[Paper Review] Boltzmann vs Gibbs: a finite-size match

L. Ferrari|arXiv (Cornell University)|Jan 19, 2015

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper investigates the physical differences between Boltzmann and Gibbs definitions of entropy in finite-size systems, showing that measurable discrepancies between the two approaches decay as $N^{-1/2}$, not $1/N$, making experimental discrimination feasible. Using a two-level gas and an Ising model, it demonstrates that the divergence in predictions—particularly in critical behavior and heat capacity—can be detected in small systems, offering a concrete path to test foundational statistical mechanics principles.

ABSTRACT

The long standing contrast between Boltzmann's and Gibbs' approach to statistical thermodynamics has been recently rekindled by Dunkel and Hilbert [1], who criticize the notion of negative absolute temperature (NAT), as a misleading consequence of Boltzmann's definition of entropy. A different definition, due to Gibbs, has been proposed, which forbids NAT and makes the energy equipartition rigorous in arbitrary sized systems. The two approaches, however, are shown to converge to the same results in the thermodynamical limit. A vigorous debate followed ref.[1], with arguments against [2,3] and in favor [4,5,6,7] of Gibbs' entropy. In an attempt to leave the speculative level and give the discussion some deal of concreteness, we analyze the practical consequences of Gibbs' definition in two finite-size systems: a non interacting gas of N atoms with two-level internal spectrum, and an Ising model of N interacting spins. It is shown that for certain measurable quantities, the difference resulting from Boltzmann's and Gibbs' approach vanishes as the inverse square rrot of N, much less rapidly than the 1/N slope expected. As shown by numerical estimates, this makes the experimental solution of the controversy a feasible task.

Motivation & Objective

• To resolve the long-standing debate over Boltzmann vs. Gibbs entropy definitions by testing their physical consequences in finite systems.
• To determine whether measurable differences exist between the two entropy definitions in small systems, beyond the thermodynamic limit.
• To propose concrete experimental tests—particularly in sub-millimetric magnetic particles—where the two approaches yield distinct predictions.
• To analyze the critical behavior of the Ising model under both entropy definitions, focusing on the stability of the critical temperature $T_M$.
• To clarify the role of the Helmholtz free energy minimization in equilibrium thermodynamics under Gibbs' entropy, countering objections from peer review.

Proposed method

• Analyzes a non-interacting gas of $N$ two-level atoms and an Ising model of $N$ interacting spins to compare predictions from Boltzmann and Gibbs entropy.
• Uses the Gibbs entropy $S_G(E) = \ln\left[\sum_\eta \Theta(E - H(\eta))\right]$ and Boltzmann entropy $S_B(E) = \ln\left[\sum_\eta \delta(H(\eta) - E)\right]$ to compute thermodynamic quantities.
• Applies Laplace's method and asymptotic analysis to evaluate integrals involving the density of states, approximating the partition function and entropy in the large-$N$ limit.
• Derives the heat capacity and magnetization behavior under both definitions, identifying divergences and critical points such as $T_M$.
• Evaluates the stability of the Weiss ferromagnetic transition under external fields, predicting a destabilization threshold proportional to $1/\sqrt{N}$.
• Compares extremal values of the Helmholtz free energy $\Psi$ under both definitions, distinguishing between minima (equilibrium) and maxima (non-equilibrium) states.

Experimental results

Research questions

• RQ1 Do Boltzmann and Gibbs entropy definitions yield measurable differences in finite systems, and if so, at what rate do these differences vanish as $N \to \infty$?
• RQ2 Can the critical temperature $T_M$ of the Ising model be destabilized by weak external fields, and does this depend on the entropy definition?
• RQ3 Is the minimization of the Helmholtz free energy $\Psi$ a valid equilibrium criterion under Gibbs' entropy, especially in microcanonical-like systems?
• RQ4 Can experimental detection of discrepancies between the two entropy definitions be achieved in systems of order $N \sim 10^3$ to $10^6$ particles?
• RQ5 Does Gibbs' entropy eliminate negative absolute temperature (NAT) while preserving correct thermodynamic behavior in small systems?

Key findings

• The difference between predictions from Boltzmann and Gibbs entropy for measurable quantities decays as $N^{-1/2}$, not $1/N$, making experimental detection significantly more feasible.
• In the Ising model, the critical temperature $T_M$ becomes unstable under an external magnetic field of order $1/\sqrt{N}$, a signature that could be observed in sub-millimetric metallic particles.
• The heat capacity derived from Gibbs' entropy diverges at $T_M$, a feature absent in the Boltzmann approach, indicating a fundamental physical difference in critical behavior.
• The minimization of the Helmholtz free energy $\Psi$ correctly identifies equilibrium states under both definitions, with minima corresponding to stable equilibria and maxima to unstable or non-equilibrium states.
• The paper identifies a key inconsistency in the referee’s objection: the equivalence between $T = (\partial S / \partial E)^{-1}$ and $\partial \Psi / \partial m = 0$ holds for both entropy definitions, invalidating the claim that Gibbs' entropy invalidates free energy minimization.
• Numerical estimates show that detecting $N^{-1/2}$ effects is experimentally accessible with current technology, particularly in nanoscale magnetic systems under weak fields ($\sim 10^{-6}$ G).