Starting from metric of the general nonextreme stationary axisymmetric black
hole in four-dimensional spacetime, both statistical-mechanical and
thermodynamical entropies are studied. First, by means of the "brick wall"
model in which the Dirichlet condition is replaced by a scattering ansatz for
the field functions at the horizon and with Pauli-Villars regularization
scheme, an expression for the statistical-mechanical entropy arising from the
nonminimally coupled scalar fields is obtained. Then, by using the conical
singularity method Mann and Solodukhin's result for the Kerr-Newman black hole
(Phys. Rev. D54, 3932(1996)) is extended to the general stationary black hole
and the nonminimally coupled scalar field. We last shown by comparing the two
results that the statistical-mechanical entropy and one-loop correction to the
thermodynamical entropy are equivalent for coupling ξ≤0. After
renormalization, a relation between the two entropies is given.Comment: 18 pages, Latex, nofigue. Accepted by Phys. Rev.