  We show that finite systems whose Hamiltonians obey a generalized homogeneity
relation rigorously follow the nonextensive thermostatistics of Tsallis. In the
thermodynamical limit, however, our results indicate that the Boltzmann-Gibbs
statistics is always recovered, regardless of the type of potential among
interacting particles. This approach provides, moreover, a one-to-one
correspondence between the generalized entropy and the Hamiltonian structure of
a wide class of systems, revealing a possible origin for the intrinsic
nonlinear features present in the Tsallis formalism that lead naturally to
power-law behavior. Finally, we confirm these exact results through extensive
numerical simulations of the Fermi-Pasta-Ulam chain of anharmonic oscillators.

Statistical Mechanics