Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2011-03-28
Phys.Rev.E84:031101,2011
Physics
Condensed Matter
Statistical Mechanics
Scientific paper
10.1103/PhysRevE.84.031101
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, are different. The Jeans criterion is expressed in terms of a critical wavelength $\lambda_J$ while our criterion is expressed in terms of a critical polytropic index $\gamma_{4/3}$. 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 a new type of nonlinear mean field Fokker-Planck equations including quantum effects.
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