1 / 20

Introduction Second Law Weak nonlocality Ginzburg-Landau equation Schrödinger-Madelung equation

Weakly nonlocal nonequilibrium thermodynamics – fluids and beyond Peter Ván BCPL, University of Bergen, Bergen and RMKI , Department of Theoretical Physics , Budapest. Introduction Second Law Weak nonlocality Ginzburg-Landau equation Schrödinger-Madelung equation

arlais
Download Presentation

Introduction Second Law Weak nonlocality Ginzburg-Landau equation Schrödinger-Madelung equation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Weakly nonlocal nonequilibrium thermodynamics –fluids and beyond Peter Ván BCPL, University of Bergen, Bergen andRMKI, Department of Theoretical Physics, Budapest • Introduction • Second Law • Weak nonlocality • Ginzburg-Landau equation • Schrödinger-Madelung equation • Digression: Stability and statistical physics • Discussion

  2. Nonequilibrium thermodynamics science of temperature Thermodynamics science of macroscopic energy changes Thermodynamics • general framework of any • Thermodynamics (?) macroscopic continuum • theories • General framework: • Second Law • fundamental balances • objectivity - frame indifference reversibility – special limit

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

  4. 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 ???

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

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

  7. Irreversible thermodynamics – traditional approach: • basic state: • constitutive state: • constitutive functions: J= Solution! currents andforces Heat conduction: a=e

  8. 1 2 Weakly nonlocal internal variables Ginzburg-Landau (variational): • Variational (!) • Second Law?

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

  10. constitutive state space Liu equations:

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

  12. reversible pressure Potential form: Euler-Lagrange form Variational origin

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

  14. 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

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

  16. 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

  17. References: • Ván, P., Exploiting the SecondLaw in weakly nonlocal continuum physics, PeriodicaPolytechnica, Ser. Mechanical Engineering, 2005, 49/1, p79-94, (cond-mat/0210402/ver3). • Ván, P. and Fülöp, T., Weakly nonlocal fluid mechanics - the Schrödingerequation, Proceedings of the Royal Society,London A, 2006, 462, p541-557, (quant-ph/0304062). • 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) • Ván, P., Weakly nonlocalcontinuum theories of granular media: restrictions from theSecond Law, International Journal of Solids andStructures, 2004, 41/21, p5921-5927, (cond-mat/0310520). • 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). • 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). • P. Ván. Unique additive information measures - Boltzmann-Gibbs-Shannon, Fisher and beyond. Physica A, 365:28(33), 2006. (cond-mat/0409255) • P. Ván, A. Berezovski, and Engelbrecht J. Internal variables and dynamic degrees of freedom.2006. (cond-mat/0612491)

  18. Thank you for your attention!

More Related