This paper is a survey paper describing the current progress in Hilbert’s Sixth problem. It gives an overview of the proof of Lanford’s Theorem [36,52], which describes how to derive the Boltzmann equation as the first marginal of the limit of hard sphere dynamics. This limit, taken in both the diameter of the particles and the number of them, is forced to satisfy , and is called the Boltzmann-Grad limit.

Included in the survey are summaries of results of the results F. Golse and L. Saint-Raymond relating limit of rescaled Boltzmann equations to solutions to incompressible Navier-Stokes [39,40]. Also, the survey describes the result of T. Bodineau, I. Gallagher, and L. Saint-Raymond relating a limit of rescaled solutions to the linearized Boltzmann equation and the heat equation [10].

The following references are found in the bibliography of the paper and are related to extending a “Lanford type” theorem to more general situations where the particles are of differing masses or sizes.

This paper describes the following physical scenario: spherical, hard particles of diameter and mass 1, and one smooth, large particle which has mass . It considers the Boltzmann-Grad limit (for dimension ) , and shows that for any time , the distribution of the in phase space is asymptotically close to a Gibbs measure times a solution to the linear Boltzmann equation [Theorem 1.1].

Note it is one large particle and many small particles in this scenario, and is fundamentally a deterministic setup.

The physical scenario for this paper is the following: A large particle of fixed mass is suspended in a “heat bath” comprised of infinitely many small particles of mass . These smaller particles only interact with the larger particle, and their dynamics are chosen so that their velocity distribution is preserved up until they collide with the larger particle.

This model uses infinite point-particles, non-Hamiltonian dynamics, and is inherently probabilistic.

[31] [Dürr, Goldstein, Lebowitz, 1981] A Mechanical Model of Brownian Motion

[32] [Dürr, Goldstein, Lebowitz, 1981] A Mechanical Model of Brownian Motion of a Convex Body

Both these papers consider the scenario of one large particle (or convex body) of mass interacting with infinitely many point particles of mass . The dynamics of the smaller particles are ballistic and only interact with the large particle (or convex body).

[45] [Holley, 1971] The Motion of a Heavy Particle in an Infinite One Dimensional Gas of Hard Spheres

This paper covers exactly the same physical scenario as [31,32] above, but in only one dimension.