Infinite self-gravitating systems

The evolution of systems of a very large number of classical point particles interacting solely by Newtonian gravity is a paradgmatic problem for the statistical physics of long range interacting systems. It is also a very relevant limit for real physical problems studied in astrophysics and cosmology, ranging from the formation of galaxies to the evolution of the largest scale structures in the universe. Indeed current theories of the universe postulate that most of the particle-like clustering matter in the universe is “cold” and “dark”, i.e. very non-relativistic and interacting essentially only by gravity, which means that the purely Newtonian approximation is very good one over a very great range of temporal and spatial scales. In practice, as we discuss further below, the very large numerical simulations used to follow the evolution of the structures in the universe, starting from the initial conditions on the density perturbations inferred from measurements of the fluctuations in the cosmic microwave background, are completely Newtonian. While simulations in this context have developed impressively in size and sophistication over the last three decades, the results they provide remain essentially phenomenological in the sense that our analytical understanding of them is very limited.
Our first aim is to clarify, in a statistical physics language, precisely what the relevant problem is which is currently studied in the context of the problem of structure formation in cosmology. The essential point is that it corresponds simply to a specific infinite volume limit of the Newtonian problem. A well defined limit of the resulting equations of motion is given by the case that the universe does not expand. This case, we explain, corresponds to the most natural definition of the usual thermodynamic limit of the Newtonian problem. We note that this limit is distinct from the “mean-field” or “dilute” one sometimes considered in the literature, and in which, under certain circumstances, one can define thermodynamic equilibria. In the limit we consider the system has no thermodynamic equilibrium, and the system is a lways intrinsically out of equilibrium.