Department of Mathematics and Physics, Nottingham Trent University, Nottingham, UK, NG1 4FQ |
By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that the collision invariant is of class $ \mathscr{C}^{1} $. This work extends that of Saint-Raymond and Wilkinson (Communications on Pure and Applied Mathematics (2018), 71(8), pp. 1494–1534), in which the authors characterise collision invariants only in the case of the so-called canonical physical scattering family. Indeed, our method extends to the case of non-canonical physical scattering, whose existence was reported in Wilkinson (Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055–2083). Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson as we place no symmetry hypotheses on the underlying non-spherical particles which make up the gas under consideration. The techniques established in this paper also yield a new proof of the result of Boltzmann for collision invariants of class $ \mathscr{C}^{1} $ in the classical case of hard spheres.
| [1] |
L. Arkeryd,
On the Boltzmann equation. II. The full initial value problem, Arch. Rational Mech. Anal., 45 (1972), 17-34.
doi: 10.1007/BF00253393. |
| [2] |
P. Ballard,
The dynamics of discrete mechanical systems with perfect unilateral constraints, Arch. Rational Mech. Anal., 154 (2000), 199-274.
doi: 10.1007/s002050000105. |
| [3] |
J. Bochnak, M. Coste and M. F. Roy, Real Algebraic Geometry, Springer-Verlag, Berlin, 1987. |
| [4] |
L. Boltzmann, Wissenschaftliche Abhandlungen, Cambridge University Press, 2012. |
| [5] |
C. Cercignani,
Are there more than five linearly-independent collision invariants for the Boltzmann equation?, J. Statist. Phys., 58 (1990), 817-823.
doi: 10.1007/BF01026552. |
| [6] |
C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, Springer New York, 2012. |
| [7] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [8] |
A. Donev, S. Torquato and F. H. Stillinger,
Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles: II. Applications to ellipses and ellipsoids, J. Comput. Phys., 202 (2005), 765-793.
doi: 10.1016/j.jcp.2004.08.025. |
| [9] |
M. Eaton and M. Perlman,
Generating $\mathrm{O}(n)$ with reflections, Pacific J. Math., 73 (1977), 73-80.
|
| [10] |
T. H. Gronwall,
A functional equation in the kinetic theory of gases, Ann. of Math., 17 (1915), 1-4.
doi: 10.2307/2007210. |
| [11] |
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality, 2$^{nd}$ edition, Springer, 2009.
doi: 10.1007/978-3-7643-8749-5. |
| [12] |
Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge
University Press, 2005.
doi: 10.1017/CBO9780511801228. |
| [13] |
L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-92847-8. |
| [14] |
L. Saint-Raymond and M. Wilkinson,
On collision invariants for linear scattering, Comm. Pure Appl. Math., 71 (2018), 1494-1534.
doi: 10.1002/cpa.21761. |
| [15] |
C. Truesdell and R. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas: Treated as a Branch of Rational Mechanics, New York-London, 1980. |
| [16] |
M. Wilkinson,
On the non-uniqueness of physical scattering for hard non-spherical particles, Arch. Rational Mech. Anal, 235 (2020), 2055-2083.
doi: 10.1007/s00205-019-01460-y. |
show all references
| [1] |
L. Arkeryd,
On the Boltzmann equation. II. The full initial value problem, Arch. Rational Mech. Anal., 45 (1972), 17-34.
doi: 10.1007/BF00253393. |
| [2] |
P. Ballard,
The dynamics of discrete mechanical systems with perfect unilateral constraints, Arch. Rational Mech. Anal., 154 (2000), 199-274.
doi: 10.1007/s002050000105. |
| [3] |
J. Bochnak, M. Coste and M. F. Roy, Real Algebraic Geometry, Springer-Verlag, Berlin, 1987. |
| [4] |
L. Boltzmann, Wissenschaftliche Abhandlungen, Cambridge University Press, 2012. |
| [5] |
C. Cercignani,
Are there more than five linearly-independent collision invariants for the Boltzmann equation?, J. Statist. Phys., 58 (1990), 817-823.
doi: 10.1007/BF01026552. |
| [6] |
C. Cercignani, The Boltzmann Equation and its Applications, Applied Mathematical Sciences, Springer New York, 2012. |
| [7] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [8] |
A. Donev, S. Torquato and F. H. Stillinger,
Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles: II. Applications to ellipses and ellipsoids, J. Comput. Phys., 202 (2005), 765-793.
doi: 10.1016/j.jcp.2004.08.025. |
| [9] |
M. Eaton and M. Perlman,
Generating $\mathrm{O}(n)$ with reflections, Pacific J. Math., 73 (1977), 73-80.
|
| [10] |
T. H. Gronwall,
A functional equation in the kinetic theory of gases, Ann. of Math., 17 (1915), 1-4.
doi: 10.2307/2007210. |
| [11] |
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality, 2$^{nd}$ edition, Springer, 2009.
doi: 10.1007/978-3-7643-8749-5. |
| [12] |
Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge
University Press, 2005.
doi: 10.1017/CBO9780511801228. |
| [13] |
L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-92847-8. |
| [14] |
L. Saint-Raymond and M. Wilkinson,
On collision invariants for linear scattering, Comm. Pure Appl. Math., 71 (2018), 1494-1534.
doi: 10.1002/cpa.21761. |
| [15] |
C. Truesdell and R. Muncaster, Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas: Treated as a Branch of Rational Mechanics, New York-London, 1980. |
| [16] |
M. Wilkinson,
On the non-uniqueness of physical scattering for hard non-spherical particles, Arch. Rational Mech. Anal, 235 (2020), 2055-2083.
doi: 10.1007/s00205-019-01460-y. |
| [1] |
Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065 |
| [2] |
Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417 |
| [3] |
Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic and Related Models, 2015, 8 (2) : 309-333. doi: 10.3934/krm.2015.8.309 |
| [4] |
Özlem Orhan, Teoman Özer. New conservation forms and Lie algebras of Ermakov-Pinney equation. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 735-746. doi: 10.3934/dcdss.2018046 |
| [5] |
Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 |
| [6] |
Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124 |
| [7] |
Joris Vankerschaver, Eva Kanso, Jerrold E. Marsden. The geometry and dynamics of interacting rigid bodies and point vortices. Journal of Geometric Mechanics, 2009, 1 (2) : 223-266. doi: 10.3934/jgm.2009.1.223 |
| [8] |
José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 |
| [9] |
Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic and Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034 |
| [10] |
Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic and Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381 |
| [11] |
Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121 |
| [12] |
Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 |
| [13] |
Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249 |
| [14] |
Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373 |
| [15] |
Dmitriy Chebanov. New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies. Conference Publications, 2013, 2013 (special) : 105-113. doi: 10.3934/proc.2013.2013.105 |
| [16] |
Nicolai Sætran, Antonella Zanna. Chains of rigid bodies and their numerical simulation by local frame methods. Journal of Computational Dynamics, 2019, 6 (2) : 409-427. doi: 10.3934/jcd.2019021 |
| [17] |
Giulio G. Giusteri, Alfredo Marzocchi, Alessandro Musesti. Nonlinear free fall of one-dimensional rigid bodies in hyperviscous fluids. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2145-2157. doi: 10.3934/dcdsb.2014.19.2145 |
| [18] |
Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic and Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014 |
| [19] |
Adriano Festa, Andrea Tosin, Marie-Therese Wolfram. Kinetic description of collision avoidance in pedestrian crowds by sidestepping. Kinetic and Related Models, 2018, 11 (3) : 491-520. doi: 10.3934/krm.2018022 |
| [20] |
Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic and Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019 |
2021 Impact Factor: 1.398
[Back to Top]
