Phase Transitions in "Small" Systems - A Challenge for Thermodynamics

Traditionally, phase transitions are defined in the thermodynamic limit only. We propose a new formulation of equilibrium thermo-dynamics that is based entirely on mechanics and reflects just the {\em geometry and topology} of the N-body phase-space as function of the conserved quantities, energy, particle number and others. This allows to define thermo-statistics {\em without the use of the thermodynamic limit}, to apply it to ``Small'' systems as well and to define phase transitions unambiguously also there. ``Small'' systems are systems where the linear dimension is of the characteristic range of the interaction between the particles. Also astrophysical systems are ``Small'' in this sense. Boltzmann defines the entropy as the logarithm of the area $W(E,N)=e^{S(E,N)}$ of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant $D(E,N)=\partial^2S/\partial E^2*\partial^2S/\partial N^2-(\partial^2S/\partial E\partial N)^2$ allows the classification of phase transitions {\em without taking the thermodynamic limit}. The topology gives further a simple and transparent definition of the {\em order parameter.} Attention: Boltzmann's entropy S(E) as defined here is different from the information entropy and can even be non-extensive and convex.Comment: 8 pages, 4 figures, Invited paper for CRIS200

Geometric Foundation of Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit

A geometric foundation thermo-statistics is presented with the only axiomatic assumption of Boltzmann's principle S(E,N,V)=k\ln W. This relates the entropy to the geometric area e^{S(E,N,V)/k} of the manifold of constant energy in the finite-N-body phase space. From the principle, all thermodynamics and especially all phenomena of phase transitions and critical phenomena can unambiguously be identified for even small systems. The topology of the curvature matrix C(E,N) of S(E,N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Within Boltzmann's principle, Statistical Mechanics becomes a geometric theory addressing the whole ensemble or the manifold of all points in phase space which are consistent with the few macroscopic conserved control parameters. This interpretation leads to a straight derivation of irreversibility and the Second Law of Thermodynamics out of the time-reversible, microscopic, mechanical dynamics. This is all possible without invoking the thermodynamic limit, extensivity, or concavity of S(E,N,V). The main obstacle against the Second Law, the conservation of the phase-space volume due to Liouville is overcome by realizing that a macroscopic theory like Thermodynamics cannot distinguish a fractal distribution in phase space from its closure.Comment: 26 pages, 6 figure