Equilibrium CBE in Galactic Dynamics

• The equilibrium collisionless Boltzmann equation defines a framework to model phase-space dynamics of stellar and dark matter systems in dynamical equilibrium.
• Recent methodologies, including neural networks and differentiable programming, enable high-resolution solutions and correction of observational biases from surveys like Gaia DR3.
• Applications such as the ClearPotential framework yield precise gravitational potential mappings and dark matter density estimates with resolutions of 50–300 pc.

The equilibrium collisionless Boltzmann equation (CBE) is the foundational equation governing the behavior of stellar and dark matter systems under the assumption of dynamical equilibrium and negligible collisionality. It underpins the modeling of galactic structure, the inference of gravitational potentials, and the interpretation of large-scale astrometric surveys within a rigorous phase-space framework.

1. Mathematical Definition and Physical Interpretation

The collisionless Boltzmann equation in steady state reads: $$v \cdot \nabla_x f(x,v) - \nabla_x \Phi(x) \cdot \nabla_v f(x,v) = 0$$ where $f(x,v)$ is the phase-space distribution function, $\Phi(x)$ is the gravitational potential, $x$ is the spatial coordinate, and $v$ is the velocity. The equation states that, in equilibrium, the distribution function remains constant along the characteristics determined by the Hamiltonian flow: $$\frac{df}{dt} = 0 \qquad \text{(for all phase-space trajectories)}$$ Collisions are neglected, making the framework suitable for systems with long mean free paths, such as stellar disks, halos, and globular clusters at scales well above individual star-star interactions.

2. Methodologies for Solving the Equilibrium CBE

Historically, direct solutions to the full six-dimensional steady-state CBE have been intractable due to the complexity of the phase-space and limited observational completeness. Common approaches have relied on moment-based reductions (e.g., Jeans equations), heavy binning, or assumed forms for $f(x,v)$ and $\Phi(x)$, often with imposed symmetry (axisymmetry or sphericity). This introduces systematic uncertainties, resolution loss, and can obscure non-axisymmetric and small-scale features.

Recent advances leverage machine learning methodologies to obtain model-free, high-resolution solutions. Notably, the ClearPotential framework (Putney et al., 10 Dec 2025) employs normalizing flows to learn the observed phase-space density $f_\mathrm{obs}(x,v)$ and neural networks for both the gravitational potential $\Phi(x)$ and a spatial "efficiency" function $\epsilon(x)$ correcting for incompleteness, such as dust extinction or survey limits. The equilibrium CBE is enforced directly by minimizing the mean-square residual of the CBE over a large set of Monte Carlo samples drawn from the learned $f_\mathrm{obs}$, with additional regularization on the physicality of solutions (e.g., positive-definite mass density).

3. Selection Effects and Correction Mechanisms

Observational datasets such as Gaia DR3 introduce spatial incompleteness due to survey limits and extinction, necessitating correction factors. In the ClearPotential methodology, a neural network $\epsilon(x)$ models the spatially dependent selection function. The corrected density is then $$f_\mathrm{corr}(x,v) = f_\mathrm{obs}(x,v) / \epsilon(x)$$. The CBE residual is minimized in this corrected space: $$v \cdot \nabla \ln f_\mathrm{obs} - v \cdot \nabla \ln \epsilon(x) - \nabla \Phi(x) \cdot \partial_v \ln f_\mathrm{obs} = 0$$ where gradients are computed analytically via automatic differentiation, exploiting the parametric representation of $f_\mathrm{obs}(x,v)$ and $\Phi(x)$ as neural networks. This approach allows correction for dust-driven and crowding biases without requiring external extinction maps.

4. Inference of Gravitational Potential, Accelerations, and Mass Density

Once $\Phi(x)$ is learned, its spatial derivatives directly yield the gravitational accelerations,

$a(x) = - \nabla_x \Phi(x)$

and the mass density follows via Poisson's equation,

$\rho(x) = \frac{1}{4\pi G} \nabla^2_x \Phi(x)$

In the machine-learning-based unsupervised solution, these quantities are available everywhere in the surveyed volume, offering a continuous, differentiable, and high-resolution mapping. Spatial resolution is set by correlational properties of the data and the architecture, achieving $\sim$50–300 pc scale, depending on the region sampled (e.g., disk vs. halo near the Sun) (Putney et al., 10 Dec 2025).

5. Empirical Results and Astrophysical Constraints

Application to Gaia DR3 Red Clump stars within 4 kpc of the Sun yields the most precise, fully data-driven 3D mapping of the Galactic potential to date. The ClearPotential results include:

• Local dark matter density at $R_\odot = 8.122\,\mathrm{kpc}$: $$(0.84 \pm 0.08) \times 10^{-2}\,M_\odot\,\mathrm{pc}^{-3}$$, consistent with independent direct detection constraints.
• Characterization of the dark halo as tilted and oblate, with weak evidence for a central core.
• Tightest constraints yet on a thin dark disk: e.g., for $h_\mathrm{DD}=20\,\mathrm{pc}$, $$\Sigma_{\mathrm{DD}}<0.38\,M_\odot\,\mathrm{pc}^{-2}$$ (95% CL), an order-of-magnitude improvement over prior work.
• Detailed non-stationarity (disequilibrium) mapping, showing the local Milky Way is largely in equilibrium with mild evidence for phase-space structure within the disk (Putney et al., 10 Dec 2025).

Empirical evaluation includes cross-validating learned accelerations with line-of-sight measurements from binary pulsars, with only 3 of 24 objects exceeding $2\sigma$ tension with equilibrium predictions.

6. Impact and Future Directions

Direct, unsupervised, and model-independent solutions to the equilibrium CBE present a transformative advance for galactic dynamics. The methodology,

• Eliminates the need for strong prior assumptions about symmetry or functional form.
• Enables high-resolution mapping of gravitational potentials and mass densities, including in highly extincted regions.
• Sets the stage for dynamical studies harnessing next-generation astrometric data in the Gaia era.

Future directions include adapting these techniques to non-equilibrium systems, further improving selection-function modeling, and extending coverage beyond the local volume as additional data becomes available. A plausible implication is the broader integration of differentiable CBE solvers with cosmological simulations and time-domain surveys, enhancing the fidelity and physical insight of galactic modeling (Putney et al., 10 Dec 2025).