The equation of Boltzmann (1872) is an equation intégro-differential of the kinetic theory which describes the evolution of a not very dense gas out of balance. It in particular makes it possible to show the Théorème H, and to study the relieving of gas of a state of local balance towards total balance characterized by the distribution of Maxwell speeds.

Mechanical model of gas

One considers a gas of hard spheres made up of $N$ identical atoms of mass $m$ and ray $r$. These atoms:

• is confined in a box;

• travels at constant speed between the collisions;

• elastically rebounds the ones on the others;

• rebounds elastically on the walls of the box.

Function of distribution to a particle

One notes $f\; (\backslash \; mathbf\; \{X\},\; \backslash \; mathbf\; \{U\},\; T)$ the function of distribution to a particle of gas, such as:

$DNN\; \backslash \; =\; \backslash \; F\; (\backslash \; mathbf\; \{X\},\; \backslash \; mathbf\; \{U\},\; T)\; \backslash \; d^3\; \backslash \; mathbf\; \{X\}\; \backslash \; d^3\; \backslash \; mathbf\; \{U\}$

represent the number of gas molecules located at the moment $t$ in a small volume of space $d^3\; \backslash \; mathbf\; \{X\}$ around the point $\backslash \; mathbf\; \{X\}$ and having a speed $\backslash \; mathbf\; \{U\}$ defined except for $d^3\; \backslash \; mathbf\; \{U\}$.

Equation of Boltzmann not-relativist

Operator of Liouville not-relativist

That is to say a gas placed in a field of external force macroscopic $\backslash \; mathbf\; \{F\}\; (\backslash \; mathbf\; \{X\},\; T)$ (for example, the local field of gravity). The operator of Liouville $\backslash \; hat\; \{\backslash \; mathbf\; \{L\}\}$ describing the total variation of the function of distribution to a particle $f\; (\backslash \; mathbf\; \{X\},\; \backslash \; mathbf\; \{U\},\; T)$ in the space of the phases to a particle is the linear operator defined in mechanics not-relativist by:

$\backslash \; hat\; \{\backslash \; mathbf\; \{L\}\}\; \_\; \backslash \; mathrm\; \{NR\}\; F\; \backslash \; =\; \backslash \; \backslash \; frac\; \{D\; F\}\; \{D\; T\}\; \backslash \; =\; \backslash \; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; \backslash \; +\; \backslash \; \backslash \; mathbf\; \{U\}\; \backslash \; cdot\; \backslash \; nabla\_\; \backslash \; mathbf\; \{X\}\; F\; \backslash \; +\; \backslash \; \backslash \; frac\; \{\backslash \; mathbf\; \{F\}\}\; \{m\}\; \backslash \; cdot\; \backslash \; nabla\_\; \backslash \; mathbf\; \{U\}\; F$

Equation of Boltzmann not-relativist

Because of the collisions, the function of distribution to a particle has a not-null total variation; she obeys the equation of Boltzmann:

$\backslash \; hat\; \{\backslash \; mathbf\; \{L\}\}\; \backslash \; =\; \backslash \; \backslash \; mathbf\; \{C\}$

where $\backslash \; mathbf\; \{C\}$ is the operator of collision , integral operator non-linear. Historically, Boltzmann obtained the analytical expression of this operator of collision by a fine analysis of the collisions with two bodies. It is also possible to derive the equation from Boltzmann by a suitable truncation of the equations of the Hiérarchie BBGKY.

Theorem of Lanford

Limit of Boltzmann-Grad

The limit known as of Boltzmann-Grad consists in taking the joint limit:

• of a number of atoms $N\; \backslash \; \backslash \; to\; \backslash \; +\; \backslash \; \backslash \; infty$;

• of a ray $r\; \backslash \; \backslash \; to\; \backslash \; 0$;

by maintaining the product $N\; \backslash \; r^2\; \backslash \; =\; \backslash$ cte. In particular, the volume excluded tends towards zero within this limit: $N\; \backslash \; r^3\; \backslash \; \backslash to\; \backslash \; 0$

Theorem of Lanford (1973)

Lanford showed rigorously that a gas of hard spheres diluted in $\backslash \; mathbb\; \{R\}\; ^3$ obeys the equation of Boltzmann within the limit of Boltzmann-Grad, at least for a very short time, equal only to one fifth of the average run time of an atom.

In spite of this restriction over the duration, this rigorous mathematical theorem is very important conceptually, since the equation of Boltzmann involves the Théorème H, by the way whose Boltzmann was shown to practice “doubtful mathematics”. It does not remain about it less than it remains to be shown that this result remains true for macroscopic times, as when the atoms are confined in one limps.

Equation of relativistic Boltzmann

The operator of Liouville is written in General relativity:

$\backslash \; hat\; \{\backslash \; mathbf\; \{L\}\}\; \_\; \backslash \; mathrm\; \{GR.\}\; \backslash \; =\; \backslash \; \backslash \; sum\_\; \backslash \; alpha\; p^\; \backslash \; partial\; alpha\; \backslash \; frac\; \{\backslash \}\; \{\backslash \; partial\; x^\; \backslash \; alpha\}\; \backslash \; -\; \backslash \; \backslash \; sum\_\; \{\backslash \; alpha\; \backslash \; beta\; \backslash \; gamma\}\; \backslash \; Gamma^\; \{\backslash \; alpha\}\; \{\}\; \_\; \{\backslash \; beta\; \backslash \; gamma\}\; p^\; \backslash \; alpha\; p^\; \backslash \; partial\; gamma\; \backslash \; frac\; \{\backslash \}\; \{\backslash \; partial\; p^\; \backslash \; alpha\}$

where $p^\; \backslash \; alpha$ is the quadri-impulse and the $\backslash \; Gamma^\; \{\backslash \; alpha\}\; \{\}\; \_\; \{\backslash \; beta\; \backslash \; gamma\}$ are the Symboles of Christoffel.

Hydrodynamic limits

Sixth problem of Hilbert

“ the book of Mr. Boltzmann on the Principles of Mechanics encourages us to establish and discuss from the mathematical point of view in a complete and rigorous way the methods based on the idea of passage in extreme cases, and which atomic design lead us to the laws of the movement continued. ” David Hilbert (1900).

Derivation of the Navier-Stokes equations

It is possible to derive the Navier-Stokes equations starting from the equation of Boltzmann.

Related articles

• kinetic Theory of the gases

• Physical statistics out of balance
• Theorem H
• Hierarchy BBGKY
• Equation of Vlassov
• Equation of Poisson-Boltzmann
• Navier-Stokes equations