Conditions for negative specific heat in systems of attracting classical particles

We identify conditions for the presence of negative specific heat in non-relativistic self-gravitating systems and similar systems of attracting particles. The method used, is to analyse the Virial theorem and two soluble models of systems of attracting particles, and to map the sign of the specific heat for different combinations of the number of spatial dimensions of the system, $D$($\geq 2$), and the exponent, $\nu$($\neq 0$), in the force potential, $\phi=Cr^\nu$. Negative specific heat in such systems is found to be present exactly for $\nu=-1$, at least for $D \geq 3$. For many combinations of $D$ and $\nu$ representing long-range forces, the specific heat is positive or zero, for both models and the Virial theorem. Hence negative specific heat is not caused by long-range forces as such. We also find that negative specific heat appears when $\nu$ is negative, and there is no singular point in a certain density distribution. A possible mechanism behind this is suggested.Comment: 11 pages, 1 figure, Published version (including correlation between positive specific heat and singular points

Exact General Solutions to Extraordinary N-body Problems

We solve the N-body problems in which the total potential energy is any function of the mass-weighted root-mean-square radius of the system of N point masses. The fundamental breathing mode of such systems vibrates non-linearly for ever. If the potential is supplemented by any function that scales as the inverse square of the radius there is still no damping of the fundamental breathing mode. For such systems a remarkable new statistical equilibrium is found for the other coordinates and momenta, which persists even as the radius changes continually.Comment: 15 pages, LaTeX. Accepted for publication in Proc. Roy. Soc.

Relaxation to a Perpetually Pulsating Equilibrium

Paper in honour of Freeman Dyson on the occasion of his 80th birthday. Normal N-body systems relax to equilibrium distributions in which classical kinetic energy components are 1/2 kT, but, when inter-particle forces are an inverse cubic repulsion together with a linear (simple harmonic) attraction, the system pulsates for ever. In spite of this pulsation in scale, r(t), other degrees of freedom relax to an ever-changing Maxwellian distribution. With a new time, tau, defined so that r^2d/dt =d/d tau it is shown that the remaining degrees of freedom evolve with an unchanging reduced Hamiltonian. The distribution predicted by equilibrium statistical mechanics applied to the reduced Hamiltonian is an ever-pulsating Maxwellian in which the temperature pulsates like r^-2. Numerical simulation with 1000 particles demonstrate a rapid relaxation to this pulsating equilibrium.Comment: 9 pages including 4 figure

From Quasars to Extraordinary N-body Problems

We outline reasoning that led to the current theory of quasars and look at George Contopoulos's place in the long history of the N-body problem. Following Newton we find new exactly soluble N-body problems with multibody forces and give a strange eternally pulsating system that in its other degrees of freedom reaches statistical equilibrium.Comment: 13 pages, LaTeX with 1 postscript figure included. To appear in Proceedings of New York Academy of Sciences, 13th Florida Workshop in Nonlinear Astronomy and Physic