Microcanonical Thermostatistics, the basis for a New Thermodynamics, "heat can flow from cold to hot", and nuclear multifragmentation. The correct treatment of Phase Separation after 150 years of statistical mechanics

Equilibrium statistics of finite Hamiltonian systems is fundamentally described by the microcanonical ensemble (ME). Canonical, or grand-canonical partition functions are deduced from this by Laplace transform. Only in the thermodynamic limit are they equivalent to ME for homogeneous systems. Therefore ME is the only ensemble for non-extensive/inhomogeneous systems like nuclei or stars where the $\lim_{N\to \infty,\rho=N/V=const}$ does not exist. Conventional canonical thermo-statistic is inapplicable for non-extensive systems. This has far reaching fundamental and quite counter-intuitive consequences for thermo-statistics in general: Phase transitions of first order are signaled by convexities of $S(E,N,Z,...)$ \cite{gross174}. Here the heat capacity is {\em negative}. In these cases heat can flow from cold to hot! The original task of thermodynamics, the description of boiling water in heat engines can now be treated. Consequences of this basic peculiarity for nuclear statistics as well for the fundamental understanding of Statistical Mechanics in general are discussed. Experiments on hot nuclei show all these novel phenomena in a rich variety. The close similarity to inhomogeneous astro physical systems will be pointed out. \keyword{Microcanonical statistics, first order transitions, phase separation, steam engines, nuclear multifragmentation, negative heat capacity}Comment: 6 pages, 3 figures, Invited plenary talk at VI Latin American Symposium on Nuclear Physics and Applications, Iguaz\'u, Argentina. October 3 to 7, 200

Microcanonical Thermostatistics as Foundation of Thermodynamics. The microscopic origin of condensation and phase separations

Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 150 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 canonical 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. This is elaborated in the present article. It turns out that the Boltzmann-Planck statistics is much richer and gives fundamental insight into statistical mechanics and especially into entropy. This can even be done to some extend rigorously and analytically. The microcanonical entropy has a very simple physical meaning: It measures the microscopic uncertainty that we have about the system, i.e. the number of points in $6N$-dim phase, which are consistent with our information about the system. It can rigorously be split into an ideal-gas part and a configuration part which contains all the physics and especially is responsible for all phase transitions. The deep and essential difference between ``extensive'' and ``intensive'' control parameters, i.e. microcanonical and canonical statistics, is exemplified by rotating, self-gravitating systems.Comment: Invited paper for the conference "Frontiers of Quantum and Mesoscopic Thermodynamics", Prague 26-29 July 2004, 9 pages, 3 figures A detailed discussion of Clausius original papers on entropy are adde