From chaos to cosmology: insights gained from 1D gravity

Abstract

The gravitational force controls the evolution of the universe on several scales. It is responsible for the formation of galaxies from the primordial matter distribution and the formation of planets from solar nebulae. Because the gravitational force is singular and has infinite range, making predictions based on fully three-dimensional models may be challenging. One-dimensional (1D) Newtonian gravity models were proposed as toy models for understanding the dynamics of gravitational systems. They can be integrated exactly and were used for computer simulations starting in the 1960s, providing the first demonstration of violent relaxation and the rapid development of long-lived quasi-stationary states. The present review provides the bases of the physics of 1D gravitational systems. It is divided into two main parts, the first concerning the approach to equilibrium and the second applications to cosmology. Each part is self-contained and can be read independently of the other. In the first part, we provide an introduction to the equilibrium thermodynamics of the one-dimensional gravitational sheet (OGS) system in the Vlasov limit. Both fixed and periodic boundary conditions are considered. The relaxation to equilibrium of the OGS is studied through numerical simulations which establish the role played by quasi-stationary states and violent relaxation. We also survey existing work on the Lyapunov exponents of the OGS and on the chaotic dynamics of 1D systems with few particles, focusing on the 1D three-body problem. The second part summarizes work on dynamical structure formation in cosmology using 1D systems. By transforming to comoving coordinates, which follow the global expansion of the universe, the 1D approach provides a useful laboratory for studying structure formation in various cosmological scenarios, from Einstein-de Sitter and $\Lambda$CDM to more recent, alternative cosmological models. A key result is the appearance of scale-free behavior with fractal dimension, which can be reliably studied in 1D for large systems over many epochs. Finally, an appendix gives some details on the numerical simulation methods used in these studies.

Lifetime of locally stable states near a phase transition in the Thirring model

“…A salient property of long-range interacting systems is that they are intrinsically non-additive, leading to the possibility of ensemble inequivalence [15,16,19,22,24,25] and to the emergence of an additional term in the Gibbs-Duhem relation [26][27][28][29]. Their dynamics also presents interesting features, since these systems may remain trapped in long-lived quasi-stationary states [30][31][32] before evolving towards equilibrium, and exhibit non-equilibrium phenomena such as anomalous relaxation and diffusion [33][34][35][36][37]. Interestingly, the relaxation timescale of non-equilibrium quasi-stationary states (which are stable under Vlasov dynamics) increases algebraically with N, the number of particles in the system [31,38,39].…”

A Monte Carlo Method for Calculating Lynden-Bell Equilibrium in Self-Gravitating Systems

“…We start by testing our MC algorithm on simple one-and two-level waterbag distributions for which the LB entropy function can be maximized exactly. We undertake this investigation within the framework of the one-dimensional self-gravitating model (ODSGM), a model that has been extensively studied in the field of stellar dynamics since the seminal works of Lecar [33] and Hohl [34,35], up to the more recent works of Miller [36][37][38]. It has also found applications in cosmological models explored by Joyce and collaborators [39][40][41].…”