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Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation
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 |
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). |
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. |
[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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