Boltzmann or Gibbs Entropy? <br/>Thermostatistics of Two Models with Few Particles

Abstract

We study the statistical mechanics of small clusters (N ~ 10-100) for two-level systems and harmonic oscillators. Both Boltzmann's and Gibbs's definitions of entropy are used. The properties of the studied systems are evaluated numerically but exactly; this means that Stirling's approximation was not used in the calculation and that the discrete nature of energy was taken into account. Results show that, for the two-level system, using Gibbs entropy prevents temperatures from assuming negative values; however, they reach very high values that are not plausible in physical terms. In the case of harmonic oscillators, there are no significant differences when using either definition of entropy. Both systems show that for N = 100 the exact results evaluated with statistical mechanics coincide with those found in the thermodynamic limit. This suggests that thermodynamics can be applied to systems as small as these.

Relation between Boltzmann and Gibbs entropy and example with multinomial distribution

“…For clarity, it is also important to mention the reference [7], where a numerical calculation of the Boltzmann entropy and the so-called Gibbs volume entropy, as candidates for the microcanonical entropy, is done on small clusters for two level systems and harmonic oscilators. However, in view of recent discussions [8][9][10][11][12] on the question which of these two definitions of microcanonical entropy is justified on thermodynamic basis, numerical results of reference [7] are not decisive. The Gibbs volume entropy, and the Gibbs entropy defined in the sense that is used in our paper, coincide only if the microstates of energy less than or equal to the value E are only allowed and they have equal probabilities.…”

Thermodynamics of finite systems: a key issues review

“…Other authors, myself included, have disputed these points [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. The most fundamentally important issue in the debate is whether entropy is additive in thermodynamics, but the one that has aroused the most interest is whether negative temperatures provide a consistent description of systems with a decreasing density of states in some energy region.…”

Resolving the debate about proposed expressions for the classical entropy

“…Some workers in the field have pointed to weaknesses in the Boltzmann definition (defined in section III) [1,2], and claimed that the Gibbs entropy (defined in section IV) [3][4][5] must replace it [6][7][8][9][10][11][12][13][14][15][16]. Others have defended the use of the Boltzmann entropy and pointed to flaws in the Gibbs entropy [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”