Harold Grad and the Boltzmann Equation Amelia Carolina Sparavigna

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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.

In the Internet Archive, non-profit digital library which is providing free public access to collections of digitized materials, we can always find publications of great value for studies concerning the history of science. During a research made for finding articles and books on Ludwig Boltzmann, one of the fathers of the equilibrium statistical mechanics and, according to [1], the father of the non-equilibrium statistical mechanics, I found an interesting publication. It is entitled "Theory of the Boltzmann Equation", written by Harold Grad and published in 1964 by the Courant Institute of Mathematical Sciences, New York University. This publication diverted my research from Boltzmann into an investigation concerning the Grad's works, in particular his interest in the study of Boltzmann Equation (it could not be otherwise, for a person who investigates the thermal transport by means of this equation [2-6]). Harold Grad (1923 - 1986) was born in New York City. He was a mathematician who specialized in statistical mechanics applied to plasma physics and magnetohydrodynamics [7,8]. Grad became bachelor in 1943, after his studies in Electrical Engineering at the Cooper Union for the Advancement of Science and Art, a College in Lower Manhattan, New York. Then he obtained his Master's Degree in 1945 from New York University. Grad did his doctoral work under Richard Courant and graduated in 1948, with a thesis on the "Approximation in the Boltzmann Equation by Moments". After the doctoral work, he continued to work at the Courant Institute of Mathematical Sciences of New York University, from 1948 as associate professor and from 1957 as professor. Grad founded the Magneto-fluid Dynamics Division of the Institute [8,9]. . Actually, as told in [8], Harold Grad had developed in his thesis new methods for the solution of the Boltzmann equation, which allowed a simplified treatment of the dynamics of gases [9]. As explained in [10], this method was obtained after the explicit introduction of a hierarchy of models interpolating between the kinetic theory and the fluid dynamics, to reach a level higher than the Navier-Stokes system, by means of the so-called thirteen-moments equations [10]. As "architect of general systems" [10], in a few years he was recognized as a leader in the kinetic theory of gases. In 1958, he published a monograph on this subject that became a fundamental article for following researches [11], and in 1963, he contributed with a significant paper on the asymptotic theory of Boltzmann equation [10,24]. He was present also in statistical mechanics, where he contributed to the discussion on entropy and molecular chaos [10]. For what concerns his studies on plasma physics - he was a supporter of many early fusion 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". For this reason, according to Grad, the validity of the Boltzmann equation "must be based on studies of internal mathematical consistency as well as on external confrontation with experiment". He also noted that, at the time, this was particularly true for the existence of nonlinear solutions. "There are powerful existence theorems for spatially homogeneous problems in which fluid dynamics plays no role. But the question of the existence of spatially dependent solutions is certainly as difficult as the corresponding question in nonlinear fluid dynamics, and this connection is, in fact, the most important heuristic argument for a belief in the reasonable behavior of solutions of the Boltzmann equation in the large". Another difficulty evidenced by Grad is the following: "there is no evidence at this time that even the linear equation has any validity for potentials of infinite total cross-section. We would expect no physical differences between an infinite range but rapidly decaying intermolecular potential and a closely approximating cutoff potential; but the mathematics seems to be extraordinarily sensitive to this point … (we must distinguish solution of the Boltzmann equation from solution of the integral equation which arises in transport theory and which is known to be insensitive to "small" changes in the potential)". It is clear from these sentences that the approach of a mathematician, as Grad was, to the Boltzmann equation is different from that of a physicist, who is involved in the study of the transport theory, such as the Boltzmann equation for the thermal transport [2]. In [36], Grad is reporting the researches he made concerning the "significance of the Hilbert and Chapman-Enskog theories as formal asymptotic expansions of solutions of the Boltzmann equation". He tells that this significance was described qualitatively in 1958 [11], but in the Section 3 of [36], it is described in full mathematical detail (to see how Hilbert was involved in the problem, I suggest article in Ref.44, where the reader can find the discussions on Chapman-Enskog expansion and the method of moments too). 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. [13] Grad, H. (1949). Note on N-dimensional Hermite polynomials. Communications on Pure and Applied Mathematics, 2(4), 325-330. [14] Grad, H. (1949). Resonance burning in rocket motors. Communications on Pure and Applied Mathematics, 2(1), 79-102. [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. [34] Grad, H. (1965). On Boltzmann’s H-theorem. Journal of the Society for Industrial and Applied Mathematics, 13(1), 259-277. [35] Grad, H. (1964). U.S. Patent No. 3,141,826. Washington, DC: U.S. Patent and Trademark Office. [36] Grad, H. (1964). Theory of the Boltzmann equation. Courant Institute of Mathematical Sciences, New York University. https://archive.org/details/theoryofboltzman00grad [37] Grad, H. (1966). The guiding center plasma, Publisher New York: Courant Institute of Mathematical Sciences, New York University. [38] Grad, H. (1953). Thermonuclear reaction rates in and electrical discharge, Publisher New York: Courant Institute of Mathematical Sciences, New York University. [39] Grad, H. (1957). Theory of cusped geometries. I: General Survey. Publisher New York: Courant Institute of Mathematical Sciences, New York University. [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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The full citation for this Article is: Sparavigna, A. C. (2018). Harold Grad and the Boltzmann Equation. PHILICA.COM Article number 1211.