On the dispute between Boltzmann and Gibbs entropy

Abstract

The validity of the concept of negative temperature has been recently challenged by arguing that the Boltzmann entropy (that allows negative temperatures) is inconsistent from a mathematical and statistical point of view, whereas the Gibbs entropy (that does not admit negative temperatures) provides the correct definition for the microcanonical entropy. Here we prove that the Boltzmann entropy is thermodynamically and mathematically consistent. Analytical results on two systems supporting negative temperatures illustrate the scenario we propose. In addition we numerically study a lattice system to show that negative temperature equilibrium states are accessible and obey standard statistical mechanics prediction. 05.20.Gg, 05.30.Ch PACS numbers

A consistent description of fluctuations requires negative temperatures

“…It is interesting to notice that T B < 0 (and the corresponding clusterization) is not a peculiarity of the divergence of G(r) in r = 0, nor of the long range nature of the interaction: indeed, it can be obtained with any arbitrary G(r) having a maximum (even finite) in r = 0, and vanishing at large r, provided that the domain is bounded. The presence of spatial order at high values of energy, in the form of discrete breathers, has been observed also in the discrete non-linear Schrödinger equation and analogous systems [10,31]. In Section IV we introduce a different, in a way simpler, model which still exhibits spatial order at small negative temperatures.…”

“…Two different definitions of temperature in equilibrium statistical mechanics have been recently the subject of an intense debate [1][2][3][4][5][6][7][8][9][10], after the publication of experimental measurements of a negative absolute temperature [11,12]. In [11] it was demonstrated the possibility to prepare a state where the observed distribution of the modified kinetic energy per atom appeared to be inverted, i.e.…”

Isotropic–Nematic Phase Transitions in Gravitational Systems

“…This achievement triggered a debate (Schneider et al 2014;Hänggi et al 2015;Frenkel & Warren 2015;Campisi 2015;Cerino et al 2015;Poulter 2016), initiated by Dunkel & Hilbert (2014), on whether negative absolute thermodynamic temperatures are observable after all or maybe the Boltzmann's entropy formula should be abandoned in favor of Gibbs' entropy, which allows only positive temperatures. However, the arguments in favor of the validity of Boltzmann's formula are more convincing at least for the case of ensembleequivalence where it was proven recently that Boltzmann's formula is appropriate and negative temperatures do occur (Buonsante et al 2016).…”

In defense of negative temperature

“…5 is that it is always appropriate to use the Gibbs entropy and reject the Boltzmann entropy. This has created a controversy with support for the proposal 7-10 as well as criticism in support of keeping the Boltzmann entropy and negative temperatures [11][12][13][14][15][16][17][18] . In ref.…”