Straight way to Thermo-Statistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit

D. H. E. Gross

Published 2001-05-16, updated 2001-07-04Version 2

Boltzmann's principle S(E,N,V)=k\ln W relates the entropy to the geometric area e^{S(E,N,V)} of the manifold of constant energy in the N-body phase space. From the principle all thermodynamics and especially all phenomena of phase transitions and critical phenomena can be deduced. The topology of the curvature matrix C(E,N) (Hessian) of S(E,N) determines regions of pure phases, regions of phase separation, and (multi-)critical points and lines. Thus, C(E,N) describes all kind of phase-transitions with all their flavor. They are linked to convex (upwards bending) intruders of S(E,N), here the canonical ensemble defined by the Laplace transform to the intensive variables becomes non-local and violates the basic conservation laws (it mixes widely different conserved quantities). Thus 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. Moreover, 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) and also without invoking any cosmological constraints. It is further shown that non-extensive Hamiltonian systems at equilibrium are described by Boltzmann's principle and not by Tsallis non-extensive statistics.