Thermodynamics of Self-Gravitating Systems with Softened Potentials

Abstract

The microcanonical statistical mechanics of a set of self-gravitating particles is analyzed in mean-field approach. In order to deal with an upper bounded entropy functional, a softened gravitational potential is used. The softening is achieved by truncating to N terms an expansion of the Newtonian potential in spherical Bessel functions. The order N is related to the softening at short distances. This regularization has the remarkable property that it allows for an exact solution of the mean field equation. It is found that for N not too large the absolute maximum of the entropy coincides to high accuracy with the solution of the Lane-Emden equation, which determines the mean field mass distribution for the Newtonian potential for energies larger than $E_c\approx -0.335G{M}^2/R$. Below this energy a collapsing phase transition, with negative specific heat, takes place. The dependence of this result on the regularizing parameter N is discussed.

16 pages, 1 table, 4 figures, REVTEX. Partially rewritten, 5 references added

References in corpus (4)

• Lyapunov instability and finite size effects in a system with long-range forces
• Chaos and Statistical Mechanics in the Hamiltonian Mean Field Model
• Anomalous diffusion as a signature of collapsing phase in two dimensional self-gravitating systems
• The effect of angular momentum conservation in the phase transitions of collapsing systems

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• Anomalously slow phase transitions in self-gravitating systems
• Gravitational instability of slowly rotating isothermal spheres
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• Statistical Mechanics of Gravitational Systems with Regular Orbits: Rigid Body Model of Vector Resonant Relaxation
• Dilute and Collapsed Phases of the Self-Gravitating Gas
• Density profiles of a self-gravitating lattice gas in one, two, and three dimensions
• Challenges in Non-Linear Gravitational Clustering
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• Scale Interactions and Galaxy Evolution
• An approach to the quasi-equilibrium state of a self-gravitating system