Collisionless self-gravitating systems such as cold dark matter halos are known to harbor universal density profiles despite the intricate nonlinear physics of hierarchical structure formation in the $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) paradigm. The origin of such states has been a persistent mystery, particularly because the physics of collisionless relaxation has remained poorly understood. To solve this long-standing problem, we develop a self-consistent quasilinear theory in action-angle space for the collisionless relaxation of inhomogeneous, self-gravitating systems by perturbing the governing Vlasov-Poisson equations. We obtain a quasilinear diffusion equation that describes the secular evolution of the mean coarse-grained distribution function $f_0$ of accreted matter in the fluctuating force field of a spherical isotropic halo. The diffusion coefficient not only depends on the fluctuation power spectrum but also on the evolving potential of the system, which reflects the self-consistency of the problem. Diffusive heating in the preassembled halo develops an $r^{-\gamma }$ cusp ($r$ is the halocentric radius) in the density profile of the accreted material. Accretion and relaxation in this $r^{-\gamma }$ inner cusp develops an $r^{-\beta }$ outer fall-off with $\beta \approx 5-2\gamma$ in the quasisteady state. Spherical collapse theory dictates that a quasi-steady outer halo must settle to $\beta \approx 3$ since then the mass enclosed within a radially moving shell barely changes with time. This implies that the quasisteady $\gamma$ must be approximately 1, which is possible in the quasilinear framework only if (i) the preassembled halo harbors an $r^{-{\gamma }_{\mathrm{P}}}$ profile with ${\gamma }_{\mathrm{P}}\gtrsim 0.5$, (ii) its fluctuations are sufficiently correlated in time (red noise), and (iii) the initial value of $\gamma$ is smaller than 1, implying that the $r^{-1}$ cusp is a neutral equilibrium. Self-consistent quasilinear relaxation therefore establishes the Navarro-Frenk-White (NFW) profile. We demonstrate for the first time how this profile emerges as a quasisteady state of collisionless relaxation.

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• Classical statistical mechanics
• Cosmology
• Dark matter
• Diffusion
• Formation & evolution of stars & galaxies
• Kinetic theory
• Nonequilibrium & irreversible thermodynamics
• Nonequilibrium statistical mechanics
• Stochastic processes
• Galactic halos
• Galaxies
• Large scale structure of the Universe
• Nonequilibrium systems
• Astronomical masses & mass distributions
• Coarse graining
• Fokker–Planck equation
• Integrable systems
• Linear response theory
• Phase space dynamics