Weakly nonlocal continuum physics

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Weakly nonlocal continuum physics the role of
the Second Law Peter Ván HAS, RIPNP,
Department of Theoretical Physics

• Introduction
• Second Law
• Weak nonlocality
• Liu procedure
• Classical irreversible thermodynamics
• Ginzburg-Landau equation
• Discussion

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Nonequilibrium thermodynamics

• Thermodynamics

science of temperature
Thermodynamics science of macroscopic
energy changes
general framework of any Thermodynamics
(?) macroscopic continuum theories

• General framework
• Second Law
• fundamental balances
• objectivity - frame indifference

reversibility special limit
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Thermo-Dynamic theory
Evolution equation
1 Statics (equilibrium properties)
2 Dynamics
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1 2 closed system
S is a Ljapunov function of the equilibrium of
the dynamic law
Constructive application
force
current
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Classical evolution equations balances
constitutive assumptions
Fourier heat conduction
Dgt0
Not so classical evolution equations balances
(?) constitutive assumptions
Ginzburg-Landau equation relaxation
nonlocality
lgt0, kgt0
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Nonlocalities
Restrictions from the Second Law. change of the
entropy current change of the entropy
Change of the constitutive space
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Basic state, constitutive state and constitutive
functions
Heat conduction Irreversible Thermodynamics
1)

• basic state
• (wanted field T(e))

• constitutive state

• constitutive functions

Fourier heat conduction
But
Guyer-Krumhansl
Cattaneo-Vernote
???
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Fluid mechanics
2)

• basic state

• constitutive state

• constitutive function

Local state Euler equation
Nonlocal extension - Navier-Stokes equation
But
Korteweg fluid
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Internal variable
3)

• basic state

• constitutive state

• constitutive function

A) Local state - relaxation
B) Nonlocal extension - Ginzburg-Landau
e.g.
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Irreversible thermodynamics traditional
approach

• basic state
• constitutive state
• constitutive functions

J
Solution!
currents and forces
Heat conduction ae
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Second Law
basic balances

• basic state
• constitutive state
• constitutive functions

Second law
(universality)
Constitutive theory balances are constraints
Method Liu procedure
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Liu procedure
LEMMA (FARKAS, 1896) Let Ai ? 0 be independent
vectors in a finite dimensional vector space V, i
1...n, and S p ?V pAi 0, i
1...n. The following statements are equivalent
for a b ? V (i) pB 0, for all p ? S. (ii)
There are non-negative real numbers ?1,..., ?n
such that
Vocabulary elements of V independent
variables, V the space of independent
variables, Inequalities in S
constraints, ?i Lagrange-Farkas multipliers.
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Usage
A1
B
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Proof S is not empty. In fact, for all k, i
?1,..., n there is a such that pkAk 1 and
pkAi 0 if i ? k. Evidently pk?S for all
k. (ii) ? (i)
if p? S. (i) ? (ii) Let S0 y?V
y Ai 0, i 1...n. Clearly Ø? S0 ? S. If y?
S0 then -y is also in S0, therefore yB 0 and
-yB 0 together. Therefore for all y? S0 it is
true that yB 0. As a consequence B is in the
set generated by Ai, that is, there are real
numbers ?1,..., ?n such that B .
These numbers are non- negative, because with
the previously defined pk? S, is
valid for all k. QED
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Vocabulary Final equality Liu
equations Final inequality residual
(dissipation) inequality.
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Usage
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Irreversible thermodynamics beyond traditional
approach

• basic state
• constitutive state
• constitutive functions

Liu and Müller validity in every time and
space points, derivatives of C are independent
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A) Liu equations
solution
B) Dissipation inequality
Spec Heat conduction ae
A)
B)
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What is explained The origin of Clausius-Duhem
inequality - form of the entropy current -
what depends on what Conditions of
applicability!! - the key is the constitutive
space
Logical reduction the number of independent
physical assumptions! Mathematician ok
but Physicist no need of such thinking, I
am satisfied well and used to my analogies no
need of thermodynamics in general Engineer con
sequences?? Philosopher
Popper, Lakatos excellent, in this way we can
refute
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Weakly nonlocal internal variables
Ginzburg-Landau (variational)

• Variational (!)
• Second Law?

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Ginzburg-Landau (thermodynamic, relocalized)
constitutive state space
constitutive functions
local state
Liu procedure (Farkass lemma)
?
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current multiplier
isotropy
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Ginzburg-Landau (thermodynamic, non relocalizable)
state space
constitutive functions
Liu procedure (Farkass lemma)
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• Discussion
• Applications
• heat conduction, one component fluid
  (Schrödinger-Madelung, ), two component fluids
  (sand), complex Ginzburg-Landau, , weakly
  non-local statistical physics,
• ? Cahn-Hilliard, Korteweg-de Vries, mechanics
  (hyperstress),
• Dynamic stability, Ljapunov function???
• Universality independent on the micro-modell
• Constructivity Liu force-current systems
• Variational principles an explanation
• Second Law

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References Discrete, stability T. Matolcsi
Ordinary thermodynamics, Publishing House of the
Hungarian Academy of Sciences, Budapest,
2005. Liu procedure Liu, I-Shih, Method of
Lagrange Multipliers for Exploitation of the
Entropy Principle, Archive of Rational Mechanics
and Analysis, 1972, 46, p131-148. Weakly
nonlocal 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). 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). 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).
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Thank you for your attention!