Reconciliation of statistical mechanics and astro-physical statistics: The errors of conventional canonical thermostatistics

Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. (Only in this case, this is equivalent to the fundamental microcanonical ensemble.) However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e., boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing that a phase transition of first order is signaled canonically by a Yang–Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann–Planck statistics. It turns out that the Boltzmann–Planck statistics are much richer and give fundamental insight into statistical mechanics and especially into entropy. This can be done to a far extend rigorously and analytically. As no extensivity, no thermodynamic limit, no concavity, no homogeneity is needed, it also applies to astro-physical systems. The deep and essential difference between ‘extensive’ and ‘intensive’ control parameters, i.e., microcanonical and canonical statistics, is exemplified by rotating, self-gravitating systems. In the present article, the necessary appearance of a convex entropy S ( E ) and negative heat capacity at phase separation in small as well macroscopic systems independently of the range of the force is pointed out. Thus the old puzzle of stellar statistics is finally solved, the appearance of negative heat capacity which is forbidden and cannot appear in the canonical formalism. To cite this article: D.H.E. Gross, C. R. Physique 7 (2006). La thermo-statistique conventionnelle considère des systèmes homogènes et infinis dans le cadre des ensembles canoniques. (Seulement dans ce cas, ceci est équivalent à l'ensemble fondamental micro-canonique.) Cependant, il y a environ 170 ans, la motivation première de la thermodynamique était la description des machines à vapeur, i.e., de l'eau bouillante. La physique essentielle en est la séparation des phases gazeuse et liquide. Bien sûr, l'eau bouillante est inhomogène, et par là ne peut pas être traitée par la thermo-statistique conventionnelle. Donc ce n'est pas surprenant qu'une transition de phase de premier ordre se caractérise en canonique par une singularité de Yang–Lee. Elle est traitée correctement uniquement par la statistique microcanonique de Boltzmann–Planck. Il s'avère que la statistique de Boltzmann–Plan