2000/11/08 by D. H. E. Gross, Gross, D. H. E.
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Physics (math-ph) #Nuclear Theory (nucl-th) #Statistical Mechanics (cond-mat.stat-mech) #Statistical Mechanics and Entropy #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP #nlin.CD #nucl-th
paper · pdf · doi:10.48550/arxiv.cond-mat/0011130
5 pages, no figures
arxiv created 2000/11/08 · openalex publication_date 2000/11/08 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.