Statistical mechanics and computer simulation of systems with attractive positive power-law potentials

We study many-body systems in $d$ dimensions interacting with a purely attractive pair potential $\sim |{\mathbf{x}}_i-{\mathbf{x}}_j{|}^{\nu }$, where ${\mathbf{x}}_i$ is the position vector of particle $i$, and $\nu$ is a positive parameter. We derive the temperature in microcanonical equilibrium for arbitrary $\nu$ and $d$ and, for $d=1\mathrm{}$, the corresponding velocity distribution for a finite number $N$ of particles. The latter reduces to the Maxwell-Boltzmann distribution in the infinite-particle limit. The one-dimensional particle distribution of the equilibrium cluster in the mean-field limit is computed numerically for various potential parameters $\nu$. We test these theoretical expressions by comparing them to extensive computer simulation results of one-dimensional systems and find close agreement for $\nu =1\mathrm{}$ (the sheet model) and $\nu =4.5\mathrm{}$. In similar simulations for $\nu =1.5\mathrm{}$ the macroscopic relaxation time exceeded the length of our simulation runs and the system did not relax towards the known microcanonical equilibrium state. We also compute full Lyapunov spectra for the linear sheet model and find that the Kolmogorov-Sinai entropy starts to increase linearly with $N$ for $N>\mathrm{}10\mathrm{}$.