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.

[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).

[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.

[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.

[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).

[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.

[7]

S. Chandrasekhar, "Liquid Crystals,", Second Edition. Cambridge University Press, (1992).

[8]

P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems,, Commun. Math. Sci., 3 (2005), 531.

[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.

[10]

M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics,", Oxford University Press, (1988).

[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).

[12]

I. E. Dzyaloshinskiĭ and G. E. Volovik, Poisson brackets in condensed matter physics,, Ann. Phys., 125 (1980), 67.

[13]

A. C. Eringen, A unified continuum theory of liquid crystals,, ARI, 50 (1997), 73.

[14]

K. H. Fischer, Ferromagnetic modes in spin glasses and dilute ferromagnets,, Z. Physik B, 39 (1980), 37.

[15]

F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv App Math, 42 (2009), 176.

[16]

F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Equivalent theories of liquid crystal dynamics,, , ().

[17]

F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Euler-Poincaré approaches to nematodynamics,, Acta Appl. Math., 120 (2012). doi: 10.1007/s10440-012-9719-x.

[18]

F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems,, Phys. D, 239 (2010), 1929.

[19]

F. Gay-Balmaz, C. Tronci and C. Vizman, Geodesic flows on the automorphism group of principal bundles,, , ().

[20]

P. G. de Gennes, Short range order effects in the isotropic phase of nematics and cholesterics,, Mol. Cryst. Liq. Cryst., 12 (1971), 193. doi: 10.1080/15421407108082773.

[21]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals,", 2nd edn. Oxford University Press, (1993).

[22]

J. Gibbons, D. D. Holm and B. A. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics,, Physica D, 6 (1983), 179.

[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.

[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. doi: 10.1103/PhysRevB.16.2154.

[25]

D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, in, (2002), 113.

[26]

D. D. Holm, Hamiltonian dynamics of a charged fluid, including electro- and magnetohydrodynamics,, Phys. Lett. A, 114 (1986), 137.

[27]

D. D. Holm, Hamiltonian dynamics and stability analysis of neutral electromagnetic fluids with induction,, Physica D, 25 (1987), 261.

[28]

D. D. Holm, R. I. Ivanov and J. R. Percival, $G$-Strands,, J. Nonlinear Sci., 22 (2012), 517. doi: 10.1007/s00332-012-9135-4.

[29]

D. D. Holm and B. A. Kupershmidt, Hamiltonian formulation of ferromagnetic hydrodynamics,, Phys. Lett. A, 129 (1988), 93.

[30]

D. D. Holm and B. A. Kupershmidt, Poisson structures of superfluids,, Phys. Lett. A, 91 (1982), 425.

[31]

D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids,, J. Math Phys., 29 (1988), 21. doi: 10.1063/1.528176.

[32]

D. D. Holm and B. A. Kupershmidt, Yang-Mills magnetohydrodynamics,, Phys. Rev. D, 30 (1984), 2557.

[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.

[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.

[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. doi: 10.1098/rspa.2010.0043.

[36]

D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions,", Oxford University Press, (2009).

[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.

[38]

D. D. Holm and C. Tronci, The geodesic Vlasov equation and its integrable moment closures,, J. Geom. Mech., 1 (2009), 181.

[39]

A. A. Isayev, Hamiltonian formalism in the theory of quadruple magnet,, Low Temp. Phys., 23 (1997), 933.

[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.

[41]

A. Kadič and D. G. B. Edelen, A gauge theory of dislocations and disclinations,, Lect. Notes Phys., 174 (1983).

[42]

Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma,", M. I. T. Press, (1967).

[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.

[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).

[45]

F. M. Leslie, Some topics in continuum theory of nematics,, Philos. Trans. Soc. London Ser. A, 309 (1983), 155.

[46]

M. Marklund and P. J. Morrison, Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations,, Phys. Lett. A, 305 (2011), 2362.

[47]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Springer-Verlag, (1994).

[48]

J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics,, Trans. Amer. Math. Soc., 281 (1984), 147. doi: 10.1090/S0002-9947-1984-0719663-1.

[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.

[50]

R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations,, Lett. Math. Phys., 8 (1984), 59.

[51]

R. Montgomery, J. Marsden and T. Ratiu, Gauged Lie-Poisson structures,, Contemp. Math., 28 (1984), 101. doi: 10.1090/conm/028/751976.

[52]

K. Penc and M. Läuchli, Spin nematic phases in quantum spin systems,, Springer Ser. Solid-State Sci., 164 (2011), 331. doi: 10.1007/978-3-642-10589-0_13.

[53]

L. Onsager, The effects of shape on the interaction of colloidal particles,, Ann. N. Y. Acad. Sci., 51 (1949), 627.

[54]

R. G. Spencer, The Hamiltonian structure of multi-species fluid electrodynamics,, AIP Conf. Proc., 88 (1982), 121. doi: 10.1063/1.33630.

[55]

R. G. Spencer and A. N. Kaufman, Hamiltonian structure of two-fluid plasma dynamics,, Phys. Rev. A (3), 25 (1982), 2437.

[56]

H. Stark and T. C. Lubensky, Poisson-bracket approach to the dynamics of nematic liquid crystals,, Phys. Rev. E, 67 (2003).

[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).

[58]

H. Tsunetsugu and M. Arikawa, Spin nematic phase in $S=1$ triangular antiferromagnets,, J. Phys. Soc. Jpn., 75 (2006).

[59]

J. A. C. Veerman and D. Frenkel, Phase behavior of disklike hard-core mesogens,, Phys. Rev. A, 45 (1992), 5632.

[60]

G. E. Volovik and I. E. Dzyaloshinskiĭ, Additional localized degrees of freedom in spin glasses,, Sov. Phys. JETP, 48 (1978), 555.

[61]

G. E. Volovik and E. I. Kats, Nonlinear hydrodynamics of liquid crystals,, Sov. Phys. JETP, 54 (1981), 122.

show all references

References:
[1]

A. F. Andreev and I. A. Grishchuk, Spin nematics,, Sov. Phys. JETP, 60 (1984), 267.

[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).

[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.

[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.

[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).

[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.

[7]

S. Chandrasekhar, "Liquid Crystals,", Second Edition. Cambridge University Press, (1992).

[8]

P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems,, Commun. Math. Sci., 3 (2005), 531.

[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.

[10]

M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics,", Oxford University Press, (1988).

[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).

[12]

I. E. Dzyaloshinskiĭ and G. E. Volovik, Poisson brackets in condensed matter physics,, Ann. Phys., 125 (1980), 67.

[13]

A. C. Eringen, A unified continuum theory of liquid crystals,, ARI, 50 (1997), 73.

[14]

K. H. Fischer, Ferromagnetic modes in spin glasses and dilute ferromagnets,, Z. Physik B, 39 (1980), 37.

[15]

F. Gay-Balmaz and T. S. Ratiu, The geometric structure of complex fluids,, Adv App Math, 42 (2009), 176.

[16]

F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Equivalent theories of liquid crystal dynamics,, , ().

[17]

F. Gay-Balmaz, T. S. Ratiu and C. Tronci, Euler-Poincaré approaches to nematodynamics,, Acta Appl. Math., 120 (2012). doi: 10.1007/s10440-012-9719-x.

[18]

F. Gay-Balmaz and C. Tronci, Reduction theory for symmetry breaking with applications to nematic systems,, Phys. D, 239 (2010), 1929.

[19]

F. Gay-Balmaz, C. Tronci and C. Vizman, Geodesic flows on the automorphism group of principal bundles,, , ().

[20]

P. G. de Gennes, Short range order effects in the isotropic phase of nematics and cholesterics,, Mol. Cryst. Liq. Cryst., 12 (1971), 193. doi: 10.1080/15421407108082773.

[21]

P. G. de Gennes and J. Prost, "The Physics of Liquid Crystals,", 2nd edn. Oxford University Press, (1993).

[22]

J. Gibbons, D. D. Holm and B. A. Kupershmidt, The Hamiltonian structure of classical chromohydrodynamics,, Physica D, 6 (1983), 179.

[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.

[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. doi: 10.1103/PhysRevB.16.2154.

[25]

D. D. Holm, Euler-Poincaré dynamics of perfect complex fluids,, in, (2002), 113.

[26]

D. D. Holm, Hamiltonian dynamics of a charged fluid, including electro- and magnetohydrodynamics,, Phys. Lett. A, 114 (1986), 137.

[27]

D. D. Holm, Hamiltonian dynamics and stability analysis of neutral electromagnetic fluids with induction,, Physica D, 25 (1987), 261.

[28]

D. D. Holm, R. I. Ivanov and J. R. Percival, $G$-Strands,, J. Nonlinear Sci., 22 (2012), 517. doi: 10.1007/s00332-012-9135-4.

[29]

D. D. Holm and B. A. Kupershmidt, Hamiltonian formulation of ferromagnetic hydrodynamics,, Phys. Lett. A, 129 (1988), 93.

[30]

D. D. Holm and B. A. Kupershmidt, Poisson structures of superfluids,, Phys. Lett. A, 91 (1982), 425.

[31]

D. D. Holm and B. A. Kupershmidt, The analogy between spin glasses and Yang-Mills fluids,, J. Math Phys., 29 (1988), 21. doi: 10.1063/1.528176.

[32]

D. D. Holm and B. A. Kupershmidt, Yang-Mills magnetohydrodynamics,, Phys. Rev. D, 30 (1984), 2557.

[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.

[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.

[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. doi: 10.1098/rspa.2010.0043.

[36]

D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions,", Oxford University Press, (2009).

[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.

[38]

D. D. Holm and C. Tronci, The geodesic Vlasov equation and its integrable moment closures,, J. Geom. Mech., 1 (2009), 181.

[39]

A. A. Isayev, Hamiltonian formalism in the theory of quadruple magnet,, Low Temp. Phys., 23 (1997), 933.

[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.

[41]

A. Kadič and D. G. B. Edelen, A gauge theory of dislocations and disclinations,, Lect. Notes Phys., 174 (1983).

[42]

Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma,", M. I. T. Press, (1967).

[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.

[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).

[45]

F. M. Leslie, Some topics in continuum theory of nematics,, Philos. Trans. Soc. London Ser. A, 309 (1983), 155.

[46]

M. Marklund and P. J. Morrison, Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations,, Phys. Lett. A, 305 (2011), 2362.

[47]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Springer-Verlag, (1994).

[48]

J. E. Marsden, T. S. Ratiu and A. Weinstein, Semidirect products and reduction in mechanics,, Trans. Amer. Math. Soc., 281 (1984), 147. doi: 10.1090/S0002-9947-1984-0719663-1.

[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.

[50]

R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations,, Lett. Math. Phys., 8 (1984), 59.

[51]

R. Montgomery, J. Marsden and T. Ratiu, Gauged Lie-Poisson structures,, Contemp. Math., 28 (1984), 101. doi: 10.1090/conm/028/751976.

[52]

K. Penc and M. Läuchli, Spin nematic phases in quantum spin systems,, Springer Ser. Solid-State Sci., 164 (2011), 331. doi: 10.1007/978-3-642-10589-0_13.

[53]

L. Onsager, The effects of shape on the interaction of colloidal particles,, Ann. N. Y. Acad. Sci., 51 (1949), 627.

[54]

R. G. Spencer, The Hamiltonian structure of multi-species fluid electrodynamics,, AIP Conf. Proc., 88 (1982), 121. doi: 10.1063/1.33630.

[55]

R. G. Spencer and A. N. Kaufman, Hamiltonian structure of two-fluid plasma dynamics,, Phys. Rev. A (3), 25 (1982), 2437.

[56]

H. Stark and T. C. Lubensky, Poisson-bracket approach to the dynamics of nematic liquid crystals,, Phys. Rev. E, 67 (2003).

[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).

[58]

H. Tsunetsugu and M. Arikawa, Spin nematic phase in $S=1$ triangular antiferromagnets,, J. Phys. Soc. Jpn., 75 (2006).

[59]

J. A. C. Veerman and D. Frenkel, Phase behavior of disklike hard-core mesogens,, Phys. Rev. A, 45 (1992), 5632.

[60]

G. E. Volovik and I. E. Dzyaloshinskiĭ, Additional localized degrees of freedom in spin glasses,, Sov. Phys. JETP, 48 (1978), 555.

[61]

G. E. Volovik and E. I. Kats, Nonlinear hydrodynamics of liquid crystals,, Sov. Phys. JETP, 54 (1981), 122.

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