The formation of structures in the Universe is one of the major questions in Cosmology. The growth of sructure in the linear regime of low amplitude fluctuations is well understood analytically, but N-body simulations remain the main tool to probe the non-linear regime, where fluctuations are large. This problem, as old as Newton, can be seen essentially as a simple well-posed problem of out of equilibrium statistical mechanics. In this context, however, it has been relatively neglected, primarily because of the intrinsic difficulties associated with the attractive long-range nature of the gravitational potential and its singular behavior at vanishing separation. Approaching this question from the more general perspective of statistical physics of out-of-equilibrium dynamics of systems with long-range interaction highlights a different look on this problem.
In Cosmology perturbative approaches to the problem, which treat the very limited range of low to modest amplitude deviations from uniformity, have been developed. While such simulations constitute a very powerful and essential tool, they lack the valuable guidance, which a fuller analytic understanding of the problem would provide. Approaching the problem in the context of statistical mechanics, it is natural to start by reducing as much as possible the complexity of the analogous cosmological problem. In an attempt to progress in this direction, simplified models may provide insight and qualitative understanding of both the cosmological motivation of gravitational clustering. These simplified models can show similar behavior of what is obtained in more realistic simulations, which in turn resemble those in the expanding universe. For instance, one of the most important properties of the evolution of gravitational clustering in an infinite universe is its self-similarity, along with the importance of the initial conditions on the multifractal evolution of the halo structures.
D. Benhaiem, M. Joyce* and F. Sicard, Exponents of non-linear clustering in scale-free one-dimensional cosmological simulations, Mont. Not. R. Astron. Soc. 429 (4), 3423 (2013)
F. Sicard* and M. Joyce*, Non-linear gravitational clustering of cold matter in an expanding universe: indications from 1D toy models, Mont. Not. R. Astron. Soc. 413, 1439 (2011)
F. Sicard*, Out-of-equilibrium dynamics in infinite one-dimensional self-gravitating systems, Ph.D. Thesis, University Pierre & Marie Curie (2010)
A. Gabrielli, M. Joyce* and F. Sicard, One dimensional gravity in infinite point distributions, Phys. Rev. E 80, 041108 (2009)
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