

R

mUQM

2

2

2

Hydrodynamic form:

VQM Pv

R

R

mUQM

2

2

2

0 v

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

),,,(C vv ),,,,(Cwnl vv

)C(),C(),C(s Pjs

Liu procedure (Farkas’s lemma):

constitutive state

constitutive functions

0 v

0)C()C(s s j0Pv )C(

... Pvjs2

)(s),(s2

e

vv

2),(s),,(s

2

e

vv

),( v basic state

0:s2

ss2

1 22

s

vIP

rv PPP

reversible pressurerP

Potential form: Qr U P

)()( eeQ ssU Euler-Lagrange form

Variational origin

Schrödinger-Madelung fluid

222),,(

22v

v

SchM

SchMs

2

8

1 2rSchM IP

(Fisher entropy)

Bernoulli equation

Schrödinger equation

v ie

Landau fluid

2

)(),(

2 LanLans

22

2 IP Lan

rLan

22

1 2 LanLanU

Alternate fluid

2)(),(

AltAlts

)(

42IP Altr

Alt

2Alt

AltU

Korteweg fluids:

22)( IP prKor

– Isotropy

))(,(),( 2 ss

– Extensivity (mean, density)

– Additivity

),(),())(,( 22112121 ssDs

Unique under physically reasonable conditions.

Origin of quantum potential – weakly nonlocal statistics:

ln

)())(,(

2

22 ks

FisherBoltzmann-Gibbs-Shannon

Extreme Physical Information (EPI) principle (Frieden, 1998)

– Mass-scale invariance (particle interpretation)

),(),( ss

Two component weakly nonlocal fluid

2211density of the solid componentvolume distribution function

),,( v

),,,,,( vv C

constitutive functions

)C(),C(),C(s s Pj

basic state

constitutive state

00 v

0Pv )C(0)C()C(s s j

Constraints: )3(),2(),2(),1(),1(

.)(

,)(

,)(

,s

,s

,s

,s

,s

,s

s54s

s5s

s5s

5

4

3

2

1

0PIj

0Pj

0Pj

0

vv

v

v

.s

,s

,s

,s

0

0

0

0

isotropic, second order

Liu equations

Solution:

2

)(),(

2),(m),(s),,,,(s

22

e

vv

).,,()(),()( 1 vjPvj CmCs

Simplification:

0:)s(:)m( vIPv

.p

s,),,(,1m2e1

0vj

0:)2

)(p(

2

vIP

Pr

Coulomb-Mohr

vLPPP vr

isotropy: Navier-Stokes like + ...

Entropy inequality:

Properties

1 Other models: a) Goodman-Cowin

2)2)(p( 2r IP

h configurational force balance

b) Navier-Stokes type: somewhere

2)( s

2)(2

pt

spt

)(ln

2

11

N

S

t

s

unstable

stable

2 Coulomb-Mohr

nPnN r: NPS r:

222 )( stNS

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 stabilitye.g. phase transitions (gradient systems?)

What about quantum mechanics?

– There is a meaning of dissipation.– There is a family of equilibrium (stationary) solutions.

0v .constEUU SchM – There is a thermodynamic Ljapunov function:

dVEUL

22

22

1

2),(

v

v

semidefinite in a gradient (Soboljev ?) space