Statistical mechanics of D0-branes and black hole thermodynamics

We consider a system of D0-branes in toroidally compactified space with interactions described by a Born-Infeld-type generalisation of the leading v^2 + v^4/r^{D-4} terms (D is the number of non-compact directions in M-theory, including the longitudinal one). This non-linear action can be interpreted as an all-loop large N super Yang-Mills effective action and has a remarkable scaling property. We first study the classical dynamics of a brane probe in the field of a central brane source and observe the interesting difference between the D=5 and D > 5 cases: for D >5 the center acts as a completely absorbing black hole of effective size proportional to a power of the probe energy, while for D=5 there is no absorption for any impact parameter. A similar dependence on D is found in the behaviour of the Boltzmann partition function Z of an ensemble of D0-branes. For D=5 (i.e. for compactification on 6-torus) Z is convergent at short distances and is analogous to the ideal gas one. For D > 5 the system has short-distance instability. For sufficiently low temperature Z is shown to describe the thermodynamics of a Schwarzschild black hole in D > 5 dimensions, supporting recent discussions of black holes in Matrix theory.