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.
Second Law of Thermodynamics, Macroscopic Observables within Boltzmann's Principle but without Thermodynamic Limit
Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit
A mathematical theorem as the basis for the second law: Thomson's formulation applied to equilibrium
