ar X iv : h ep - t h / 94 12 21 1 v 1 2 3 D ec 1 99 4 BLACK HOLE ENTROPY

According to the thermodynamical analogy in black hole physics, the entropy of a black hole in the Einstein theory of gravity equals S BH = A H /(4l 2 P), where A H is the area of a black hole surface and l P = (¯ hG/c 3) 1/2 is the Planck length [1, 2]. In black hole physics the Bekenstein-Hawking entropy S BH plays essentially the same role an in the usual thermodynamics. In particular it allows one to estimate what part of the internal energy of a black hole can be transformed into work. Four laws of black hole physics which form the basis in the thermodynamical analogy were formulated in [3]. The generalized second law [1, 2, 4] (see also [5, 6, 7, 8] and references therein) implies that when a black hole is a part of the thermodynamical system the total entropy (i.e. the sum of the entropy of a black hole and the entropy of the surrounding matter) does not decrease. The success of the thermodynamical analogy in black hole physics allows one to hope that this analogy may be is even deeper and it is possible to develop statistical-mechanical foundation of black hole thermodynamics. Thermodynamical and statistical-mechanical definitions of the entropy are logically different. Thermodynamical entropy S T D is defined by the response of the free energy F of the system on the change of its temperature: dF = −S T D dT. (1) (This definition applied to a black hole determines its Bekenstein-Hawking entropy.) Statistical-mechanical entropy S SM is defined as S SM = −Tr(ˆ ρ lnˆρ), (2) wherê ρ is the density matrix describing the internal state of the system under consideration. It is also possible to introduce the informational entropy S