Introduction Second Law Weak nonlocality Ginzburg-Landau equation Schrödinger-Madelung equation
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Transcript of Introduction Second Law Weak nonlocality Ginzburg-Landau equation Schrödinger-Madelung equation
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– Introduction • Second Law
• Weak nonlocality
– Ginzburg-Landau equation
– Schrödinger-Madelung equation
– Digression: Stability and statistical physics
– Discussion
Weakly nonlocal nonequilibrium thermodynamics –fluids and beyond
Peter Ván BCPL, University of Bergen, Bergen and
RMKI, Department of Theoretical Physics, Budapest
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general framework of anyThermodynamics (?) macroscopic continuum
theories
Thermodynamics science of macroscopic energy changes
Thermodynamics
science of temperature
Nonequilibrium thermodynamics
reversibility – special limit
General framework: – Second Law – fundamental balances– objectivity - frame indifference
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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.change of the entropy currentchange of the entropy
Change of the constitutive space
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Basic state, constitutive state and constitutive functions:
ee q
– basic state:(wanted field: T(e))
e
)(Cq),( eeC
Heat conduction – Irreversible Thermodynamics
),( ee ))(),(( eTeT T q )())(),((),( eTeTeTee q
Fourier heat conduction:
But: qq LT qqq 21LLT Cattaneo-VernoteGuyer-Krumhansl
– constitutive state:– constitutive functions:
,...),,,,( 2eeeee ???
1)
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fa
a
s
a
sLa
Internal variable
– basic state: aa– constitutive state:
– constitutive function:
A) Local state - relaxation
0 fda
ds
da
dsLa
2)
B) Nonlocal extension - Ginzburg-Landau
aaa 2,,
),( aaa
sL
alaslaaasaas )('ˆ,
2)(ˆ),( 2 e.g.
)(Cf
)0)('ˆ( as
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)(C ),( v C
Local state – Euler equation
0
0
Pv
v
3)
– basic state:– constitutive state:– constitutive function:
Fluid mechanics
Nonlocal extension - Navier-Stokes equation:v
se
p1
),,()()( 2
IP
vIvvP 2))((),( p
But: 22)( IP prKor
),,,( 2 vC),( v
)(CP
Korteweg fluid
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Irreversible thermodynamics – traditional approach:
0
J
0ja
sa
– basic state:
– constitutive state:– constitutive functions:
a
Jj ,, sa
),( aa C
Te
s qqJ
Heat conduction: a=e
0
a
js
as
01
2 T
TT
0)(
a
jja
aaa
jaa
Jasssss
s aaa
J=
currents and forces
aLj
s
a
Solution!
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Ginzburg-Landau (variational):
dVaasas ))(2
)(ˆ()( 2
))('ˆ( aasla – Variational (!) – Second Law?– ak
aassa )('ˆ
sla a
Weakly nonlocal internal variables
dVaasas ))(2
)(ˆ()( 2
sla a
1
2
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Ginzburg-Landau (thermodynamic, non relocalizable)
fa
0 Js
),,( 2aaa
J),,( sf
Liu procedure (Farkas’s lemma)
),( aas ),()()( 0 aaCfa
sC
jJ
0
fa
s
a
ss
a
s
a
sLa
constitutive state space
constitutive functions 0 fa
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),,( aaaC xxx
),(
),(0
;;
33 aajfsJfJ
aass
ss
xaxx
xa
aa
x
xx
x
Liu equations:
0)(
fa
s
a
s
xxs
0)()(
)()()(
2211
33321
fafJafJ
fJasasasa
xxxxx
xxxxtxxtxt
)()( 321
321321
afafafafa
aJaJaJasasas
xxxxxxtxt
xxxxxxxxxtxxtxt
))()(())()(()()( CfaCCfaCCJCs xtxtxxt
constitutive state space
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Korteweg fluids (weakly nonlocal in density, second grade)
),,( v C ),,,( v wnlC
)(),(),( CCCs PJ
Liu procedure (Farkas’s lemma):
constitutive state
constitutive functions
0 v
0)()( CCs J0Pv )C(
...J)(ess ),(),( ess
),( v basic state
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0:s2
ss2
1 22
s
vIP
rv PPP
reversible pressurerP
Potential form: nlr U P
)()( eenl ssU Euler-Lagrange form
Variational origin
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Spec.: Schrödinger-Madelung fluid2
22),(
SchM
SchMs
2
8
1 2IP rSchM
(Fisher entropy)
Potential form: Qr U P
Bernoulli equation
Schrödinger equation
v ie
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R1: Thermodynamics = theory of material stability
In quantum fluids:– 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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2
xD)(xU
2
Mov1.exe
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– Isotropy– Extensivity (mean, density)
– Additivity
Entropy is unique under physically reasonable conditions.
R2: Weakly nonlocal statistical physics:
Boltzmann-Gibbs-Shannon
)()( ss
)()()( 2121 sss
ln)( ks
))(,(),( 2 ss
),(),())(,( 22112121 ssDs
2
22 )(
ln))(,(
ks
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-2 -1 1 2x
0.2
0.4
0.6
0.8
1
1.2
R
18
,12
),5,2,5.1,3.1,2.1,1.1,1(4 111
mkk
k
k
k
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Discussion:
– Applications: – heat conduction (Guyer-Krumhansl), Ginzburg-Landau, Cahn-Hilliard, one component fluid (Schrödinger-Madelung, etc.), two component fluids (gradient phase trasitions), … , weakly nonlocal statistical physics,… – ? Korteweg-de Vries, mechanics (hyperstress), …
– Dynamic stability, Ljapunov function?– Universality – independent on the micro-modell– Constructivity – Liu + force-current systems– Variational principles: an explanation
Thermodynamics – theory of material stability
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References:
1. Ván, P., Exploiting the Second Law in weakly nonlocal continuum physics, Periodica Polytechnica, Ser. Mechanical Engineering, 2005, 49/1, p79-94, (cond-mat/0210402/ver3).
2. Ván, P. and Fülöp, T., Weakly nonlocal fluid mechanics - the Schrödinger equation, Proceedings of the Royal Society, London A, 2006, 462, p541-557, (quant-ph/0304062).
3. P. Ván and T. Fülöp. Stability of stationary solutions of the Schrödinger-Langevin equation. Physics Letters A, 323(5-6):374(381), 2004. (quant-ph/0304190)
4. Ván, P., Weakly nonlocal continuum theories of granular media: restrictions from the Second Law, International Journal of Solids and Structures, 2004, 41/21, p5921-5927, (cond-mat/0310520).
5. Cimmelli, V. A. and Ván, P., The effects of nonlocality on the evolution of higher order fluxes in non-equilibrium thermodynamics, Journal of Mathematical Physics, 2005, 46, p112901, (cond-mat/0409254).
6. V. Ciancio, V. A. Cimmelli, and P. Ván. On the evolution of higher order fluxes in non-equilibrium thermodynamics. Mathematical and Computer Modelling, 45:126(136), 2007. (cond-mat/0407530).
7. P. Ván. Unique additive information measures - Boltzmann-Gibbs-Shannon, Fisher and beyond. Physica A, 365:28(33), 2006. (cond-mat/0409255)
8. P. Ván, A. Berezovski, and Engelbrecht J. Internal variables and dynamic degrees of freedom. 2006. (cond-mat/0612491)
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Thank you for your attention!