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Poisson Brackets and Derivation of Equations of Motion of Soft-Matter Quasicrystals

  • Tian-You Fan
Chapter
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 260)

Abstract

The previous chapters provided knowledge for us to understanding soft-matter quasicrystals, an understanding quantitatively is needed to set up the equations of motion of the matter.

References

  1. 1.
    A. Einstein, Ueber die von der molekularkinetischen Theorie der Waerme geforderte Bewegung von in ruhenden Fluessigkeiten suspendierten Teilchen. Ann d Phys 17(4), 549–560 (1905)ADSCrossRefMATHGoogle Scholar
  2. 2.
    Perrin J B, The Atoms, Nabu Press, New York, 2010 (English translation, by Hammick D L)Google Scholar
  3. 3.
    D. Forster, Hydrodynamic Fluctuation, Broken Symmetry and Correlation Functions, vol. 47. Frontier in Physics, A Lecture Note and Reprint Series (W A Benjamin, Incorporated, Massachusetts, 1975)Google Scholar
  4. 4.
    J. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995)CrossRefGoogle Scholar
  5. 5.
    L.D. Landau, M.E. Lifshitz, in Fluid Mechanics, Theory of Elasticity (Pergamon, Oxford, 1998)Google Scholar
  6. 6.
    L.D. Landau, The theory of superfluidity of heilium II, Zh. Eksp. Teor. Fiz, II, 592. J. Phys. USSR 5, 71–90 (1941)Google Scholar
  7. 7.
    L.D. Landau, E.M. Lifshitz, in Zur Theorie der Dispersion der magnetische Permeabilitaet der ferromagnetische Koerpern. Physik Zeitschrift fuer Sowjetunion 8(2), 158–164 (1935)Google Scholar
  8. 8.
    I.E. Dzyaloshinskii, G.E. Volovick, Poisson brackets in condensed matter physics. Ann. Phys. (NY) 125(1), 67–97 (1980)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    I.E. Dzyaloshinskii, G.E. Volovick, On the concept of local invariance in spin glass theory. J. de Phys. 39(6), 693–700 (1978)CrossRefGoogle Scholar
  10. 10.
    G.E. Volovick, Additional localized degrees of freedom in spin glasses. Zh. Eksp. Teor. Fiz. 75(7), 1102–1109 (1978)Google Scholar
  11. 11.
    P.C. Martin, O. Paron, P.S. Pershan, Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids. Phys. Rev. A 6(6), 2401–2420 (1972)ADSCrossRefGoogle Scholar
  12. 12.
    P.D. Fleming, C. Cohen, Hydrodynamics of solids. Phys. Rev. B 13(2), 500–516 (1976)ADSCrossRefGoogle Scholar
  13. 13.
    T.C. Lubensky, S. Ramaswamy, J. Toner, Hydrodynamics of icosahedral quasicrystals. Phys. Rev. B 32(11), 7411–7444 (1985)CrossRefGoogle Scholar
  14. 14.
    T.C. Lubensky, Symmetry, elasticity and hydrodynamics of quasiperioic structures, in Aperiodic Crystals, vol. I, ed. by M.V. Jaric (Academic Press, Boston, 1988), pp. 199–280Google Scholar
  15. 15.
    T.Y. Fan, Poisson bracket method and it applications to quasicrystals, liquid crystals and a class of soft matter. Acta. Mech. Sin. 45(4), 548–559 (2013). (in Chinese)Google Scholar
  16. 16.
    S.B. Rochal, V.L. Lorman, Minimal model of the phonon-phason dynamics in icosahedral quasicrystals and its application to the problem of internal friction in the i-AlPdMn alloy. Phys. Rev. B 66(14), 144204 (2002)ADSCrossRefGoogle Scholar
  17. 17.
    G. Coddens, On the problem of the relation between phason elasticity and phason dynamics in quasicrystals. Eur. Phys. J. B 54(1), 37–65 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    T.Y. Fan, Equation system of generalized hydrodynamics of soft-matter quasicrystals. Appl. Math. Mech. 37(4), 331–347 (2016). (in Chinese)MathSciNetGoogle Scholar
  19. 19.
    T.Y. Fan, Generalized hydrodynamics of soft-matter second kind two-dimensional quasicrystals. Appl. Math. Mech. 38(2), 189–199 (2017). (in Chinese)Google Scholar
  20. 20.
    T.Y. Fan, Z.Y. Tang, Three-dimensional hydrodynamics of soft-matter quasicrystals. Appl. Math. Mech. 38 (2017). (in press, in Chinese)Google Scholar
  21. 21.
    H. Cheng, T.Y. Fan, H. Wei, Solution of hydrodynamics of 5- and 10-fold symmetry quasicrystals. Appl. Math. Mech. 37(10), 1393–1404 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    H. Cheng, T.Y. Fan, H. Wei, Characters of deformation and motion of possible soft-matter quasicrystals with 5- and 10-fold symmetries. Appl. Phys. A (2017, in reviewing)Google Scholar
  23. 23.
    Wang F, Hu H Y, Fan T Y and Cheng H, Hydrodynamic analysis of octagonal soft-matter quasicrystals, Appl Math Mech, 2017, acceptedGoogle Scholar
  24. 24.
    H. Cheng, T.Y. Fan, Flow of soft-matter quasicrystals past a circular cylinder. Unpublished workGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. and Beijing Institution of Technology Press 2017

Authors and Affiliations

  1. 1.Beijing Institute of TechnologyBeijingChina

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