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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 any Thermodynamics (?) macroscopiccontinuum 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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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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Internal variable – basic state: – constitutive state: – constitutive function: A) Local state - relaxation 2) B) Nonlocal extension - Ginzburg-Landau e.g.

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Local state – Euler equation 3)3) – basic state: – constitutive state: – constitutive function: Fluid mechanics Nonlocal extension - Navier-Stokes equation: But: Korteweg fluid

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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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Ginzburg-Landau (variational): – Variational (!) – Second Law? – Weakly nonlocal internal variables 1 2

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

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Liu equations: constitutive state space

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Korteweg fluids ( weakly nonlocal in density, second grade) Liu procedure (Farkas’s lemma): constitutive state constitutive functions basic state

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reversible pressure Potential form: Euler-Lagrange form Variational origin

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Spec.: Schrödinger-Madelung fluid (Fisher entropy) Potential form: Bernoulli equation Schrödinger equation

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R1: Thermodynamics = theory of material stability 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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–Isotropy –Extensivity (mean, density) –Additivity Entropy is unique under physically reasonable conditions. R2: Weakly nonlocal statistical physics: Boltzmann-Gibbs-Shannon

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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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