Post on 08-Jan-2016
description
– 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
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
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
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)
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
)(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
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!
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
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
),,( 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
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
0:s2
ss2
1 22
s
vIP
rv PPP
reversible pressurerP
Potential form: nlr U P
)()( eenl ssU Euler-Lagrange form
Variational origin
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
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
2
xD)(xU
2
Mov1.exe
– 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
-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
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
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)
Thank you for your attention!