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

D. H. E. Gross|arXiv (Cornell University)|May 16, 2001

Statistical Mechanics and Entropy35 references3 citations

TL;DR

This paper presents a geometric derivation of equilibrium thermodynamics, phase transitions, and the Second Law of Thermodynamics directly from Newtonian mechanics without requiring the thermodynamic limit. By interpreting entropy as the logarithm of the phase space volume (Boltzmann's principle), it shows that curvature topology of the entropy manifold fully determines phase behavior and irreversibility arises from ensemble-level spreading, not single trajectories.

ABSTRACT

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.

Motivation & Objective

• To establish a direct, geometric derivation of equilibrium thermodynamics from microscopic reversible dynamics without relying on the thermodynamic limit.
• To resolve the paradox of irreversibility in time-reversible systems by showing that ensemble-level spreading—not individual trajectories—drives entropy increase.
• To demonstrate that phase transitions, critical points, and phase separation emerge naturally from the curvature topology of the entropy manifold S(E,N).
• To argue that non-extensive systems at equilibrium are described by Boltzmann statistics, not Tsallis statistics, even without extensivity or concavity of entropy.
• To show that coarse-graining is not an ad hoc assumption but a necessary consequence of averaging over unobserved degrees of freedom in macroscopic measurements.

Proposed method

• Uses Boltzmann’s principle S(E,N,V) = ln W to define entropy as the logarithm of the geometric volume of the constant-energy manifold in 6N-dimensional phase space.
• Analyzes the Hessian matrix C(E,N) of the entropy function S(E,N) to determine curvature topology, which classifies regions of pure phases, phase separation, and critical points.
• Applies the Legendre transform to connect microcanonical and canonical ensembles, showing its breakdown at phase transitions due to non-locality and multi-modality.
• Models the time evolution of the phase space ensemble as a spreading manifold, with entropy increasing due to geometric spreading over accessible states.
• Uses box-counting measure instead of Hausdorff measure to reflect macroscopic averaging over unobserved degrees of freedom, introducing irreversibility naturally.
• Demonstrates that Liouville’s theorem preserves phase space volume under Hamiltonian dynamics, but macroscopic observables depend on coarse-grained measures, leading to effective irreversibility.

Experimental results

Research questions

• RQ1 How can the Second Law of Thermodynamics be derived from time-reversible microscopic dynamics without invoking the thermodynamic limit?
• RQ2 What geometric features of the entropy manifold S(E,N) determine phase transitions, critical points, and phase separation?
• RQ3 Why does the canonical ensemble fail to describe systems at phase transitions, and how does this relate to the breakdown of the Legendre transform?
• RQ4 Can non-extensive systems at equilibrium be described by Boltzmann statistics rather than Tsallis statistics, even without extensivity?
• RQ5 What is the role of coarse-graining in irreversibility, and why is the box-counting measure more appropriate than the Hausdorff measure in statistical mechanics?

Key findings

• The curvature topology of the entropy manifold S(E,N) fully determines all phase transition phenomena, including critical points and lines, without requiring the thermodynamic limit.
• Irreversibility arises from the irreversible spreading of the entire ensemble over accessible phase space, not from individual trajectories, even in finite-N systems.
• The canonical ensemble becomes multi-modal and non-local at phase transitions, signaling a breakdown of the one-to-one correspondence of the Legendre transform.
• The Second Law is derived directly from Newtonian mechanics via ensemble-level geometric spreading, with entropy increasing due to the growth of the accessible phase space volume.
• Box-counting measure, not Hausdorff measure, is physically appropriate because it reflects macroscopic averaging over unobserved degrees of freedom, making coarse-graining a natural consequence of incomplete measurement.
• Non-extensive systems at equilibrium are governed by Boltzmann’s principle, not Tsallis statistics, even when the system is small or non-self-averaging.