On the Hagedorn Transition and Collective Dynamics of D0-branes

Banks, Fischler, Klebanov and Susskind have proposed a model for black hole thermodynamics based on the principle that the entropy is of order the number of particles at the phase transition point in a Boltzmann gas of D0-branes. We show that the deviations from Boltzmann scaling found in $d<6$ noncompact spatial dimensions have a simple explanation in the analysis of self-gravitating random walks due to Horowitz and Polchinski. In the special case of $d=4$ we find evidence for the onset of a phase transition in the Boltzmann gas analogous to the well-known Hagedorn transition in a gas of free strings. Our result relies on an estimate of the asymptotic density of states in a dilute gas of D0-branes.