Second Law of Thermodynamics, but Without a Thermodynamic Limit

Abstract

A geometric foundation of thermo-statistics is presented with only the axiomatic assumption of Boltzmann’s principle S(E, N, V) = k In W. This relates the entropy to the geometric area e ^S(E,N,V) of the manifold of constant energy in the (finite) N-body phase space. From this principle, all of thermodynamics and especially all phenomena of phase transitions and critical phenomena can be unambiguously identified for even small systems. 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 like Thermodynamics cannot distinguish a fractal distribution in phase space from its closure.