Collisionless Self-Gravitating Systems

• Collisionless self-gravitating systems are ensembles of stars or dark matter particles that interact purely via long-range gravitational forces, described by the Vlasov–Poisson equations.
• They evolve rapidly into quasi-stationary states through mechanisms such as violent relaxation and phase mixing, establishing equilibrium profiles critical for understanding galaxy formation.
• Advanced numerical techniques, including direct phase-space integration and emerging quantum algorithms, are key to accurately simulating their non-collisional dynamics and probing instability criteria.

A collisionless self-gravitating system is a large ensemble of particles—such as stars in galaxies or dark matter particles in cosmological halos—which interact solely through collective gravitational forces and for which direct two-body (collisional) encounters are negligible on the characteristic dynamical timescale. The evolution of such systems is governed by mean-field kinetic theory: the coupled Vlasov (also called collisionless Boltzmann) and Poisson equations. This framework underpins classical, post-Newtonian, relativistic, and generalized gravity descriptions, and provides the basis for both the dynamical theory and equilibrium (statistical) mechanics of galaxies and cosmological structures.

1. Kinetic Description and Governing Equations

A collisionless self-gravitating system is described by the one-particle phase-space distribution function $f(\mathbf{x}, \mathbf{v}, t)$, where the density in physical space is $$\rho(\mathbf{x}, t) = \int f(\mathbf{x}, \mathbf{v}, t)\, d^3 v$$. The kinetic equation is the Vlasov (collisionless Boltzmann) equation: $$\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}} f - \nabla_{\mathbf{x}} \Phi \cdot \nabla_{\mathbf{v}} f = 0,$$ where $\Phi(\mathbf{x}, t)$ is the self-consistent gravitational potential. The potential is itself determined through the Poisson equation: $\nabla^2 \Phi = 4\pi G \rho(\mathbf{x}, t).$ The Vlasov–Poisson system admits a wide class of stationary and evolutionary solutions, supports phase mixing (leading to irreversible macroscopic behavior), and is the foundation for Jeans, Landau, and Lynden-Bell-type analyses (Banik, 2024, Halle et al., 2017, Capozziello et al., 2011). In regimes of strong gravity or for scalar fields and modified gravity, this system is extended (e.g., to Einstein–Vlasov or Vlasov–Poisson–scalar systems) (Rein, 2023, Ignat'ev, 2015).

2. Equilibrium and Quasi-Stationary States

Collisionless systems generically evolve on short timescales to virialized quasi-steady states via violent relaxation. Stationary solutions satisfy $f = f(\mathbf{x}, \mathbf{v})$ such that the collisionless Boltzmann equation vanishes; by Jeans’ theorem, any function of the isolating integrals of motion is a stationary solution. Common examples include ergodic distributions $f(E)$, anisotropic DFs $f(E, L)$, and the full polytrope and King model families (Halle et al., 2017, He, 2011, Agón et al., 2011).

Statistical mechanics of collisionless systems is subtle due to the non-additivity and lack of global ergodicity. Nevertheless, by extremizing suitable entropy-like functionals (e.g., the Boltzmann–Gibbs functional under local ergodicity, or microcanonical occupancy in energy space), one derives families of equilibrium energy distributions such as the DARKexp solution,

$$n(E) \propto \exp\bigl[-\beta(E - \Phi_0)\bigr] - 1,$$

which provides a statistical-mechanical basis for the near-universality of dark matter halo profiles (Williams et al., 2022). Local ergodicity replacing global ergodicity is crucial for the applicability of the standard Boltzmann–Gibbs entropy in the statistical treatment (He, 2011).

3. Dynamical Evolution and Relaxation Phenomena

The evolution of collisionless self-gravitating systems from arbitrary initial conditions is governed by collective modes, phase mixing, and non-collisional relaxation mechanisms. This evolution can be characterized in three main phases (Halle et al., 2017, Benhaiem et al., 2019, Banik, 2024):

• Violent relaxation: Rapid redistribution of energies via the time-dependent mean-field leads to the formation of quasi-steady profiles. This is not a truly time-reversible Vlasov regime, as demonstrated by entropy production during this phase (Silva et al., 2017).
• Phase mixing: Fine-grained features in phase space wind up and can be smeared out in coarse-grained observables, driving the system towards equilibrium. Coarse-grained entropy generally increases on the dynamical timescale, but no strict H-theorem guarantees strict monotonicity (Barnes et al., 2019, Barnes et al., 2013).
• Secular evolution: Over longer timescales, collisional effects (two-body relaxation) become relevant only in finite N systems (e.g., clusters, globular clusters), with associated timescale $t_\text{rel} \sim [N/\ln N] t_\text{dyn}$.

Certain instabilities, such as radial orbit instability (ROI), can break symmetry and further change the phase-space structure, especially at low virial ratio ($\eta \ll 1$) or steep initial density profiles (Halle et al., 2017).

4. Modifications in General Relativity and Modified Gravity

Collisionless self-gravitating systems can be formulated in post-Newtonian, relativistic, or modified gravity frameworks, crucial for dense stellar systems, galactic nuclei, or cosmological collapse (Rein, 2023, Ramos-Caro et al., 2012, Agón et al., 2011, Ignat'ev, 2015, André et al., 2014, Capozziello et al., 2011):

• Post-Newtonian theory: The kinetic theory is extended to first post-Newtonian (1PN) order, yielding corrections to both the Boltzmann equation and the gravitational field equations. Energy and angular momentum are modified, and new terms arise in the equilibrium DFs and macroscopic quantities (Agón et al., 2011, Ramos-Caro et al., 2012).
• Relativistic kinetic description: The Einstein–Vlasov system governs the co-evolution of the distribution function and spacetime geometry. Spherical, static equilibrium families (polytropes and shells) have been constructed, with linear and nonlinear stability characterized by energy–Casimir analysis and Birman–Schwinger methods (Rein, 2023).
• $f(R)$ and other modified gravity theories: The field equations acquire higher-order spatial derivatives, leading to modified dispersion relations for gravitational instability (Jeans analysis). In Palatini $f(R)$ gravity, the critical Jeans scale for instability is lowered compared to Newtonian gravity, implying that a broader range of wavelengths is unstable and the mass threshold for collapse is decreased (André et al., 2014, Capozziello et al., 2011).

5. Instabilities, Dispersion Relations, and Structure Formation

Linear stability analysis of homogeneous collisionless self-gravitating systems yields the Jeans instability criterion. For a Maxwellian background,

$k_J^2 = \frac{4\pi G \rho_0}{\sigma^2}$

sets the threshold: modes with $k < k_J$ have growing solutions. In modified gravity (e.g., metric and Palatini $f(R)$), the effective Jeans wavenumber increases, and the minimum collapsed mass decreases: $k_\mathrm{crit}^2 = \begin{cases} 1.3171\,k_J^2 & \text{(Palatini %%%%12%%%%)}\ 1.2638\,k_J^2 & \text{(metric %%%%13%%%%)}\ k_J^2 & \text{(Newtonian)} \end{cases}$

$$\frac{M_\mathrm{crit}^{\mathrm{Palatini}}}{M_J^{\mathrm{Newton}}} = 0.662$$

These results imply a greater propensity toward gravitational collapse in the presence of higher-order gravitational interactions, with small-scale, high-$k$ perturbations collapsing more readily due to a "stiffening" of the effective gravitational force (André et al., 2014, Capozziello et al., 2011).

In the nonlinear regime, formation of caustics and cold phase-space spirals are generic, with density profiles near caustics exhibiting universal inverse-square-root singularities (Nityananda, 4 Jan 2026, Colombi, 2014).

6. Numerical Methods and Simulation Advances

Simulation of collisionless self-gravitating systems at scale requires direct integration of the Vlasov–Poisson equations in 6D phase space, as pioneered by positive flux conservation (PFC) and semi-Lagrangian codes (Yoshikawa et al., 2012). Key algorithmic advances:

• Direct integration approaches: High-resolution phase-space grids (e.g., $64^6$ cells) with conservative, positivity-preserving solvers avoid shot noise and artificial relaxation of N-body methods. These allow precise studies of linear instabilities, equilibrium stability (e.g., King models), and merging events (Yoshikawa et al., 2012).
• Quantum algorithms: State-of-the-art quantum reservoir and operator-splitting algorithms encode the distribution in qubit registers, advancing the state and computing gravity via a small set of dominant Fourier modes. These approaches can, in principle, achieve exponential speedup in velocity space dimensionality and have been validated in 1D test cases including Jeans instability (Yamazaki et al., 2023).
• Reduced models: One- and two-dimensional analogues, including Hermite–Legendre expansion and Lagrangian perturbation theory, provide analytic insight into collisionless relaxation, entropy production, and caustic structure formation (Barnes et al., 2013, Colombi, 2014, Barnes et al., 2019).

7. Applications, Physical Implications, and Open Problems

Collisionless self-gravitating theory is fundamental to the understanding of galaxy and cluster formation, dark matter halo structure, and dynamical evolution of stellar systems. Non-equilibrium and secular processes (e.g., phase spirals, dynamical friction, core stalling of SMBHs, bar and spiral formation) are well described within Vlasov–Poisson kinetics generalized to multiple regimes: impulsive, resonant, and adiabatic (Banik, 2024, Benhaiem et al., 2019). Recent simulations have confirmed the emergence of universal quasi-equilibria, the inevitability of entropy increase through phase mixing and violent relaxation, and the conditions for catastrophic gravitational collapse—or the onset of macroscopic instabilities in strongly relativistic or anisotropic systems (Silva et al., 2017, Rein, 2023, Ignat'ev, 2015).

Open challenges include:

• Rigorous statistical mechanics (and entropy maximization) for long-range, non-additive systems with global non-ergodicity;
• The role of velocity-space structure and singularities in cold collapse and caustic formation;
• The complete nonlinear instability criteria in relativistic (Einstein–Vlasov) models and the transition from stability to instability as parameterized by, e.g., redshift;
• The impact of modified gravity on the early stages of structure formation, star formation thresholds, and cosmological evolution.

Collisionless self-gravitating systems remain a cornerstone of dynamical astronomy, cosmology, and gravitational theory, with ongoing advances connecting fundamental kinetic theory to large-scale phenomena and the deepest puzzles in galactic and cosmological structure (He, 2011, André et al., 2014, Williams et al., 2022, Fardeau et al., 2023, Nityananda, 4 Jan 2026, Banik, 2024, Rein, 2023).