Comptes Rendus

Putting physics into equations: a problem of mathematics?

"In 1900, at the Second International Congress of Mathematicians, David Hilbert set forth a list of 23 mathematical challenges for the coming century, known as the "Hilbert problems." The sixth of these problems concerns the foundations of classical mechanics. This problem was motivated by the recent work of Ludwig Boltzmann who introduced a description of gases intermediate between atomic models based on Newtonian mechanics, and continuous fluid models.
A natural question is whether these different models give consistent predictions, and Hilbert suggested using methods based on the idea of "boundary crossing" to provide a mathematical answer to this question.
Despite more than a century of effort, the problem raised by Hilbert remains largely unsolved." (Laure Saint-Raymond)

The presentation of May 23, 2023 by Laure Saint-Raymond (Meeting 5 to 7) aims to explain the difficulties encountered in this quest and to show how they can bring an interesting light on physics.

This collection proposes to gather a part of the notes published on this transverse subject in the Comptes Rendus Mathématique, Physique et Mécanique. 

*VI. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.

Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach, Hertz, Boltzmann and Volkmann. It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Thus Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely one might try to derive the laws of the motion of rigid bodies by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. For the question as to the equivalence of different systems of axioms is always of great theoretical interest.

If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. At the same time Lie's a principle of subdivision can perhaps be derived from profound theory of infinite transformation groups. The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed.

Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain such proof always forces us most effectually to an exact formulation of the axioms.

So far we have considered only questions concerning the foundations of the mathematical sciences. Indeed, the study of the foundations of a science is always particularly attractive, and the testing of these foundations will always be among the foremost problems of the investigator. Weierstrass once said, "The final object always to be kept in mind is to arrive at a correct understanding of the foundations of the science. ... But to make any progress in the sciences the study of particular problems is, of course, indispensable." In fact, a thorough understanding of its special theories is necessary to the successful treatment of the foundations of the science. Only that architect is in the position to lay a sure foundation for a structure who knows its purpose thoroughly and in detail. So we turn now to the special problems of the separate branches of mathematics and consider first arithmetic and algebra.

Limite de diffusion linéaire pour un système déterministe de sphères dures

Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure

Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 411-419

Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

Arsénio, Diogo; Saint-Raymond, Laure

Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 357-360

Velocity averaging in L 1 for the transport equation

Golse, François; Saint-Raymond, Laure

Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 557-562

Calcul pseudodifférentiel en grande dimension et limites thermodynamiques

Royer, Christophe

Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 413-418

The Boltzmann–Grad limit of the periodic Lorentz gas in two space dimensions

Caglioti, Emanuele; Golse, François

Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 477-482

From hard spheres dynamics to the Stokes–Fourier equations: An L2 analysis of the Boltzmann–Grad limit

Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure

Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 623-627

Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics

Golse, François; Paul, Thierry

Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 177-197

Derivation of the Schrödinger–Poisson equation from the quantum 𝐍-body problem

Bardos, Claude; Erdös, Laszlo; Golse, François; Mauser, Norbert; Yau, Horng-Tzer

Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 515-520

A microscopic view of the Fourier law

Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure

Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 402-418

Role of conserved quantities in Fourier's law for diffusive mechanical systems

Olla, Stefano

Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 429-441

The vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow

Bardos, Claude; Titi, Edriss S.; Wiedemann, Emil

Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 757-760

Observation estimate for kinetic transport equations by diffusion approximation

Bardos, Claude; Phung, Kim Dang

Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 640-664

A formalism for the differentiation of conservation laws

Bardos, Claude; Pironneau, Olivier

Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 839-845

Global existence of weak solutions to some micro-macro models

Lions, Pierre-Louis; Masmoudi, Nader

Comptes Rendus. Mathématique, Volume 345 (2007) no. 1, pp. 15-20

Stratified radiative transfer for multidimensional fluids

Golse, François; Pironneau, Olivier

Comptes Rendus. Mécanique, Volume 350 (2022) no. S1, pp. 1-15

Regularity of solutions for the Boltzmann equation without angular cutoff

Alexandre, Radjesvarane; Morimoto, Yoshinore; Ukai, Seiji; Xu, Chao-Jiang; Yang, Tong

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 747-752

On steady-state preserving spectral methods for homogeneous Boltzmann equations

Filbet, Francis; Pareschi, Lorenzo; Rey, Thomas

Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 309-314

About Kacʼs program in kinetic theory

Mischler, Stéphane; Mouhot, Clément

Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1245-1250

Lattice Fluid Dynamics: Thirty-five Years Down the Road

Succi, Sauro

Comptes Rendus. Mécanique, Volume 350 (2023) no. S1, pp. 1-12

A hierarchy of reduced models to approximate Vlasov–Maxwell equations for slow time variations

Assous, Franck; Furman, Yevgeni

Comptes Rendus. Mécanique, Volume 348 (2021) no. 12, pp. 969-981

High order asymptotic-preserving schemes for the Boltzmann equation

Dimarco, Giacomo; Pareschi, Lorenzo

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 481-486

Un modèle ES–BGK pour des mélanges de gaz

Brull, Stéphane

Comptes Rendus. Mathématique, Volume 351 (2013) no. 19-20, pp. 775-779

A boundary matching micro/macro decomposition for kinetic equations

Lemou, Mohammed; Méhats, Florian

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 479-484

Kinetic theory and Bose–Einstein condensation

Connaughton, Colm; Pomeau, Yves

Comptes Rendus. Physique, Volume 5 (2004) no. 1, pp. 91-106