Boltzmann-Grad limit

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The Boltzmann–Grad limit of a many-particle system is an approximation of a dynamical system with respect to the small parameter defined as the ratio of the range of the interaction potential to the mean free path, which is assumed fixed. This limit was first studied by H. Grad [a1] in connection with the problem of justifying the Boltzmann equation. In modern terminology, the Boltzmann–Grad limit is one among the so-called large-scale limits applied to the derivation of kinetic equations for the dynamics of particles (cf. also Kinetic equation).

The asymptotics of the solution of the Cauchy problem for the BBGKY hierarchy for many-particle systems interacting via short-range potentials in the Boltzmann–Grad limit can be described by a certain hierarchy of equations usually called the Boltzmann hierarchy. This hierarchy of equations has the property to preserve the "initial chaos" : if the initial data is factorized, then this is true at any time (propagation of chaos). As a consequence, the equation determining the evolution of the initial state possessing the factorization condition is a closed equation for a one-particle distribution function and is the Boltzmann equation.

The rigorous validity of the Boltzmann equation in the Boltzmann–Grad limit was proved for a hard-sphere system in [a2], [a5], [a4], [a3].

The concept of Boltzmann–Grad limit is directly connected with the irreversibility problem in statistical mechanics: the irreversible Boltzmann equation can be rigorously obtained from reversibility in time dynamics.

General references for this area are [a2], [a5], [a4], [a1], [a3].


[a1] H. Grad, "Principles of the kinetic theory of gases" , Handbuch Physik , 12 , Springer (1958) pp. 205–294
[a2] C. Cercignani, "On the Boltzmann equation for rigid spheres" Transp. Theory Stat. Phys. , 2 (1972) pp. 211–225
[a3] O.E. Lanford, "Time evolution of large classical dynamical system" , Lecture Notes Physics , 38 , Springer (1975) pp. 1–111
[a4] V. Gerasimenko, D. Petrina, "Mathematical problems of statistical mechanics of a hard-sphere system" Russian Math. Surveys , 45 : 3 (1990) pp. 159–211
[a5] C. Cercignani, V. Gerasimenko, D. Petrina, "Many-particle dynamics and kinetic equations" , Kluwer Acad. Publ. (1997)
How to Cite This Entry:
Boltzmann-Grad limit. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Gerasimenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article