# American Institute of Mathematical Sciences

September  2012, 4(3): 333-363. doi: 10.3934/jgm.2012.4.333

## Hybrid models for perfect complex fluids with multipolar interactions

 1 Department of Mathematics, University of Surrey, Guildford GU2 7XH

Received  November 2010 Revised  October 2011 Published  October 2012

Multipolar order in complex fluids is described by statistical correlations. This paper presents a novel dynamical approach, which accounts for microscopic effects on the order parameter space. Indeed, the order parameter field is replaced by a statistical distribution function that is carried by the fluid flow. Inspired by Doi's model of colloidal suspensions, the present theory is derived from a hybrid moment closure for Yang-Mills Vlasov plasmas. This hybrid formulation is constructed under the assumption that inertial effects dominate over dissipative phenomena (perfect complex fluids), so that the total energy is conserved and the Hamiltonian approach is adopted. After presenting the basic geometric properties of the theory, the effect of Yang-Mills fields is considered and a direct application is presented to magnetized fluids with quadrupolar order (spin nematic phases). Hybrid models are also formulated for complex fluids with symmetry breaking. For the special case of liquid crystals, the moment method can be applied to the hybrid formulation to study to the dynamics of cubatic phases.
Citation: Cesare Tronci. Hybrid models for perfect complex fluids with multipolar interactions. Journal of Geometric Mechanics, 2012, 4 (3) : 333-363. doi: 10.3934/jgm.2012.4.333
##### References:
 [1] A. F. Andreev and I. A. Grishchuk, Spin nematics, Sov. Phys. JETP, 60 (1984), 267-271. Google Scholar [2] R. D. Batten, F. H. Stillinger and S. Torquato, Phase behavior of colloidal superballs: Shape interpolation from spheres to cubes, Phys. Rev. E, 81 (2010), 061105. Google Scholar [3] S. Blenk, H. Ehrentraut and W. Muschik, Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation, Phys. A, 174 (1991), 119-138.  Google Scholar [4] S. Blenk, H. Ehrentraut and W. Muschik, Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution, Int. J. Engng Sci., 30 (1992), 1127-1143.  Google Scholar [5] G. Brodin, M. Marklund, J. Zamanian, AA. Ericsson and P. L. Mana, Effects of the $g$ factor in semiclassical kinetic plasma theory, Phys. Rev. Lett., 101 (2008), 245002. Google Scholar [6] H. Cendra, D. D. Holm, M. J. W. Hoyle and J. E. Marsden, The Maxwell-Vlasov equations in Euler-Poincaré form, J. Math. Phys., 39 (1998), 3138-3157.  Google Scholar [7] S. Chandrasekhar, "Liquid Crystals," Second Edition. Cambridge University Press, Cambridge, 1992. Google Scholar [8] P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems, Commun. Math. Sci., 3 (2005), 531-544.  Google Scholar [9] D. A. Dem'yanenko and M. Yu. Kovalevskiĭ, Classification of the equilibrium states of magnets with vector and quadrupole order parameters, Low Temp. Phys., 33 (2007), 965-973. Google Scholar [10] M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics," Oxford University Press, 1988. Google Scholar [11] P. D. Duncan, M. Dennison, A. J. Masters and M. R. Wilson, Theory and computer simulation for the cubatic phase of cut spheres, Phys. Rev. E, 79 (2009), 031702. Google Scholar [12] I. E. Dzyaloshinskiĭ and G. E. Volovik, Poisson brackets in condensed matter physics, Ann. Phys., 125 (1980), 67-97.  Google Scholar [13] A. C. Eringen, A unified continuum theory of liquid crystals, ARI, 50 (1997), 73-84. Google Scholar [14] K. H. Fischer, Ferromagnetic modes in spin glasses and dilute ferromagnets, Z. Physik B, 39 (1980), 37-46. Google Scholar [15] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv App Math, 42 (2009), 176-275.  Google Scholar [16] F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Equivalent theories of liquid crystal dynamics,, , ().   Google Scholar [17] F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Euler-Poincaré approaches to nematodynamics, Acta Appl. Math., 120 (2012), 127151. doi: 10.1007/s10440-012-9719-x.  Google Scholar [18] F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Phys. D, 239 (2010), 1929-1947.  Google Scholar [19] F. Gay-Balmaz, C. Tronci and C. Vizman, Geodesic flows on the automorphism group of principal bundles,, , ().   Google Scholar [20] P. G. de Gennes, Short range order effects in the isotropic phase of nematics and cholesterics, Mol. Cryst. Liq. Cryst., 12 (1971), 193-214. doi: 10.1080/15421407108082773.  Google Scholar [21] P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," 2nd edn. Oxford University Press, Oxford, 1993. Google Scholar [22] J. Gibbons, D. D. Holm and B. A. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics, Physica D, 6 (1983), 179-194.  Google Scholar [23] J. Gibbons, D. D. Holm and C. Tronci, Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket, Phys. Lett. A, 372 (2008), 4184-4196.  Google Scholar [24] B. I. Halperin and W. M. Saslow, Hydrodynamic theory of spin waves in spin glasses and other systems with noncollinear spin orientations, Phys. Rev. B, 16 (1977), 2154-2162. doi: 10.1103/PhysRevB.16.2154.  Google Scholar [25] D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in "Geometry, Mechanics, and Dynamics,'' Springer, New York, (2002), 113-167  Google Scholar [26] D. D. Holm, Hamiltonian dynamics of a charged fluid, including electro- and magnetohydrodynamics, Phys. Lett. A, 114 (1986), 137-141.  Google Scholar [27] D. D. Holm, Hamiltonian dynamics and stability analysis of neutral electromagnetic fluids with induction, Physica D, 25 (1987), 261-287.  Google Scholar [28] D. D. Holm, R. I. Ivanov and J. R. Percival, $G$-Strands, J. Nonlinear Sci., 22 (2012), 517-551. doi: 10.1007/s00332-012-9135-4.  Google Scholar [29] D. D. Holm and B. A. Kupershmidt, Hamiltonian formulation of ferromagnetic hydrodynamics, Phys. Lett. A, 129 (1988), 93-100.  Google Scholar [30] D. D. Holm and B. A. Kupershmidt, Poisson structures of superfluids, Phys. Lett. A, 91 (1982), 425-430.  Google Scholar [31] D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids, J. Math Phys., 29 (1988), 21-30. doi: 10.1063/1.528176.  Google Scholar [32] D. D. Holm and B. A. Kupershmidt, Yang-Mills magnetohydrodynamics, Phys. Rev. D, 30 (1984), 2557-2560.  Google Scholar [33] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.  Google Scholar [34] D. D. Holm, J. E. Marsden, T. S. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1-116.  Google Scholar [35] D. D. Holm, V. Putkaradze and C. Tronci, Double bracket dissipation in kinetic theory for particles with anisotropic interactions, Proc. R. Soc. A, 466 (2010), 2991-3012. doi: 10.1098/rspa.2010.0043.  Google Scholar [36] D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions," Oxford University Press, 2009.  Google Scholar [37] D. D. Holm and C. Tronci, Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions, Proc. R. Soc. A, 465 (2008), 457-476.  Google Scholar [38] D. D. Holm and C. Tronci, The geodesic Vlasov equation and its integrable moment closures, J. Geom. Mech., 1 (2009), 181-208.  Google Scholar [39] A. A. Isayev, Hamiltonian formalism in the theory of quadruple magnet, Low Temp. Phys., 23 (1997), 933-935. Google Scholar [40] A. A. Isaev, M. Yu. Kovalevskiĭ and S. V. Peletminskiĭ, Hamiltonian approach in the theory of condensed media with spontaneously broken symmetry, Phys Part Nuclei, 27 (1996), 179-203. Google Scholar [41] A. Kadič and D. G. B. Edelen, A gauge theory of dislocations and disclinations, Lect. Notes Phys., 174 (1983). Google Scholar [42] Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma," M. I. T. Press, Cambridge, Massachusetts, 1967. Google Scholar [43] P. S. Krishnaprasad and J. E. Marsden, Hamiltonian structure and stability for rigid bodies with flexible attachments, Arch. Rational Mech. Anal., 98 (1987), 71-93.  Google Scholar [44] A. Läuchli, J. C. Domenge, C. Lhuillier, P. Sindzingre and M. Troyer, Two-step restoration of $SU(2)$ symmetry in a frustrated ring-exchange magnet, Phys. Rev. Lett., 95 (2005), 137206. Google Scholar [45] F. M. Leslie, Some topics in continuum theory of nematics, Philos. Trans. Soc. London Ser. A, 309 (1983), 155-165. Google Scholar [46] M. Marklund and P. J. Morrison, Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations, Phys. Lett. A, 305 (2011), 2362-2365.  Google Scholar [47] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, 1994.  Google Scholar [48] J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147-177. doi: 10.1090/S0002-9947-1984-0719663-1.  Google Scholar [49] J. E. Marsden, A. Weinstein, T. Ratiu, R. Schimd and R. G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 289-340.  Google Scholar [50] R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations, Lett. Math. Phys., 8 (1984), 59-67.  Google Scholar [51] R. Montgomery, J. Marsden and T. Ratiu, Gauged Lie-Poisson structures, Contemp. Math., 28 (1984), 101-114. doi: 10.1090/conm/028/751976.  Google Scholar [52] K. Penc and M. Läuchli, Spin nematic phases in quantum spin systems, Springer Ser. Solid-State Sci., 164 (2011), 331-362. doi: 10.1007/978-3-642-10589-0_13.  Google Scholar [53] L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N. Y. Acad. Sci., 51 (1949), 627-659. Google Scholar [54] R. G. Spencer, The Hamiltonian structure of multi-species fluid electrodynamics, AIP Conf. Proc., 88 (1982), 121-126. doi: 10.1063/1.33630.  Google Scholar [55] R. G. Spencer and A. N. Kaufman, Hamiltonian structure of two-fluid plasma dynamics, Phys. Rev. A (3), 25 (1982), 2437-2439.  Google Scholar [56] H. Stark and T. C. Lubensky, Poisson-bracket approach to the dynamics of nematic liquid crystals, Phys. Rev. E, 67 (2003), 061709.  Google Scholar [57] J. Sudan, A. Lüscher and A. M. Läuchli, Emergent multipolar spin correlations in a fluctuating spiral: the frustrated ferromagnetic spin-$1/2$ Heisenberg chain in a magnetic field, Phys. Rev. B, 80 (2009), 140402(R). Google Scholar [58] H. Tsunetsugu and M. Arikawa, Spin nematic phase in $S=1$ triangular antiferromagnets, J. Phys. Soc. Jpn., 75 (2006), 083701. Google Scholar [59] J. A. C. Veerman and D. Frenkel, Phase behavior of disklike hard-core mesogens, Phys. Rev. A, 45 (1992), 5632-5648. Google Scholar [60] G. E. Volovik and I. E. Dzyaloshinskiĭ, Additional localized degrees of freedom in spin glasses, Sov. Phys. JETP, 48 (1978), 555-559. Google Scholar [61] G. E. Volovik and E. I. Kats, Nonlinear hydrodynamics of liquid crystals, Sov. Phys. JETP, 54 (1981), 122-126. Google Scholar

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##### References:
 [1] A. F. Andreev and I. A. Grishchuk, Spin nematics, Sov. Phys. JETP, 60 (1984), 267-271. Google Scholar [2] R. D. Batten, F. H. Stillinger and S. Torquato, Phase behavior of colloidal superballs: Shape interpolation from spheres to cubes, Phys. Rev. E, 81 (2010), 061105. Google Scholar [3] S. Blenk, H. Ehrentraut and W. Muschik, Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation, Phys. A, 174 (1991), 119-138.  Google Scholar [4] S. Blenk, H. Ehrentraut and W. Muschik, Macroscopic constitutive equations for liquid crystals induced by their mesoscopic orientation distribution, Int. J. Engng Sci., 30 (1992), 1127-1143.  Google Scholar [5] G. Brodin, M. Marklund, J. Zamanian, AA. Ericsson and P. L. Mana, Effects of the $g$ factor in semiclassical kinetic plasma theory, Phys. Rev. Lett., 101 (2008), 245002. Google Scholar [6] H. Cendra, D. D. Holm, M. J. W. Hoyle and J. E. Marsden, The Maxwell-Vlasov equations in Euler-Poincaré form, J. Math. Phys., 39 (1998), 3138-3157.  Google Scholar [7] S. Chandrasekhar, "Liquid Crystals," Second Edition. Cambridge University Press, Cambridge, 1992. Google Scholar [8] P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems, Commun. Math. Sci., 3 (2005), 531-544.  Google Scholar [9] D. A. Dem'yanenko and M. Yu. Kovalevskiĭ, Classification of the equilibrium states of magnets with vector and quadrupole order parameters, Low Temp. Phys., 33 (2007), 965-973. Google Scholar [10] M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics," Oxford University Press, 1988. Google Scholar [11] P. D. Duncan, M. Dennison, A. J. Masters and M. R. Wilson, Theory and computer simulation for the cubatic phase of cut spheres, Phys. Rev. E, 79 (2009), 031702. Google Scholar [12] I. E. Dzyaloshinskiĭ and G. E. Volovik, Poisson brackets in condensed matter physics, Ann. Phys., 125 (1980), 67-97.  Google Scholar [13] A. C. Eringen, A unified continuum theory of liquid crystals, ARI, 50 (1997), 73-84. Google Scholar [14] K. H. Fischer, Ferromagnetic modes in spin glasses and dilute ferromagnets, Z. Physik B, 39 (1980), 37-46. Google Scholar [15] F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids, Adv App Math, 42 (2009), 176-275.  Google Scholar [16] F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Equivalent theories of liquid crystal dynamics,, , ().   Google Scholar [17] F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Euler-Poincaré approaches to nematodynamics, Acta Appl. Math., 120 (2012), 127151. doi: 10.1007/s10440-012-9719-x.  Google Scholar [18] F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems, Phys. D, 239 (2010), 1929-1947.  Google Scholar [19] F. Gay-Balmaz, C. Tronci and C. Vizman, Geodesic flows on the automorphism group of principal bundles,, , ().   Google Scholar [20] P. G. de Gennes, Short range order effects in the isotropic phase of nematics and cholesterics, Mol. Cryst. Liq. Cryst., 12 (1971), 193-214. doi: 10.1080/15421407108082773.  Google Scholar [21] P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals," 2nd edn. Oxford University Press, Oxford, 1993. Google Scholar [22] J. Gibbons, D. D. Holm and B. A. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics, Physica D, 6 (1983), 179-194.  Google Scholar [23] J. Gibbons, D. D. Holm and C. Tronci, Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket, Phys. Lett. A, 372 (2008), 4184-4196.  Google Scholar [24] B. I. Halperin and W. M. Saslow, Hydrodynamic theory of spin waves in spin glasses and other systems with noncollinear spin orientations, Phys. Rev. B, 16 (1977), 2154-2162. doi: 10.1103/PhysRevB.16.2154.  Google Scholar [25] D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids, in "Geometry, Mechanics, and Dynamics,'' Springer, New York, (2002), 113-167  Google Scholar [26] D. D. Holm, Hamiltonian dynamics of a charged fluid, including electro- and magnetohydrodynamics, Phys. Lett. A, 114 (1986), 137-141.  Google Scholar [27] D. D. Holm, Hamiltonian dynamics and stability analysis of neutral electromagnetic fluids with induction, Physica D, 25 (1987), 261-287.  Google Scholar [28] D. D. Holm, R. I. Ivanov and J. R. Percival, $G$-Strands, J. Nonlinear Sci., 22 (2012), 517-551. doi: 10.1007/s00332-012-9135-4.  Google Scholar [29] D. D. Holm and B. A. Kupershmidt, Hamiltonian formulation of ferromagnetic hydrodynamics, Phys. Lett. A, 129 (1988), 93-100.  Google Scholar [30] D. D. Holm and B. A. Kupershmidt, Poisson structures of superfluids, Phys. Lett. A, 91 (1982), 425-430.  Google Scholar [31] D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids, J. Math Phys., 29 (1988), 21-30. doi: 10.1063/1.528176.  Google Scholar [32] D. D. Holm and B. A. Kupershmidt, Yang-Mills magnetohydrodynamics, Phys. Rev. D, 30 (1984), 2557-2560.  Google Scholar [33] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.  Google Scholar [34] D. D. Holm, J. E. Marsden, T. S. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1-116.  Google Scholar [35] D. D. Holm, V. Putkaradze and C. Tronci, Double bracket dissipation in kinetic theory for particles with anisotropic interactions, Proc. R. Soc. A, 466 (2010), 2991-3012. doi: 10.1098/rspa.2010.0043.  Google Scholar [36] D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions," Oxford University Press, 2009.  Google Scholar [37] D. D. Holm and C. Tronci, Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions, Proc. R. Soc. A, 465 (2008), 457-476.  Google Scholar [38] D. D. Holm and C. Tronci, The geodesic Vlasov equation and its integrable moment closures, J. Geom. Mech., 1 (2009), 181-208.  Google Scholar [39] A. A. Isayev, Hamiltonian formalism in the theory of quadruple magnet, Low Temp. Phys., 23 (1997), 933-935. Google Scholar [40] A. A. Isaev, M. Yu. Kovalevskiĭ and S. V. Peletminskiĭ, Hamiltonian approach in the theory of condensed media with spontaneously broken symmetry, Phys Part Nuclei, 27 (1996), 179-203. Google Scholar [41] A. Kadič and D. G. B. Edelen, A gauge theory of dislocations and disclinations, Lect. Notes Phys., 174 (1983). Google Scholar [42] Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma," M. I. T. Press, Cambridge, Massachusetts, 1967. Google Scholar [43] P. S. Krishnaprasad and J. E. Marsden, Hamiltonian structure and stability for rigid bodies with flexible attachments, Arch. Rational Mech. Anal., 98 (1987), 71-93.  Google Scholar [44] A. Läuchli, J. C. Domenge, C. Lhuillier, P. Sindzingre and M. Troyer, Two-step restoration of $SU(2)$ symmetry in a frustrated ring-exchange magnet, Phys. Rev. Lett., 95 (2005), 137206. Google Scholar [45] F. M. Leslie, Some topics in continuum theory of nematics, Philos. Trans. Soc. London Ser. A, 309 (1983), 155-165. Google Scholar [46] M. Marklund and P. J. Morrison, Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations, Phys. Lett. A, 305 (2011), 2362-2365.  Google Scholar [47] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Springer-Verlag, 1994.  Google Scholar [48] J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc., 281 (1984), 147-177. doi: 10.1090/S0002-9947-1984-0719663-1.  Google Scholar [49] J. E. Marsden, A. Weinstein, T. Ratiu, R. Schimd and R. G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 289-340.  Google Scholar [50] R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations, Lett. Math. Phys., 8 (1984), 59-67.  Google Scholar [51] R. Montgomery, J. Marsden and T. Ratiu, Gauged Lie-Poisson structures, Contemp. Math., 28 (1984), 101-114. doi: 10.1090/conm/028/751976.  Google Scholar [52] K. Penc and M. Läuchli, Spin nematic phases in quantum spin systems, Springer Ser. Solid-State Sci., 164 (2011), 331-362. doi: 10.1007/978-3-642-10589-0_13.  Google Scholar [53] L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N. Y. Acad. Sci., 51 (1949), 627-659. Google Scholar [54] R. G. Spencer, The Hamiltonian structure of multi-species fluid electrodynamics, AIP Conf. Proc., 88 (1982), 121-126. doi: 10.1063/1.33630.  Google Scholar [55] R. G. Spencer and A. N. Kaufman, Hamiltonian structure of two-fluid plasma dynamics, Phys. Rev. A (3), 25 (1982), 2437-2439.  Google Scholar [56] H. Stark and T. C. Lubensky, Poisson-bracket approach to the dynamics of nematic liquid crystals, Phys. Rev. E, 67 (2003), 061709.  Google Scholar [57] J. Sudan, A. Lüscher and A. M. Läuchli, Emergent multipolar spin correlations in a fluctuating spiral: the frustrated ferromagnetic spin-$1/2$ Heisenberg chain in a magnetic field, Phys. Rev. B, 80 (2009), 140402(R). Google Scholar [58] H. Tsunetsugu and M. Arikawa, Spin nematic phase in $S=1$ triangular antiferromagnets, J. Phys. Soc. Jpn., 75 (2006), 083701. Google Scholar [59] J. A. C. Veerman and D. Frenkel, Phase behavior of disklike hard-core mesogens, Phys. Rev. A, 45 (1992), 5632-5648. Google Scholar [60] G. E. Volovik and I. E. Dzyaloshinskiĭ, Additional localized degrees of freedom in spin glasses, Sov. Phys. JETP, 48 (1978), 555-559. Google Scholar [61] G. E. Volovik and E. I. Kats, Nonlinear hydrodynamics of liquid crystals, Sov. Phys. JETP, 54 (1981), 122-126. Google Scholar
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