  We propose dual thermodynamics corresponding to black hole mechanics with the
identifications E' -> A/4, S' -> M, and T' -> 1/T in Planck units. Here A, M
and T are the horizon area, mass and Hawking temperature of a black hole and
E', S' and T' are the energy, entropy and temperature of a corresponding dual
quantum system. We show that, for a Schwarzschild black hole, the dual
variables formally satisfy all three laws of thermodynamics, including the
Planck-Nernst form of the third law requiring that the entropy tend to zero at
low temperature. This is in contrast with traditional black hole
thermodynamics, where the entropy is singular. Once the third law is satisfied,
it is straightforward to construct simple (dual) quantum systems representing
black hole mechanics. As an example, we construct toy models from one
dimensional (Fermi or Bose) quantum gases with N ~ M in a Planck scale box. In
addition to recovering black hole mechanics, we obtain quantum corrections to
the entropy, including the logarithmic correction obtained by previous papers.
The energy-entropy duality transforms a strongly interacting gravitational
system (black hole) into a weakly interacting quantum system (quantum gas) and
thus provides a natural framework for the quantum statistics underlying the
holographic conjecture.
