Statistical mechanics, structure of matter Definition

BGK collision operator

Builds on Boltzmann equation Maxwell–Boltzmann distribution Kinetic theory of gases

Definition

Let $f(v) \ge 0$ be a velocity distribution on $\R^3$ with $\int (1 + \abs{v}^2) f \, dv < \infty$ and $\int f \, dv > 0$ , not concentrated at a single velocity. Let $n$ , $u$ and $T$ be its density, bulk velocity and temperature, as in the kinetic theory of gases. Let $M[f]$ be the Maxwell–Boltzmann distribution with the same $n$ , $u$ and $T$ . Let $\nu > 0$ be a collision frequency, which may depend on $n$ and $T$ . The BGK collision operator is $Q_{BGK}(f) = \nu \big( M[f] - f \big).$ The BGK equation is the Boltzmann equation with $Q_{BGK}$ in place of the Boltzmann collision operator: $\partial_t f + v \cdot \nabla_x f = \nu (M[f] - f)$ .

Remarks

Since $M[f]$ and $f$ have the same moments, $\int (1, v, \abs{v}^2)\, Q_{BGK}(f) \, dv = 0$ : mass, momentum and energy are conserved. The H-theorem holds: $\log M[f]$ is a combination of $1$ , $v$ and $\abs{v}^2$ , so $\int Q_{BGK}(f) \log M[f] \, dv = 0$ , and therefore $\int Q_{BGK}(f) \log f \, dv = \nu \int (M[f] - f)(\log f - \log M[f]) \, dv \le 0$ . The operator is nonlinear in $f$ , through the dependence of $M[f]$ on the moments of $f$ . In the Navier–Stokes limit, the BGK model gives Prandtl number $1$ , while the Boltzmann equation for a monatomic gas gives a value close to $2/3$ .

Example: if $f$ is itself a Maxwellian, then $M[f] = f$ and $Q_{BGK}(f) = 0$ .

Non-example: the relaxation $\nu (M_0 - f)$ towards a fixed Maxwellian $M_0$ is not the BGK operator. It does not conserve mass, momentum and energy when $f$ has moments different from those of $M_0$ .

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Tag	`DS66JCJS7Y`
URI	`m2://DS66JCJS7Y`
Kind	definition
Writing	W0 stub
MSC	82C40, 76P05, 35Q20
Defines	BGK collision operator
Uses	Boltzmann equation, Maxwell–Boltzmann distribution, Kinetic theory of gases
Changed	3 November 2026

BGK collision operator

Definition

Remarks

Reading path

Further reading

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