What is thermodynamics and what is it for? II. Continuum physics

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What is thermodynamics and what is it for? II.
Continuum physics constitutive theory Peter
Ván HAS, RIPNP, Department of Theoretical Physics

• Introduction
• Constitutive space and constitutive functions
• Classical irreversible thermodynamics
• Weakly non-local extensions
• Internal variables, heat conduction and fluids
• Discussion

Centre of Nonlinear Studies, Tallinn, Estonia,
19/6/2006.
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Thermo-Dynamic theory
Dynamic law
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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Why nonequilibrium thermodynamics?

• Thermodynamics

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

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

reversibility special limit
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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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Nonlocalities
Restrictions from the Second Law. change of the
entropy current change of the entropy
Change of the constitutive space
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Second Law
basic balances
(and more)

• basic state
• constitutive state
• constitutive functions

Second law
(universality)
Constitutive theory
Method Liu procedure
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Irreversible thermodynamics

• basic state
• constitutive state
• constitutive functions

Liu procedure (Farkas lemma)
A) Liu equations
B) Dissipation inequality
Heat conduction ae
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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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Weakly nonlocal extended thermodynamics
state space
constitutive space
constitutive functions
local state
Liu procedure (Farkass lemma)
solution?
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extended (Gyarmati) entropy
entropy current (Nyíri) (B current multiplier)

gradient
Guyer-Krumhansl equation
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Korteweg fluids (weakly nonlocal in density,
second grade)
basic state
constitutive state
constitutive functions
Liu procedure (Farkass lemma)
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reversible pressure
Potential form
Euler-Lagrange form
Variational origin
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Schrödinger-Madelung fluid
Bernoulli equation
Schrödinger equation
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Thermodynamics theory of material stability

• Ideas
• Phase transitions in gradient systems?
• In quantum fluids
• There is a family of equilibrium (stationary)
  solutions.

• There is a thermodynamic Ljapunov function

semidefinite in a gradient (Soboljev ?) space
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Conclusions

• Dynamic stability, Ljapunov function???
• Universality independent on the micro-modell
• Constructivity Liu force-current systems
• Variational principles an explanation
• Second Law
• Problems, perspectives
• objectivity (material frame indifference)
• mechanics (hyperstress and strain)!
• electrodynamics (special relativity)

But heat conduction, two component fluids
(sand), Cahn-Hilliard, complex Ginzburg-Landau,
Korteweg-de Vries, . , weakly non-local
statistical physics,
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Thank you for your attention!