Geometry of the energy landscape of the self-gravitating ring

Abstract

We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapse one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent $\lambda$ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of $\lambda$ and is also quantitatively correct at small energy densities and, in the limit $N\to \infty$, in the whole homogeneous phase.

arXiv comment20 pages, 12 figures

References in corpus (23)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Geometric approach to Hamiltonian dynamics and statistical mechanics
• Riemannian theory of Hamiltonian chaos and Lyapunov exponents
• Geometry of dynamics, Lyapunov exponents and phase transitions
• Classification of phase transitions and ensemble inequivalence, in systems with long range interactions
• Phase transitions and configuration space topology
• Analytic estimation of Lyapunov exponent in a mean-field model undergoing a phase transition
• Thermodynamics of the self-gravitating ring model
• Phase transitions and topology changes in configuration space
• Origin of scaling structure and non-gaussian velocity distribution in self-gravitating ring model
• Geometry of dynamics and phase transitions in classical lattice phi^4 theories
• Topological aspects of geometrical signatures of phase transitions
• Weak and strong chaos in FPU models and beyond
• Phase transitions induced by saddle points of vanishing curvature
• Hamiltonian dynamics and geometry of phase transitions in classical XY models
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Energy landscape and phase transitions in the self-gravitating ring model
• Curvature of the energy landscape and folding of model proteins
• Caloric curve of star clusters
• Solvable model of a self-gravitating system
• Phase transitions in simplified models with long-range interactions
• Geometry of the energy landscape and folding transition in a simple model of a protein
• Generalized Lyapunov exponents of the random harmonic oscillator: cumulant expansion approach

Cited by in corpus (4)

• Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions
• $N$-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited
• Coarse-grained collisionless dynamics with long-range interactions
• Ensemble Inequivalence and Maxwell Construction in the Self-Gravitating Ring Model