A unified stochastic random walk theory for Kolmogorov spectrum, halo mass functions, and halo density profiles

We develop a unified stochastic framework for a broad class of nonequilibrium systems and apply it to turbulence and cosmic structure formation. The evolution of spectra and distributions in these systems are modeled as transport processes in an appropriate ”scale space”, described by random walks and Fokker–Planck equations with scale-dependent waiting times. We reformulate the turbulent cascade as a random walk in wavenumber space with a k-dependent waiting time \(\tau _{gk}\propto k^{-{\gamma }}\) . With self-consistency condition determined value \(\gamma =2/3\) , we recover the Kolmogorov -5/3 law and obtain the full energy spectrum including the inertial-dissipation crossover, exponential cutoff, and spectrum constant without invoking dimensional analysis directly on the energy spectrum. For a bottom-up hierarchical halo structure formation, low-mass haloes collapse early and retain strong primordial imprints, while prolonged assembly of large haloes near the characteristic mass \(m_h^*\) tends to erase this memory and establishes a universal spectrum \(P(k)\propto k^{-1}\) . We model this process as an analogous random walk of dark matter haloes in mass space \(m_h\) with a mass-dependent waiting time \(\tau _g\propto m_h^{-\lambda }\) . The model yields a stretched Gaussian halo mass function that reproduces the Press-Schechter form, with both small-scale scaling \(f_M\propto m_h^{-\lambda }\) and exponential cutoff at \(m_h>m_h^*\) set by exponent \(\lambda \) . The framework naturally produces a double- \(\lambda \) mass function, where \(\lambda \) reflects the primordial spectrum at low masses and approaches \(\lambda \) =2/3 near \(m_h^*\) where gravity dominates. For individual haloes, a spatial random walk with waiting time \(\tau _{gr}\propto r^{-\gamma }\) generates a double- \(\gamma \) density profile that reproduces the Einasto profile, with both small-scale scaling \(\rho _r\propto r^{-2\gamma }\) and exponential cutoff at \(r>r_s\) set by exponent \(\gamma \) . In this case, \(\gamma \) is fixed by the initial spectrum in the inner core regions and approaches \(\gamma \) =2/3 and \(\rho _r\propto r^{-{4/3}}\) in the outer regions when gravity dominates. Our results demonstrate that this simple stochastic framework offers a common theoretical description of both turbulent cascades and cosmic structure formation.