# American Institute of Mathematical Sciences

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

## Collisionless kinetic theory of rolling molecules

 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.
Citation: Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic and 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. [2] S. A. Chaplygin, Theory of motion of non-holonomic systems: Reducing factor theorem, in "Collected Works, Vol. 1, Gostekhizdat," (1948), 15-25. [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. [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. [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. [6] A. M. Bloch, "Nonholonomic Mechanics and Control," Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [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. [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. [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. [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. [11] R. Kutteh and R. B. Jones, Rigid body molecular dynamics with nonholonomic constraints: Molecular thermostat algorithms, Phys. Rev. E, 61 (2000), 3186-3198. [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. [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. [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. [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. [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. [17] J. D. Ramshaw, Remarks on non-Hamiltonian statistical mechanics, Europhys. Lett., 59 (2002), 319-323. doi: 10.1209/epl/i2002-00196-9. [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. [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, P02013, 20 pp. [20] B. Kim and V. Putkaradze, Ordered and disordered dynamics in monolayers of rolling particles, Physical Review Letters, 105 (2010), 244302. [21] S. Hochgerner, Stochastic Chaplygin systems, Rep. Math. Phys., 66 (2010), 385-401. doi: 10.1016/S0034-4877(10)80010-2. [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. [23] F. E. Low, A Lagrangian formulation of the Boltzmann-Vlasov equation for plasmas, Proc. Roy. Soc. London Ser. A, 248 (1958), 282-287. [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. [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. [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. [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). [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. [29] D. Schneider, Non-holonomic Euler-Poincaé equations and stability in Chaplygin's sphere, Dynamical Systems, 17 (2002), 87-130. doi: 10.1080/02681110110112852. [30] J. J. Duistermaat, Chapligyn sphere, preprint, arXiv:math/0409019v1, 2004. [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. [32] W. Pabst, Micropolar materials, Ceramics, 49 (2005), 170-180. [33] A. C. Eringen, "Microcontinuum Field Theories," Volume I and II, Springer-Verlag, 2001. [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. [35] Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma," M.I.T. Press, Cambridge, Mass., 1967. [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. [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. [38] J.-A. Carrillo M. Bostan, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods Appl. Sci., to appear, (2012).

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##### 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. [2] S. A. Chaplygin, Theory of motion of non-holonomic systems: Reducing factor theorem, in "Collected Works, Vol. 1, Gostekhizdat," (1948), 15-25. [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. [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. [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. [6] A. M. Bloch, "Nonholonomic Mechanics and Control," Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [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. [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. [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. [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. [11] R. Kutteh and R. B. Jones, Rigid body molecular dynamics with nonholonomic constraints: Molecular thermostat algorithms, Phys. Rev. E, 61 (2000), 3186-3198. [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. [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. [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. [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. [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. [17] J. D. Ramshaw, Remarks on non-Hamiltonian statistical mechanics, Europhys. Lett., 59 (2002), 319-323. doi: 10.1209/epl/i2002-00196-9. [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. [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, P02013, 20 pp. [20] B. Kim and V. Putkaradze, Ordered and disordered dynamics in monolayers of rolling particles, Physical Review Letters, 105 (2010), 244302. [21] S. Hochgerner, Stochastic Chaplygin systems, Rep. Math. Phys., 66 (2010), 385-401. doi: 10.1016/S0034-4877(10)80010-2. [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. [23] F. E. Low, A Lagrangian formulation of the Boltzmann-Vlasov equation for plasmas, Proc. Roy. Soc. London Ser. A, 248 (1958), 282-287. [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. [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. [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. [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). [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. [29] D. Schneider, Non-holonomic Euler-Poincaé equations and stability in Chaplygin's sphere, Dynamical Systems, 17 (2002), 87-130. doi: 10.1080/02681110110112852. [30] J. J. Duistermaat, Chapligyn sphere, preprint, arXiv:math/0409019v1, 2004. [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. [32] W. Pabst, Micropolar materials, Ceramics, 49 (2005), 170-180. [33] A. C. Eringen, "Microcontinuum Field Theories," Volume I and II, Springer-Verlag, 2001. [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. [35] Y. L. Klimontovich, "The Statistical Theory of Non-equilibrium Processes in a Plasma," M.I.T. Press, Cambridge, Mass., 1967. [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. [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. [38] J.-A. Carrillo M. Bostan, Asymptotic fixed-speed reduced dynamics for kinetic equations in swarming, Math. Models Methods Appl. Sci., to appear, (2012).
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