On statistical mechanics of gravitational systems

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Abstract
We give a statistical mechanical description of a special class of systems. A system of this class can be thought of as a content of a `black box'; such systems have no other properties except gravitational (and possibly thermal) ones. The microscopical description is based on the description of quantum states suggested by the loop quantum gravity. We describe quantum states of such systems as specified by spin-networks on the boundary of the box. We introduce the statistical mechanical (geometrical) entropy as the logarithm of the number of different microstates corresponding to one and the same energy of the system. Under certain assumptions, our analysis shows that the largest possible value of this entropy corresponds to the largest possible value of the energy of the system, i.e. the largest possible entropy corresponds to the case when there is a single black hole which occupies the whole interior of the system. This confirms the Bekenstein bound for the entropy of such systems. We find that this extremal entropy is a function of the area of the boundary and that it depends on the area linearly, as the Bekenstein-Hawking formula suggests.

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