Violent Relaxation

Violent relaxation is a collisionless statistical relaxation mechanism in astrophysical dynamics, formulated by British astrophysicist Donald Lynden-Bell in 1967, through which a self-gravitating system of stars or dark matter particles undergoing rapid gravitational collapse or a major merger reaches a quasi-steady virial theorem equilibrium on a dynamical crossing timescale (t_dyn ≈ 10⁷–10⁸ years). In elliptical galaxies containing N ≈ 10¹¹ stars, the classical two-body gravitational relaxation timescale (t_relax ≈ [0.1 N / ln N] t_dyn ≈ 10¹⁶–10¹⁸ years) exceeds the 13.8-billion-year age of the universe by six to eight orders of magnitude. Despite the complete absence of two-body Coulomb scattering, elliptical galaxies and dark matter halos exhibit smooth, regular surface-brightness and density profiles (such as the de Vaucouleurs R^1/4 law and Navarro–Frenk–White profile). Lynden-Bell resolved this Zwicky–Oort relaxation paradox by demonstrating that when a system is far from virial equilibrium, large-scale collective fluctuations of the global gravitational potential Φ(x, t) scatter the specific orbital energies ε of individual stars across phase space within just a few free-fall times.

In dense collisional systems such as globular clusters (N ≈ 10⁵–10⁶ stars within a half-mass radius of a few parsecs), cumulative small-angle gravitational deflections between passing pairs of stars redistribute kinetic energy on a half-mass relaxation timescale of t_rh ≈ 10⁹ years. This two-body relaxation drives mass segregation toward kinetic energy equipartition (m₁⟨v₁²⟩ = m₂⟨v₂²⟩), relaxes the velocity distribution toward a lowered isothermal King profile, and causes gradual stellar evaporation.

By contrast, in a luminous elliptical galaxy with N = 10¹¹ stars and a half-light radius of R_e ≈ 5 kiloparsecs, the dynamical crossing time is t_cross = R_e / σ_v ≈ 2 × 10⁷ years (for velocity dispersion σ_v ≈ 250 km s⁻¹), yielding a two-body relaxation time of t_relax ≈ 10¹⁷ years. Furthermore, if elliptical galaxies had relaxed via two-body encounters, high-mass giant stars would exhibit much lower velocity dispersions and stronger central concentration than low-mass dwarf stars; instead, spectroscopic observations show that stars of all masses in elliptical galaxies share an identical velocity dispersion σ_v(R) and follow the same spatial profile. A mass-independent, collective relaxation process operating on the dynamical timescale t_dyn ≈ (G ρ)^−1/2 is therefore required.

Along the trajectory of an individual star of mass m moving through a smooth, time-dependent gravitational potential Φ(x, t), the star's specific orbital energy (energy per unit mass) is:

ε(t) = E / m = ½ v² + Φ(x, t)

Taking the total time derivative of ε along the star's orbit using the equation of motion dv/dt = −∇Φ(x, t) gives:

dε / dt = v · (dv / dt) + v · ∇Φ + ∂Φ / ∂t = v · (−∇Φ + ∇Φ) + ∂Φ / ∂t = ∂Φ / ∂t

This fundamental identity reveals two properties of violent relaxation:

1. Energy Redistribution via Collective Potential Fluctuations (∂Φ/∂t ≠ 0): In a static potential (∂Φ/∂t = 0), every star conserves its specific energy ε indefinitely. However, during protogalactic collapse from a cold initial state (initial virial ratio 2K / |W| ≪ 1) or during the major merger of two disk galaxies, the system contracts rapidly, overshoots virial equilibrium, and rebounds. As the global potential well deepens and oscillates (|∂Φ/∂t| ~ Φ / t_dyn), a star plunging inward through a deepening potential well gains less kinetic energy upon exiting than it lost entering, becoming tightly bound in the core, while a star traversing an expanding potential region gains net orbital energy and is lofted into an extended halo envelope.
2. Equipartition of Velocity (Not Kinetic Energy): Because the rate of change of specific energy dε/dt = ∂Φ/∂t is independent of the particle's individual mass m, stars of 0.1 M_☉, 10 M_☉, and collisionless non-baryonic dark matter particles of 10⁻⁶⁰ M_☉ experience the exact same specific energy changes and settle into the same velocity dispersion distribution (⟨v₁²⟩ = ⟨v₂²⟩), rather than the mass-segregated kinetic energy equipartition (m₁⟨v₁²⟩ = m₂⟨v₂²⟩) produced by two-body encounters or dynamical friction.

Because violent relaxation is collisionless, the fine-grained six-dimensional phase-space distribution function f(x, v, t) obeys the collisionless Boltzmann equation (Vlasov equation), whose velocity moments generate the Jeans equations and whose isotropic equilibria are inverted via the Eddington inversion formula:

Df / Dt = ∂f / ∂t + v · ∇_xf − ∇Φ · ∇_vf = 0

By Liouville's theorem, the fine-grained phase-space density f behaves like an incompressible fluid: every Lagrangian phase-space element preserves its initial density η₀, and the fine-grained Gibbs–Boltzmann entropy S_fine = −∫ f ln f d³x d³v remains constant forever. However, as phase mixing and chaotic stretching fold the phase-space sheet into intricate, filamentary whorls on ever-smaller scales, any macroscopic observation or gravitational potential integration averages f over finite "macrocells" to yield the coarse-grained distribution function f̄(x, v).

Because phase elements are incompressible and cannot overlap in fine-grained space, the coarse-grained density f̄ inside any macrocell can never exceed the maximum initial fine-grained phase density η₀: f̄(x, v) ≤ η₀. Maximizing the coarse-grained Boltzmann entropy subject to the conservation of total mass M, total energy E, and this phase-space incompressibility bound (0 ≤ f̄ ≤ η₀) yields the Lynden-Bell distribution—a fourth statistics identical in mathematical form to the Fermi–Dirac distribution of quantum mechanics, but governing classical, distinguishable phase-space elements:

f̄(ε) = η₀ / { exp[ β(m) (ε − μ) ] + 1 }

where β(m) ∝ m makes the temperature parameter β(m)ε independent of particle mass m, and μ is a chemical potential fixing total mass. In the non-degenerate limit (f̄ ≪ η₀, Typical of outer halos where phase space is diluted by vacuum folds), the Lynden-Bell distribution reduces to a mass-independent Maxwell–Boltzmann isothermal distribution: f̄(ε) ≈ A exp(−ε / σ²), where σ² is the velocity dispersion.

In a three-dimensional unbounded space, a fully relaxed singular isothermal sphere (f̄ ∝ exp(−ε/σ²)) has infinite mass and infinite energy because its density falls only as ρ(r) ∝ r⁻² at large radii (unlike the n = 5 polytropic Plummer model). Why, then, do real elliptical galaxies and N-body collapse simulations exhibit finite masses and follow Gerard de Vaucouleurs's R^1/4 surface-brightness profile (modeled in three dimensions by the Hernquist profile, Jaffe profile, and unified Dehnen profile with 3D density ρ(r) ∝ r⁻⁴ in the outer halo, or by the curved-slope Einasto profile in cosmological halos, contrasted with the supernova-cored Burkert profile in dwarf galaxies and the axisymmetric Miyamoto–Nagai potential or integrable triaxial Stäckel potential)?

As demonstrated via N-body experiments by Sebastien Peebles (1970) and Tjeerd van Albada (1982), violent relaxation in real astrophysical collapses and mergers is incomplete because the global potential fluctuations ∂Φ/∂t damp out via phase mixing within only 2 to 3 dynamical crossing times:

1. Degenerate Core and Isothermal Intermediate Envelope: In the dense central core, rapid potential oscillations thoroughly mix orbital energies up to the phase-density ceiling η₀, forming a flat or weakly cusped core and an intermediate r⁻² isothermal zone.
2. Keplerian Outer Halo (r⁻⁴ Density Falloff): Stars tossed outward into the halo during the first collapse bounce acquire apocenters at large radii (r ≫ R_e) with high radial eccentricity and long orbital periods (T_orb ≫ t_dyn). Before these halo stars return for a second pericentric passage, the central potential has already settled to ∂Φ/∂t ≈ 0. Trapped on highly radial orbits with energies tightly clustered just below the escape threshold (ε → 0⁻) and finite non-zero distribution density N(ε) > 0 at ε = 0, Kepler's third law requires their radial space density to fall as ρ(r) ∝ r⁻⁴ (Jaffe, 1987; Tremaine, 1987), naturally reproducing the de Vaucouleurs R^1/4 profile and radially anisotropic outer velocity dispersion tensor observed in elliptical galaxies and cosmological dark matter halos.

1. 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
2. Shu, F. H. (1978). "On the statistical mechanics of violent relaxation." The Astrophysical Journal, 225, 83–94. https://doi.org/10.1086/156470
3. van Albada, T. S. (1982). "Dissipationless galaxy formation and the R^(1/4) law." Monthly Notices of the Royal Astronomical Society, 201(4), 939–955. https://doi.org/10.1093/mnras/201.4.939
4. White, S. D. M. (1996). "Violent Relaxation in Hierarchical Clustering." In Gravitational Dynamics (pp. 121–134). Cambridge University Press. https://doi.org/10.48550/arXiv.astro-ph/9602021