Harold Grad and the Boltzmann Equation

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Harold Grad and the Boltzmann Equation

Amelia Carolina Sparavigna

To cite this version:

Harold Grad and the Boltzmann Equation

Amelia Carolina Sparavigna (Department of Applied Science and Technology, Politecnico di Torino)

Published in physic.philica.com

Abstract

Grad was a mathematician who specialized in the kinetic theory of gases and in the statistical mechanics applied to plasma physics and magnetohydrodynamics. Here we discuss in particular his article entitled Theory of the Boltzmann Equation, published in 1964.

schemes - a detailed discussion is given in [10]. References to some of the articles written by Harold Grad

are given in [12-35] .

Let us consider [36], the Grad's publication that we can find in Archive (references [37-43] are mentioning some of the other Grad's works present in the same digital library). In the abstract of [36], Grad tells that he was presenting a survey of the theory of the Boltzmann equation for a dilute monoatomic gas and also an improved understanding of the significance of the approximate results that we can obtain from the equation, in the framework of a "general trend away from ad-hoc and toward more precise mathematical procedures". So we find in the Introduction, which is a survey on Boltzmann equation, several references concerning the solution of this equation. Then, he continues addressing the general problem of the existence of solutions. "The development of a comprehensive theory of existence and qualitative behavior of solutions of the Boltzmann equation serves several "practical" purposes". The first purpose is to find a help for recognizing "whether the solution of a specific problem is representative of more general cases or whether it may be exceptional. For example, … almost all the extrapolations that were drawn from a very ingenious explicit solution of an exact nonlinear Boltzmann equation have been found to be nonrepresentative of the behavior of the typical solution". To reinforce his point of view, he tells that "A more basic goal of a general theory is to determine whether or not there exist any solutions at all. Such a failure is always a conceptual possibility no matter how convincing are the physical arguments which suggest the validity of the equation and no matter how plausible are the results derived by nonrigorous approximate means". And this happens because, in the Boltzmann equation we have an "inherent sensitivity of mathematical structures to arbitrarily small perturbations, which are undetectable physically".

According to Grad, the crucial point is the necessity to give an interpretation of "the variables that arise naturally in the Hilbert and Chapman-Enskog expansions as suitably modified macroscopic properties of the gas". "As a result of this generalized interpretation, the Chapman-Enskog expansion has been found to be representative of very general solutions of the Boltzmann equation in the limit of small mean free path".

Then, we find the Section 3 of [36] devoted to the small mean free path, the Section 4 to the large mean free path and the Section 5 to the role of the spectrum in the interplay between the streaming term and the collision term of the Boltzmann equation. In reading these sections, any physicist who is working on the Boltzmann equation, fells that her/his comprehension of the Boltzmann equation has been increased. This is coming from the fact that the Grad's work is approaching the Boltzmann equation from a rather different point of view. It was that of a physical mathematician who aimed to study the equation in the framework of an "internal" mathematical consistency and an "external" confrontation with physical observable quantities.

References

[1] Ruelle, D. (2008). What physical quantities make sense in non equilibrium statistical mechanics? In Boltzmann's Legacy, Giovanni Gallavotti, Wolfgang L. Reiter, Jakob Yngvason Editors, European Mathematical Society, 2008 Page 89. [2] Sparavigna, A. C. (2016). On the Boltzmann Equation of Thermal Transport for Interacting Phonons and Electrons, Mechanics, Materials Science & Engineering Journal, 2016(5), 1-13, Magnolithe, ISSN: 2412-5954

[3] Omini, M., & Sparavigna, A. (1996). Beyond the isotropic-model approximation in the theory of thermal conductivity. Physical Review B, 53(14), 9064-9073. DOI: 10.1103/physrevb.53.9064 [4] Omini, M., & Sparavigna, A. (1997). Heat transport in dielectric solids with diamond structure. Nuovo Cimento D Serie, 19, 1537. [5] Omini, M., & Sparavigna, A. (1995). An iterative approach to the phonon Boltzmann equation in the theory of thermal conductivity. Physica B: Condensed Matter, 212(2), 101-112. DOI:10.1016/0921-4526(95)00016-3

[6] Sparavigna, A. C. (2016). The Boltzmann equation of phonon thermal transport solved in the relaxation time approximation – I – Theory. Mechanics, Materials Science & Engineering Journal, 2016(3), 1-13.

[7] https://de.wikipedia.org/wiki/Harold_Grad Retrieved January 4, 2018 [8] https://en.wikipedia.org/wiki/Harold_Grad Retrieved January 4, 2018 [9] https://web.archive.org/web/20140227165756/http://www.cims.nyu.edu/information/brochure/student.html

[10] Blank, A. A. (1987). Harold Grad, Physics Today, 40(3), 86. DOI: 10.1063/1.2819960 [11] Grad, H. (1958). Principles of the kinetic theory of gases. In Thermodynamik der Gase/Thermodynamics of Gases (pp. 205-294). Springer Berlin Heidelberg. [12] Grad, H. (1949). On the kinetic theory of rarefied gases. Communications on Pure and Applied Mathematics, 2(4), 331-407.

[15] Grad, H. (1952). The profile of a steady plane shock wave. Communications on Pure and Applied Mathematics, 5(3), 257-300.

[16] Grad, H. (1952). Statistical mechanics, thermodynamics, and fluid dynamics of systems with an arbitrary number of integrals. Communications on Pure and Applied Mathematics, 5(4), 455-494. [17] Grad, H. (1952). Statistical mechanics of dynamical systems with integrals other than energy. The Journal of Physical Chemistry, 56(9), 1039-1048. [18] Grad, H. (1960). Theory of Rarefied Gas Dynamics, In Rarefied Gas Dynamics, Edited by F. Devienne, Pergamon Press, London.

[19] Grad, H. (1960). Reducible problems in magneto-fluid dynamic steady flows. Reviews of Modern Physics, 32(4), 830.

[20] Grad, H. (1960). Plasma trapping in cusped geometries. Physical Review Letters, 4(5), 222. [21] Grad, H. (1961). The many faces of entropy. Communications on Pure and Applied Mathematics, 14(3), 323-354.

[22] Grad, H. (1961). Microscopic and macroscopic models in plasma physics (No. TID-13957; MF-19). New York Univ. New York. Inst. of Mathematical Sciences. [23] Grad, H. (1969). Plasmas, Physics Today, 22(12), 34. DOI: 10.1063/1.3035293 [24] Grad, H. (1963). Asymptotic theory of the Boltzmann equation. The Physics of Fluids, 6(2), 147-181. [25] Grad, H. (1964). Some new variational properties of hydromagnetic equilibria. The Physics of Fluids, 7(8), 1283-1292.

[26] Grad, H. (1964). U.S. Patent No. 3,141,826. Washington, DC: U.S. Patent and Trademark Office. [27] Grad, H. (1965). On Boltzmann’s H-theorem. Journal of the Society for Industrial and Applied Mathematics, 13(1), 259-277. [28] Grad, H. (1966). High frequency sound according to the Boltzmann equation. SIAM Journal on Applied Mathematics, 14(4), 935-955. [29] Grad, H. (1967). Toroidal containment of a plasma. The Physics of Fluids, 10(1), 137-154. [30] Grad, H. (1967). Levels of description in statistical mechanics and thermodynamics. In Delaware Seminar in the Foundations of Physics (pp. 49-76). Springer, Berlin, Heidelberg. [31] Grad, H., & Hogan, J. (1970). Classical Diffusion in a Tokomak. Physical Review Letters, 24(24), 1337.

[32] Grad, H. (1969). Singular and nonuniform limits of solutions of the Boltzmann equation. Transport Theory, 1, 269-308. [33] Grad, H. (1974). Singular Limits of Solutions of Boltzmann's equation. Rarefied Gas Dynamics, (8), 37.

[40] Grad, H. (1964). Accuracy and limits of applicability of solutions of equations of transport; dilute monoatomic gases, Publisher New York: Courant Institute of Mathematical Sciences, New York. [41] Singular and nonuniform limits of solutions of the Boltzmann equation, by Grad, Harold, Publication date 1967 Publisher New York: Courant Institute of Mathematical Sciences, New York University [42] Grad, H. (1971). Classical plasma diffusion, Publisher New York: Courant Institute of Mathematical Sciences, New York University. [43] Grad, H. (1959). Equations of flow in a rarefied atmosphere, Publisher New York: Courant Institute of Mathematical Sciences, New York University. . [44] Golse, F. (2005). The Boltzmann equation and its hydrodynamic limits. Evolutionary Equations, 2, 159-301.

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