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June  2013, 6(2): 429-458. doi: 10.3934/krm.2013.6.429

Collisionless kinetic theory of rolling molecules

Darryl D. Holm 1, , Vakhtang Putkaradze 2, and  Cesare Tronci 3,

1. 

Department of Mathematics, Imperial College London, London SW7 2AZ
2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada
3. 

Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom

Received  March 2012 Revised  October 2012 Published  February 2013

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems. Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.
Keywords: Nonholonomic dynamics, collisionless kinetics, Euler-Poncaré reduction..
Mathematics Subject Classification: Primary: 82C05, 70F25; Secondary: 76M6.
Citation: Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
References:
[1]

S. A. Chaplygin, On a motion of a heavy body of revolution on a horizontal plane. (translated from collected works), Theoretical Mechanics Mathematics (Russian), Gos. Izd. Tekhn.-Teoret. Lit., 1 (1948), 51-57. Google Scholar

[2]

S. A. Chaplygin, Theory of motion of non-holonomic systems: Reducing factor theorem, in "Collected Works, Vol. 1, Gostekhizdat," (1948), 15-25. Google Scholar

[3]

A. M. Bloch, Asymptotic Hamiltonian dynamics: The Toda lattice, the three-wave interaction and the nonholonomic Chaplygin sleigh, Physica D, 141 (2000), 297-315. doi: 10.1016/S0167-2789(00)00046-4.  Google Scholar

[4]

A. V. Borisov and I. S. Mamaaev, The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 73 (2009), 156-161. doi: 10.1016/j.jappmathmech.2009.04.005.  Google Scholar

[5]

S. Hochgerner and L. Garcia-Naranjo, G-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball, J. Geom. Mech., 1 (2009), 35-53. doi: 10.3934/jgm.2009.1.35.  Google Scholar

[6]

A. M. Bloch, "Nonholonomic Mechanics and Control," Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[7]

A. V. Borisov and I. S. Mamaev, Chaplygin's ball rolling problem is Hamiltonian, Mat. Zametki, 70 (2001), 793-795. doi: 10.1023/A:1012995330780.  Google Scholar

[8]

A. M. Bloch and A. G. Rojo, Quantization of a nonholonomic system, Phys. Rev. Lett., 101 (2008), 030402, 4 pp. doi: 10.1103/PhysRevLett.101.030402.  Google Scholar

[9]

D. D. Holm, "Geometric Mechanics. Part II. Rotation, Translation and Rolling," Imperial College Press, London, distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.  Google Scholar

[10]

S. D. Bond, B. J. Leimkuhler and B. B. Laird, The Nosé-Poincaré method for constant temperature molecular dynamics, Journal of Computational Physics, 151 (1999), 114-134. doi: 10.1006/jcph.1998.6171.  Google Scholar

[11]

R. Kutteh and R. B. Jones, Rigid body molecular dynamics with nonholonomic constraints: Molecular thermostat algorithms, Phys. Rev. E, 61 (2000), 3186-3198. Google Scholar

[12]

P. Collins, G. S. Ezra and S. Wiggins, Phase space structure and dynamics for the hamiltonian isokinetic thermostat, J. Chem. Phys., 133 (2010), 014105. Google Scholar

[13]

J. D. Ramshaw, Remarks on entropy and irreversibility in non-Hamiltonian systems, Phys. Lett. A, 116 (1986), 110-114. doi: 10.1016/0375-9601(86)90294-X.  Google Scholar

[14]

M. E. Tuckerman, C. J. Mundy and M. L. Klein, Toward a statistical thermodynamics of steady states, Phys. Rev. Lett, 78 (1997), 2042-2045. Google Scholar

[15]

M. E. Tuckerman, C. J. Mundy and G. J. Martyna, On the classical statistical mechanics of non-hamiltonian systems, Europhys. Lett, 45 (1999), 149-155. Google Scholar

[16]

M. E. Tuckerman, Y. Liu, G. Ciccotti and G. J. Martyna, Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems, J. Chem. Phys., 115 (2001), 1678-1702. Google Scholar

[17]

J. D. Ramshaw, Remarks on non-Hamiltonian statistical mechanics, Europhys. Lett., 59 (2002), 319-323. doi: 10.1209/epl/i2002-00196-9.  Google Scholar

[18]

G. S. Ezra, On the statistical mechanics of non-Hamiltonian systems: The generalized Liouville equation, entropy, and time-dependent metrics, J. Math. Chem., 35 (2004), 29-53. doi: 10.1023/B:JOMC.0000007811.79716.4d.  Google Scholar

[19]

A. Sergi and P. V. Giaquinta, On the geometry and entropy of non-Hamiltonian phase space,, Journal of Statistical Mechanics: Theory and Experiment, 2007 ().   Google Scholar

[20]

B. Kim and V. Putkaradze, Ordered and disordered dynamics in monolayers of rolling particles, Physical Review Letters, 105 (2010), 244302. Google Scholar

[21]

S. Hochgerner, Stochastic Chaplygin systems, Rep. Math. Phys., 66 (2010), 385-401. doi: 10.1016/S0034-4877(10)80010-2.  Google Scholar

[22]

A. D. Lewis, The geometry of the gibbs-appell equations and Gauss' principle of least constraint, Rep. Math. Phys., 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0.  Google Scholar

[23]

F. E. Low, A Lagrangian formulation of the Boltzmann-Vlasov equation for plasmas, Proc. Roy. Soc. London Ser. A, 248 (1958), 282-287.  Google Scholar

[24]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Viscek model, Applied Mathematics Letters, 25 (2011), 339-343. doi: 10.1016/j.aml.2011.09.011.  Google Scholar

[25]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.  Google Scholar

[26]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131.  Google Scholar

[27]

M. Bostan and J. A. Carrillo, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods in Appl. Sciences, to appear, (2012). Google Scholar

[28]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[29]

D. Schneider, Non-holonomic Euler-Poincaé equations and stability in Chaplygin's sphere, Dynamical Systems, 17 (2002), 87-130. doi: 10.1080/02681110110112852.  Google Scholar

[30]

J. J. Duistermaat, Chapligyn sphere, preprint, arXiv:math/0409019v1, 2004. Google Scholar

[31]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Advances in Mathematics, 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[32]

W. Pabst, Micropolar materials, Ceramics, 49 (2005), 170-180. Google Scholar

[33]

A. C. Eringen, "Microcontinuum Field Theories," Volume I and II, Springer-Verlag, 2001. Google Scholar

[34]

D. V. Zenkov and A. M. Bloch, Invariant measures of nonholonomic flows with internal degrees of freedom, Nonlinearity, 16 (2003), 1793-1807. doi: 10.1088/0951-7715/16/5/313.  Google Scholar

[35]

Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma," M.I.T. Press, Cambridge, Mass., 1967. Google Scholar

[36]

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. doi: 10.1063/1.532244.  Google Scholar

[37]

D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions," Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009.  Google Scholar

[38]

J.-A. Carrillo M. Bostan, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods Appl. Sci., to appear, (2012). Google Scholar

show all references

References:
[1]

S. A. Chaplygin, On a motion of a heavy body of revolution on a horizontal plane. (translated from collected works), Theoretical Mechanics Mathematics (Russian), Gos. Izd. Tekhn.-Teoret. Lit., 1 (1948), 51-57. Google Scholar

[2]

S. A. Chaplygin, Theory of motion of non-holonomic systems: Reducing factor theorem, in "Collected Works, Vol. 1, Gostekhizdat," (1948), 15-25. Google Scholar

[3]

A. M. Bloch, Asymptotic Hamiltonian dynamics: The Toda lattice, the three-wave interaction and the nonholonomic Chaplygin sleigh, Physica D, 141 (2000), 297-315. doi: 10.1016/S0167-2789(00)00046-4.  Google Scholar

[4]

A. V. Borisov and I. S. Mamaaev, The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 73 (2009), 156-161. doi: 10.1016/j.jappmathmech.2009.04.005.  Google Scholar

[5]

S. Hochgerner and L. Garcia-Naranjo, G-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball, J. Geom. Mech., 1 (2009), 35-53. doi: 10.3934/jgm.2009.1.35.  Google Scholar

[6]

A. M. Bloch, "Nonholonomic Mechanics and Control," Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar

[7]

A. V. Borisov and I. S. Mamaev, Chaplygin's ball rolling problem is Hamiltonian, Mat. Zametki, 70 (2001), 793-795. doi: 10.1023/A:1012995330780.  Google Scholar

[8]

A. M. Bloch and A. G. Rojo, Quantization of a nonholonomic system, Phys. Rev. Lett., 101 (2008), 030402, 4 pp. doi: 10.1103/PhysRevLett.101.030402.  Google Scholar

[9]

D. D. Holm, "Geometric Mechanics. Part II. Rotation, Translation and Rolling," Imperial College Press, London, distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.  Google Scholar

[10]

S. D. Bond, B. J. Leimkuhler and B. B. Laird, The Nosé-Poincaré method for constant temperature molecular dynamics, Journal of Computational Physics, 151 (1999), 114-134. doi: 10.1006/jcph.1998.6171.  Google Scholar

[11]

R. Kutteh and R. B. Jones, Rigid body molecular dynamics with nonholonomic constraints: Molecular thermostat algorithms, Phys. Rev. E, 61 (2000), 3186-3198. Google Scholar

[12]

P. Collins, G. S. Ezra and S. Wiggins, Phase space structure and dynamics for the hamiltonian isokinetic thermostat, J. Chem. Phys., 133 (2010), 014105. Google Scholar

[13]

J. D. Ramshaw, Remarks on entropy and irreversibility in non-Hamiltonian systems, Phys. Lett. A, 116 (1986), 110-114. doi: 10.1016/0375-9601(86)90294-X.  Google Scholar

[14]

M. E. Tuckerman, C. J. Mundy and M. L. Klein, Toward a statistical thermodynamics of steady states, Phys. Rev. Lett, 78 (1997), 2042-2045. Google Scholar

[15]

M. E. Tuckerman, C. J. Mundy and G. J. Martyna, On the classical statistical mechanics of non-hamiltonian systems, Europhys. Lett, 45 (1999), 149-155. Google Scholar

[16]

M. E. Tuckerman, Y. Liu, G. Ciccotti and G. J. Martyna, Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems, J. Chem. Phys., 115 (2001), 1678-1702. Google Scholar

[17]

J. D. Ramshaw, Remarks on non-Hamiltonian statistical mechanics, Europhys. Lett., 59 (2002), 319-323. doi: 10.1209/epl/i2002-00196-9.  Google Scholar

[18]

G. S. Ezra, On the statistical mechanics of non-Hamiltonian systems: The generalized Liouville equation, entropy, and time-dependent metrics, J. Math. Chem., 35 (2004), 29-53. doi: 10.1023/B:JOMC.0000007811.79716.4d.  Google Scholar

[19]

A. Sergi and P. V. Giaquinta, On the geometry and entropy of non-Hamiltonian phase space,, Journal of Statistical Mechanics: Theory and Experiment, 2007 ().   Google Scholar

[20]

B. Kim and V. Putkaradze, Ordered and disordered dynamics in monolayers of rolling particles, Physical Review Letters, 105 (2010), 244302. Google Scholar

[21]

S. Hochgerner, Stochastic Chaplygin systems, Rep. Math. Phys., 66 (2010), 385-401. doi: 10.1016/S0034-4877(10)80010-2.  Google Scholar

[22]

A. D. Lewis, The geometry of the gibbs-appell equations and Gauss' principle of least constraint, Rep. Math. Phys., 38 (1996), 11-28. doi: 10.1016/0034-4877(96)87675-0.  Google Scholar

[23]

F. E. Low, A Lagrangian formulation of the Boltzmann-Vlasov equation for plasmas, Proc. Roy. Soc. London Ser. A, 248 (1958), 282-287.  Google Scholar

[24]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Viscek model, Applied Mathematics Letters, 25 (2011), 339-343. doi: 10.1016/j.aml.2011.09.011.  Google Scholar

[25]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.  Google Scholar

[26]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod. Meth. Appl. Sci., 21 (2011), 515-539. doi: 10.1142/S0218202511005131.  Google Scholar

[27]

M. Bostan and J. A. Carrillo, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods in Appl. Sciences, to appear, (2012). Google Scholar

[28]

A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. M. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rat. Mech. Anal., 136 (1996), 21-99. doi: 10.1007/BF02199365.  Google Scholar

[29]

D. Schneider, Non-holonomic Euler-Poincaé equations and stability in Chaplygin's sphere, Dynamical Systems, 17 (2002), 87-130. doi: 10.1080/02681110110112852.  Google Scholar

[30]

J. J. Duistermaat, Chapligyn sphere, preprint, arXiv:math/0409019v1, 2004. Google Scholar

[31]

D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Advances in Mathematics, 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[32]

W. Pabst, Micropolar materials, Ceramics, 49 (2005), 170-180. Google Scholar

[33]

A. C. Eringen, "Microcontinuum Field Theories," Volume I and II, Springer-Verlag, 2001. Google Scholar

[34]

D. V. Zenkov and A. M. Bloch, Invariant measures of nonholonomic flows with internal degrees of freedom, Nonlinearity, 16 (2003), 1793-1807. doi: 10.1088/0951-7715/16/5/313.  Google Scholar

[35]

Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma," M.I.T. Press, Cambridge, Mass., 1967. Google Scholar

[36]

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. doi: 10.1063/1.532244.  Google Scholar

[37]

D. D. Holm, T. Schmah and C. Stoica, "Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions," Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009.  Google Scholar

[38]

J.-A. Carrillo M. Bostan, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods Appl. Sci., to appear, (2012). Google Scholar

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