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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
– 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
Properties
1 Other models: a) Goodman-Cowin
2)2)(p( 2r IP
h configurational force balance
b) Navier-Stokes type: somewhere
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
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