Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles

We study the growth of perturbations in a uniformly collapsing cloud of self-gravitating Brownian particles. This problem shares analogies with the formation of large-scale structures in a universe experiencing a "big-crunch" or with the formation of stars in a molecular cloud experiencing gravitational collapse. Starting from the barotropic Smoluchowski-Poisson system, we derive a new equation describing the evolution of the density contrast in the comoving (collapsing) frame. This equation can serve as a prototype to study the process of self-organization in complex media with structureless initial conditions. We solve this equation analytically in the linear regime and compare the results with those obtained by using the "Jeans swindle" in a static medium. The stability criteria, as well as the laws for the time evolution of the perturbations, differ. The Jeans criterion is expressed in terms of a critical wavelength λ(J) while our criterion is expressed in terms of a critical polytropic index γ(4/3). In a static background, the system is stable for λ<λ(J) and unstable for λ>λ(J). In a collapsing cloud, the system is stable for γ>γ(4/3) and unstable for γ<γ(4/3). If γ=γ(4/3), it is stable for λ<λ(J) and unstable for λ>λ(J). We also study the fragmentation process in the nonlinear regime. We determine the growth of the skewness, the long-wavelength tail of the power spectrum and find a self-similar solution to the nonlinear equations valid for large times. Finally, we consider dissipative self-gravitating Bose-Einstein condensates with short-range interactions and show that, in a strong friction limit, the dissipative Gross-Pitaevskii-Poisson system is equivalent to the quantum barotropic Smoluchowski-Poisson system. This yields new types of nonlinear mean-field Fokker-Planck equations, including quantum effects.