SUPERSATURATION NUCLEATION CRYSTAL GROWTH CRYSTAL Non equilibrium Equilibrium.
Weakly nonlocal non-equilibrium thermodynamics Peter Ván Budapest University of Technology and...
20
Weakly nonlocal non- equilibrium thermodynamics Peter Ván Budapest University of Technology and Economics, Department of Chemical Physics – Non-equilibrium thermodynamics • Beyond local equilibrium and local state • Exploitations of the Second Law - Liu procedure – Examples • Guyer-Krumhansl equation • Ginzburg-Landau equation • One component fluid mechanics - quantum (?) fluids – Conclusions
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Transcript of Weakly nonlocal non-equilibrium thermodynamics Peter Ván Budapest University of Technology and...
Weakly nonlocal non-equilibrium thermodynamics
Peter VánBudapest University of Technology and Economics,
Department of Chemical Physics
– Non-equilibrium thermodynamics • Beyond local equilibrium and local state
• Exploitations of the Second Law - Liu procedure
– Examples• Guyer-Krumhansl equation
• Ginzburg-Landau equation
• One component fluid mechanics - quantum (?) fluids
– Conclusions
Classical Irreversible
Thermodynamics
Local equilibrium (~ there is no microstructure)
Beyond local equilibrium (nonlocality):
•in time (memory effects)•in space (structure effects)
dynamic variables?
Method: local state and beyond …
Space Time
Strongly nonlocal
Space integrals Memory functionals
Weakly nonlocal
Gradient dependent
constitutive functions
Rate dependent constitutive functions
Relocalized
Current multipliers Internal variables
Nonlocalities:
Restrictions from the Second Law.
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 procedureSpeciality: constructive
(universality)
(and more)
Weakly nonlocal extended thermodynamics
),,,,( 2qqq uu
)js,,( sG
Liu procedure (Farkas’s lemma):
),( qus
),,( qqj us
0 sGsuss qqj
state space
constitutive functions
0 qut
0 st s j0 Gtq
solution?
local state:
qqmqq ),(2
1)(),( 0 uusus
qqqBqqj ),,(),,( uus
extended (Gyarmati) entropy
entropy current (Nyíri)(B – current multiplier)
0)(:)( qmBqIB
Gsus
qqmB 2221 LLG qqIB 1211 LLsu
qqIqqqm 222111211 )( LLsLLt
balance? local?
Ginzburg-Landau (variational):
dVfF ))(2
)(()( 2
))('( fl
Fl
– Second Law– Variational (!)– k
dVfF ))(2
)(()( 2 )('fF
Fl
Weakly nonlocal internal variables
Ginzburg-Landau (thermodynamic, relocalized)
),,( 2
)js,,( sF
Liu procedure (Farkas’s lemma)
)(s
0' Fsss j
state space
constitutive functions
0 Ft0 st s j
),( sj
?
local state
ss ),(),( Bj
0')(' sFss BB
)( 2211 sLsLt
'' 2221 sLsLF B
'' 1211 sLsL B
isotropy
))('( fl
current multiplier
Ginzburg-Landau (thermodynamic, non relocalizable)
0 Ft
0 st s j
),,( 2
)js,,( sF
Liu procedure (Farkas’s lemma)
),(s (.)),((.) 0 Fsjjs
0)( Fsss
)( ssLt
state space
constitutive functions 0 Ft
One component weakly nonlocal fluids
and quantum mechanics
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
One component weakly nonlocal fluid
),,,(C vv ),,,,( vv wnlC
)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),(),,(
2vv ess
),( v basic state
Schrödinger-Madelung fluid
222),,(
22v
v
SchM
SchMs
2
8
1 2rSchM IP
0:s2
ss2
1 22
s
vIP
rv PPP
(Fisher entropy)
reversible pressurerP
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
One component weakly nonlocal fluid
),,( v C ),,,( v wnlC
)C(),C(),C(s Pjs
Liu procedure (Farkas’s lemma):
constitutive state
constitutive functions
0 v
0)C()C(s s j0Pv )C(
...sj)(ess ),(),( ess
),( v basic state
Schrödinger-Madelung fluid2
22),(
SchM
SchMs
2
8
1 2rSchM IP
0:s2
ss2
1 22
s
vIP
rv PPP
(Fisher entropy)
reversible pressurerP
Potential form: Qr U P
Bernoulli equation
)()( eeQ ssU Euler-Lagrange form
Schrödinger equation
v ie
Variational origin
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
Conclusions
- Not everything is a balance- Second order nonlocality is interesting- Constitutive entropy current- Force-current systems: constructivity- Variational principles
Second Law
