April  2022, 15(2): 283-315. doi: 10.3934/krm.2022008

A Lie algebra-theoretic approach to characterisation of collision invariants of the Boltzmann equation for general convex particles

Department of Mathematics and Physics, Nottingham Trent University, Nottingham, UK, NG1 4FQ

Received  October 2021 Published  April 2022 Early access  March 2022

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.

Citation: Mark Wilkinson. A Lie algebra-theoretic approach to characterisation of collision invariants of the Boltzmann equation for general convex particles. Kinetic and Related Models, 2022, 15 (2) : 283-315. doi: 10.3934/krm.2022008
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. DonevS. 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

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. DonevS. 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.

Figure 1.  A collision configuration of two hard spheres $ \mathtt{B} $ and $ \overline{\mathtt{B}} $ in $ \mathbb{R}^{3} $, each of which is congruent to a given reference set $ \mathtt{B}_{\ast}: = \{y\in\mathbb{R}^{3}\, :\, |y|\leq \frac{1}{2}\} $. The collision parameter $ n\in\mathbb{S}^{2} $ represents the direction from the centre of mass of the unbarred sphere to that of the barred
Figure 2.  A collision configuration of two compact, convex subsets $ \mathtt{P} $ and $ \overline{\mathtt{P}} $ of $ \mathbb{R}^{3} $, each of which is congruent to a given reference set $ \mathtt{P}_{\ast} $. The matrices $ R, \overline{R}\in\mathrm{SO}(3) $ represent the orientations of the two hard particles, $ n\in\mathbb{S}^{2} $ represents the direction vector connecting the centre of mass of the unbarred particle to that of the barred, and $ d_{\beta}>0 $ denotes the distance of closest approach (8)
Figure 3.  A collision configuration of two compact, convex subsets $ \mathtt{P} $ and $ \overline{\mathtt{P}} $ of $ \mathbb{R}^{2} $, each of which is congruent to a given reference set $ \mathtt{P}_{\ast} $. The elevation angle $ \psi\in\mathbb{S}^{1} $ determines the direction vector $ e(\psi): = (\cos\psi, \sin\psi) $ directed from the centre of the unbarred particle to that of the barred, $ \vartheta, \overline{\vartheta}\in\mathbb{S}^{1} $ denote the orientations of the particles, whilst $ d_{\beta}>0 $ denotes the distance of closest approach (19)
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