BhatnagarGrossKrook

The Bhatnagar–Gross–Krook (BGK) model is a simplified collision operator used in the kinetic theory of gases. Introduced in 1954 by Bhatnagar, Gross and Krook, it replaces the Boltzmann collision integral with a single relaxation term that drives the one-particle distribution function toward local equilibrium on a characteristic time scale. The model is widely used as a tractable alternative to the full Boltzmann equation and as a foundational component in lattice Boltzmann methods.

In mathematical form, the BGK model modifies the Boltzmann equation to

df/dt + v · ∇_x f = (f_eq − f)/τ,

where f = f(x, v, t) is the distribution function in phase space, f_eq is the local Maxwellian distribution determined by the macroscopic fields (density, bulk velocity, temperature), and τ is the relaxation time. The equation preserves mass, momentum, and energy locally, which ensures consistency with the macroscopic conservation laws. For many gases, the BGK operator implies a single relaxation time governing both momentum and energy relaxation.

The BGK model is valued for its simplicity and computational efficiency, and it forms the basis for many numerical schemes in rarefied gas dynamics and mesoscopic simulations, including lattice Boltzmann methods. However, it has limitations: the standard BGK form yields a fixed Prandtl number (Pr ≈ 1), which does not match the physical Prandtl numbers of most real gases. To address this, extensions such as the ES-BGK (ellipsoidal statistical BGK) and Shakhov models modify the relaxation structure to adjust Pr while retaining the model’s simplicity.

See also Boltzmann equation, lattice Boltzmann method, H-theorem.