Geometric foundation of thermo-statistics, phase transitions, second law of thermodynamics, but without thermodynamic limit

Abstract

A geometric foundation thermo-statistics is presented with the only axiomatic assumption of Boltzmann's principleS(E, N, V) = klnW. 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. Phase transitions are linked to convex (upwards bending) intruders of S(E, N), where the canonical ensemble defined by the Laplace transform to the intensive variables becomes multi-modal, non-local, (it mixes widely different conserved quantities). Here the one-to-one mapping of the Legendre transform gets lost. 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. It is the whole ensemble that spreads irreversibly over the accessible phase space not the single N-body trajectory. This is all possible without invoking the thermodynamic limit, extensivity, or concavity of S(E, N, V). Without the thermodynamic limit or at phase-transitions, the systems are usually not self-averaging, i.e. do not have a single peaked distribution in phase space. 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 such as thermodynamics cannot distinguish a fractal distribution in phase space from its closure.