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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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general framework of any Thermodynamics (?) 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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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: forcecurrent

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Classical evolution equations: balances + constitutive assumptions Fourier heat conduction Ginzburg-Landau equation: relaxation + nonlocality D>0 l>0, k>0 Not so classical evolution equations: balances (?) + constitutive assumptions

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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: – basic state: (wanted field: T(e)) Heat conduction – Irreversible Thermodynamics Fourier heat conduction: But: Cattaneo-Vernote Guyer-Krumhansl – constitutive state: – constitutive functions: ??? 1)

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Local state – Euler equation 2) – basic state: – constitutive state: – constitutive function: Fluid mechanics Nonlocal extension - Navier-Stokes equation: But: Korteweg fluid

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Internal variable – basic state: – constitutive state: – constitutive function: A) Local state - relaxation 3) B) Nonlocal extension - Ginzburg-Landau e.g.

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Irreversible thermodynamics – traditional approach: – basic state: – constitutive state: – constitutive functions: Heat conduction: a=e J= currents and forces Solution!

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Second Law: basic balances – basic state: – constitutive state: – constitutive functions: weakly nonlocal Second law: Constitutive theory: balances are constraints Method: Liu procedure (universality)

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Liu procedure LEMMA (FARKAS, 1896) Let A i 0 be independent vectors in a finite dimensional vector space V, i = 1...n, and S = {p V | p·A i 0, i = 1...n}. The following statements are equivalent for a b V: (i) p·B 0, for all p S. (ii) There are non-negative real numbers λ 1,..., λ n such that Vocabulary: elements of V – independent variables, V – the space of independent variables, Inequalities in S – constraints, λ i – Lagrange-Farkas multipliers.

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Usage: B A1A1

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Proof : S is not empty. In fact, for all k, i {1,..., n} there is a such that p k ·A k = 1 and p k ·A i = 0 if i k. Evidently p k S for all k. (ii) (i) if p S. (i) (ii) Let S 0 = {y V | y · A i = 0, i = 1...n}. Clearly S 0 S. If y S 0 then y is also in S 0, therefore y·B 0 and y·B 0 together. Therefore for all y S 0 it is true that y·B = 0. As a consequence B is in the set generated by {A i }, that is, there are real numbers λ 1,..., λ n such that B. These numbers are non- negative, because with the previously defined p k S, is valid for all k. QED

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AFFIN FARKAS: Let A i 0 be independent vectors in a finite dimensional vector space V, α i real numbers i = 1...n and S A = {p V | p · A i α i, i = 1...n}. The following statements are equivalent for a B V and a real number : (i) p · B, for all p S A. (ii) There are non-negative real numbers λ 1,..., λ n such that B = and PROOF: … Vocabulary: Final equality: – Liu equations Final inequality: – residual (dissipation) inequality.

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LIUs THEOREM: Let A i 0 be independent vectors in a finite dimensional vector space V, α i real numbers i = 1...n and S L = {p V | p · A i = α i, i = 1...n}. The following statements are equivalent for a B V and a real number : (i) p · B, for all p S L. (ii) There are real numbers λ 1,..., λ n such that B = and PROOF: A simple consequence of affine Farkas. 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: Spec: Heat conduction: a=e B) Dissipation inequality: A)B)B) solution

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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: consequences?? Philosopher: … Popper, Lakatos: excellent, in this way we can refute

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Ginzburg-Landau (variational): – Variational (!) – Second Law? – Weakly nonlocal internal variables

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Ginzburg-Landau (thermodynamic, relocalized) Liu procedure (Farkass lemma) constitutive state space constitutive functions ? local state

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isotropy current multiplier

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Ginzburg-Landau (thermodynamic, non relocalizable) Liu procedure (Farkass lemma) state space constitutive functions

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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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