  Boltzmann's principleS=k*ln W is generalized to non-equilibrium Hamiltonian
systems with possibly fractal distributions in phase space by the box-counting
volume. The probabilities P(M) of macroscopic observables M are given by the
ratio P(M)=W(M)/W of these volumes of the sub-manifold {M} of the
microcanonical ensemble with the constraint M to the one without. With this
extension of the phase-space integral the Second Law is derived without
invoking the thermodynamic limit. The irreversibility in this approach is due
to the replacement of the phase space volume of the possibly fractal
sub-manifold {M} by the volume of the closure of {M}. In contrast to
conventional coarse graining the box-counting volume is defined by the limit of
infinite resolution.
