Collisionless Boltzmann Equation

The collisionless Boltzmann equation (CBE)—also known in plasma physics and mathematical physics as the Vlasov equation—is the fundamental partial differential equation of galactic dynamics and physical cosmology that governs the time evolution of the six-dimensional phase-space distribution function f(x, v, t) for a large ensemble of stars or dark matter particles moving in their collective smooth gravitational potential Φ(x, t). Formulated for stellar systems by Sir James Jeans in 1915 (building on Ludwig Boltzmann's 1872 kinetic equation and Joseph Liouville's 1838 phase-space theorem) and independently introduced for long-range Coulomb plasmas by Anatoly Vlasov in 1938, the equation expresses the conservation of phase-space probability density along Hamiltonian trajectories when the two-body gravitational relaxation timescale t_relax exceeds the age of the system by many orders of magnitude. Coupled with the Poisson equation ∇²Φ = 4πG ∫ f d³v as the Vlasov–Poisson system, the collisionless Boltzmann equation underpins Jeans's theorem, the Jeans equations, the Eddington inversion formula, phase mixing, and violent relaxation.

Unlike molecules in a terrestrial gas that interact via short-range hard-sphere collisions, stars in galaxies and particles in dark-matter halos interact via the long-range 1/r² Newtonian gravitational force. In a self-gravitating system of N particles of mean mass m occupying a characteristic radius R with dynamical crossing time t_cross = R/v ≈ (Gρ)^−1/2, each star's trajectory is governed primarily by the smooth mean gravitational field Φ(x, t) generated by the system as a whole, perturbed slightly by cumulative small-angle two-body gravitational deflections from individual passing stars. Summing the mean-square velocity deflections over impact parameters from the strong-deflection radius b_min ≈ Gm/v² to the system size b_max ≈ R yields the Chandrasekhar two-body relaxation timescale t_relax:

t_relax ≈ (0.1N / ln Λ) t_cross where ln Λ = ln(b_max/b_min) ≈ ln N

where ln Λ is the Coulomb logarithm:

• Collisional Stellar Systems (t_relax ≲ t_age): In open clusters (N ~ 10²–10⁴, t_relax ~ 10⁷–10⁸ yr) and the cores of globular clusters (N ~ 10⁵–10⁶, t_relax ~ 10⁸–10⁹ yr), two-body scattering operates within the 13.8-Gyr Hubble time, requiring a Fokker–Planck collision term (∂f/∂t)_coll ≠ 0 to model mass segregation, evaporation, and relaxation toward a King profile.
• Collisionless Galactic Systems (t_relax ≫ t_Hubble): In a spiral or elliptical galaxy containing N ~ 10¹¹ stars with t_cross ~ 10⁸ yr, the two-body relaxation time is t_relax ~ 10¹⁷ years—seven orders of magnitude longer than the age of the universe—and > 10⁶⁵ years for cold dark matter halos. Consequently, (∂f/∂t)_coll = 0 and the system obeys the collisionless Boltzmann equation.

Let w = (x, v) = (x₁, x₂, x₃, v₁, v₂, v₃) denote coordinates in six-dimensional single-particle phase space, and let f(x, v, t) d³x d³v represent the mass (or number) of stars within the six-dimensional phase-space volume element d⁶w = d³x d³v at time t. Conservation of particle number without collisional jumps requires f to satisfy the six-dimensional continuity equation ∂f/∂t + ∂(fẋ_i)/∂x_i + ∂(fv̇_i)/∂v_i = 0.

Along a Hamiltonian orbit governed by H(x, v, t) = ½|v|² + Φ(x, t), Hamilton's equations are ẋ_i = v_i = ∂H/∂v_i and v̇_i = −∂Φ/∂x_i = −∂H/∂x_i. Because −∇_xΦ is independent of velocity v (∂v̇_i/∂v_i = 0) and v is independent of position x (∂ẋ_i/∂x_i = 0), the six-dimensional divergence of the phase-space velocity field vanishes identically (∇₆ · ẇ = 0), yielding the collisionless Boltzmann equation:

Df/Dt ≡ ∂f/∂t + v · ∇_xf − ∇_xΦ · ∇_vf = ∂f/∂t + [f, H] = 0

Coupled to the Poisson equation ∇²Φ(x, t) = 4πG ∫ f(x, v, t) d³v, the vanishing Lagrangian derivative Df/Dt = 0 expresses Liouville's theorem: the fine-grained phase-space density f(x(t), v(t), t) is conserved along every particle trajectory like an incompressible fluid, preserving all Casimir invariants C[f] = ∫ g(f) d³x d³v and the fine-grained Gibbs–Boltzmann entropy S = −∫ f ln f d³x d³v.

In spherical coordinates (r, θ, ϕ) with velocity components (v_r, v_θ, v_ϕ) inside a spherically symmetric potential Φ(r, t), the collisionless Boltzmann equation for a spherically symmetric system takes the form:

∂f/∂t + v_r(∂f/∂r) + (v_θ² + v_ϕ²)/r − ∂Φ/∂r − (v_r/r)[v_θ(∂f/∂v_θ) + v_ϕ(∂f/∂v_ϕ)] = 0

For a steady-state stellar system (∂f/∂t = 0 and ∂Φ/∂t = 0), the collisionless Boltzmann equation reduces to [f, H] = v · ∇_xf − ∇_xΦ · ∇_vf = 0. Because any function I(x, v) that is constant along every orbit ([I, H] = 0) is an integral of motion, this identity yields Jeans's theorem (Jeans, 1915; Lynden-Bell, 1962): any steady-state solution of the collisionless Boltzmann equation depends on phase-space coordinates only through integrals of motion f(I₁, I₂, I₃)—or the three canonical actions J = (J₁, J₂, J₃) as realized in a separable Stäckel potential. In a spherical potential Φ(r), ergodic distributions f(E) have isotropic velocity dispersions and can be reconstructed from ρ(r) via the Eddington inversion formula (as in the Plummer model, Hernquist profile, Jaffe profile, and Dehnen profile), while f(E, L) distributions produce anisotropic velocity dispersions.

Although the fine-grained phase-space density f is incompressible, real collisionless galaxies evolve rapidly toward macroscopic equilibrium in their coarse-grained distribution function f̄(x, v, t) (averaged over finite macrocells) via two processes:

1. Phase Mixing and Collisionless Landau Damping: Adjacent orbits with different frequencies Ω(J) shear any initial overdensity into a tightly wound phase-space spiral (such as the Gaia snail in the Milky Way disk), while collective perturbations damp via gravitational Landau damping (governing the Toomre stability criterion and Lindblad resonance wave absorption).
2. Violent Relaxation: During protogalactic collapse or a galaxy merger, collective potential fluctuations (∂Φ/∂t ≠ 0) redistribute particle energies dE/dt = ∂Φ/∂t across phase space on a dynamical crossing time while obeying the exclusion bound f̄ ≤ f_max.

Why does the collisionless Boltzmann equation have zero on the right-hand side?

In a gas with short-range collisions, molecular encounters scatter particles into and out of a phase-space element, producing a non-zero Boltzmann collision integral (∂f/∂t)_coll. In a galaxy of N ~ 10¹¹ stars, the gravitational force is dominated by the smooth global field and the two-body relaxation time t_relax ~ 10¹⁷ years greatly exceeds the age of the universe, so collisional scattering vanishes and Df/Dt = 0.

What is the difference between fine-grained and coarse-grained phase-space density?

The fine-grained distribution function f(x, v, t) is the microscopic phase-space density at a mathematical point; by Liouville's theorem, it is incompressible. The coarse-grained distribution function f̄(x, v, t) is the average of f over a finite volume element Δ³x Δ³v. As phase mixing folds occupied filaments together with unoccupied voids, f̄ decreases and coarse-grained entropy increases.

How does Jeans's theorem simplify building galaxy models?

Jeans's theorem proves that any non-negative function f(I₁, I₂, I₃) constructed from conserved integrals of motion (such as energy E, angular momentum L_z, or actions J) automatically satisfies the steady-state collisionless Boltzmann equation v · ∇_xf − ∇_xΦ · ∇_vf = 0.

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2. Vlasov, A. A. (1938). "On Vibration Properties of Electron Gas." Journal of Experimental and Theoretical Physics, 8(3), 291–318. https://doi.org/10.1070/PU1968v010n06ABEH003709
3. Lynden-Bell, D. (1962). "Stellar dynamics: Only isolating integrals should be used in Jeans' theorem." Monthly Notices of the Royal Astronomical Society, 124(1), 1–9. https://doi.org/10.1093/mnras/124.1.1
4. Lynden-Bell, D. (1967). "Statistical mechanics of violent relaxation in stellar systems." Monthly Notices of the Royal Astronomical Society, 136(1), 101–121. https://doi.org/10.1093/mnras/136.1.101
5. Binney, J., & Tremaine, S. (2008). Galactic Dynamics (2nd ed.). Princeton University Press. https://doi.org/10.1515/9781400828722