–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 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

Nonlocalities: Restrictions from the Second Law. change of the entropy current change of the entropy Change of the constitutive space

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)

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

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

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

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

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

Liu equations: constitutive state space

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

reversible pressure Potential form: Euler-Lagrange form Variational origin

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

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

–Isotropy –Extensivity (mean, density) –Additivity Entropy is unique under physically reasonable conditions. R2: Weakly nonlocal statistical physics: Boltzmann-Gibbs-Shannon

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/ /ver3). 2.Ván, P. and Fülöp, T., Weakly nonlocal fluid mechanics - the Schrödinger equation, Proceedings of the Royal Society, London A, 2006, 462, p , (quant-ph/ ). 3.P. Ván and T. Fülöp. Stability of stationary solutions of the Schrödinger-Langevin equation. Physics Letters A, 323(5-6):374(381), (quant-ph/ ) 4.Ván, P., Weakly nonlocal continuum theories of granular media: restrictions from the Second Law, International Journal of Solids and Structures, 2004, 41/21, p , (cond-mat/ ). 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/ ). 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), (cond-mat/ ). 7.P. Ván. Unique additive information measures - Boltzmann-Gibbs-Shannon, Fisher and beyond. Physica A, 365:28(33), (cond-mat/ ) 8.P. Ván, A. Berezovski, and Engelbrecht J. Internal variables and dynamic degrees of freedom (cond- mat/ )

Thank you for your attention!