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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
Why nonequilibrium thermodynamics?
Thermodynamics
Thermodynamics
Thermodynamics (?)



science of temperature
science of macroscopic
energy changes
general framework of any
macroscopic (?)
continuum (?)
theories
General framework:
– fundamental balances
– objectivity - frame indifference
– Second Law
reversibility – special limit
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
Non-equilibrium thermodynamics
a    ja  a
basic balances
– basic state:
– constitutive state:
– constitutive functions:
a  (, v,...)
a
C  (a, a , a,...)
ja (C)
weakly nonlocal
Second law:
s(C)    js (C)  s  0
(universality)
Constitutive theory
Method: Liu procedure, Lagrange-Farkas multipliers
Special: irreversible thermodynamics
Origin of quantum mechanics:
motivation – interpretation – derivation (?)
Is there any?
(Holland, 1993)
– optical analogy
– standard (probability)
– stochastic
– quantized solutions
– de Broglie – Bohm
– de Broglie-Bohm
– stochastic
– hydrodynamic
– Kaniadakis
Justified by the
consequences.
– Frieden-Plastino
(Fisher based)
– Points of views
– Hall-Reginatto
“The Theory of Everything.” – Equivalent
(Laughlin-Pines, 2000)
(for a single particle)
Schrödinger equation:


i

  V ( x)
t
2m
iS


R
e
Madelung transformation:
2
    v  0
 : R
2

v : S
m
de Broglie-Bohm form:
v  (U QM  V )
Hydrodynamic form:
v    PQM   V
 R
 2
2m R
2
U QM
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
(, v ) basic state
    v  0
C ((,,, v,,
v) , v, v)
v    P(C)  0 Cwnl
constitutive state
s(C)   js (C)  0
s(C), js (C), P(C)
constitutive functions
Liu procedure (Farkas’s lemma):
2
2
v v
ss((, ,vv, ) )se (se (), ) 
2 2
js   v  P  ...
r
P reversible pressure
2


1


2
s   P       s  s I   s  : v  0
2
2



Pv  P  P r
Potential form:
  P  UQ
r
U Q    ( se )    (   se )
Euler-Lagrange form
Variational origin
Schrödinger-Madelung fluid
(Fisher entropy)
2
 SchM    v 2

 
sSchM (  , v,  )  
2  2 
2
r
SchM
P
1
2   
2
   I    

8


   eiv
Bernoulli
equation
Schrödinger equation
Landau fluid
sLan (  ,  )   Lan
r
Lan
P
  Lan
( ) 2
2
2
2

I    
2
2

1
 
U Lan   Lan     
2
2 
Alternate fluid
s Alt (  ,  )   Alt
( )
2

 Alt
2

P 
 (I    )     
4
r
Alt
 Alt
U Alt  

2
Korteweg fluids:
r
Kor
P


  p     ( ) I       
2
2
Origin of quantum potential – weakly nonlocal statistics:
Unique under physically reasonable conditions.
– Extensivity (mean, density)
– Isotropy

s(  ,  )  s (  , ( ) 2 )
– Additivity
s( 12 , D( 12 ))  s( 1 , 1 )  s( 2 , 2 )
( )

2
s (  , ( ) )  
 k ln 
2
2

Fisher
Boltzmann-Gibbs-Shannon
Extreme Physical Information (EPI) principle (Frieden, 1998)
–
Mass-scale invariance (particle interpretation)
s( ,  )  s(  ,  )
Two component weakly nonlocal fluid
  1 1  2 2  
  0
    v  0
v    P(C)  0
s(C)    js (C)  0
 density of the solid component

volume distribution function
( , , v ) basic state
C  ( ,  , ,  , v, v)
constitutive state
s(C), js (C), P(C)
constitutive functions
Constraints:
(1), (1), (2), (2), (3)
  s  1 ,
  s  2 ,
Liu equations
 s  3 ,
 s  4 ,
 vs  5 ,
 vs  0,
(  js  5  P)s  0,
(  js  5    P)s  0,
( v js  4I  5   v P)s  0.
 s  0,
  s  0,
 s  0,
     s  0.
isotropic, second order
Solution:
v2
( )2
s(, , , , v)  se (, )  m(, )
 (, )
2
2
js (C )   m( ,  ) v  P(C )  j1 ( ,  , v).
 (mv) : P  (     sI) : v  0
Simplification:
p
m  1, j1 (, , v)  0,  se   2 .

Entropy inequality:


()2
  P  (p   
)I      : v  0
2


Pr
Coulomb-Mohr
P  P r  P v  Lv
isotropy: Navier-Stokes like + ...
Properties
1 Other models:
a) Goodman-Cowin
P r  (p  ()2  2)I  2  
    h  
b) Navier-Stokes type:
configurational force balance
 
somewhere
2 Coulomb-Mohr
N : n  P r  n
S : P r  N
S 2  (N  t ) 2  s 2
s   ( )
2
  

t  p  
  ( ) 2
 2

 1

t  p  1    (ln  )  s
 2

S
unstable
s
t
stable
N
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.
v0
U SchM  U  E  const.
– There is a thermodynamic Ljapunov function:
 v 2 1    2

  U  E dV
L(  , v)      
 2 2  2 



semidefinite in a gradient (Soboljev ?) space