Boltzmann or Gibbs Entropy?
Thermostatistics of Two Models with Few Particles


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.

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Miranda, E. (2015) Boltzmann or Gibbs Entropy?
Thermostatistics of Two Models with Few Particles. Journal of Modern Physics, 6, 1051-1057. doi: 10.4236/jmp.2015.68109.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Callen, H.B. (1985) Thermodynamics and an Introduction to Thermostatistics. Wiley, New York.
[2] Kinchin, A.I. (1949) Mathematical Foundations of Statistical Mechanics. Dover, New York.
[3] Gibbs, J.W. (1902) Elementary Principles of Statistical Mechanics. Yale University Press, New Haven. (Reprinted by Dover, New York, 1960)
[4] Campisi, M. (2005) Studies in History and Philosophy of Science Part B, 36, 275-290.
[5] Campisi, M. (2007) Physica A, 385, 501-517.
[6] Campisi, M. (2008) Studies in History and Philosophy of Science Part B, 39, 181-194.
[7] Campisi, M., Zhan, F., Talkner, P. and Hänggi, P. (2012) Physical Review Letters, 108, Article ID: 250601.
[8] Campisi, M. (2007) Microcanonical Phase Transitions in Small Systems. arXiv: 0709.1082v1 [cond-mat.stat.mech].
[9] Dunkel, J. and Hilbert, S. (2006) Physica A, 370, 390-406.
[10] Dunkel, J. and Hilbert, S. (2013) Inconsistent Thermostatistics and Negative Absolute Temperatures. arXiv:1304.2066v1 [cond-mat.stat.mech].
[11] Dunkel, J. and Hilbert, S. (2014) Nature Physics, 10, 67-72; and Supplementary Information.
[12] Frenkel, D. and Warren, P.B. (2015) American Journal of Physics, 83, 163.
[13] Swendsen, R.H. and Wang, J.-S. (2014) Negative Temperatures and the Definition of Entropy. arXiv: 1410.4619v1 [cond-mat.stat.mech]
[14] Hilbert, S., Hänggi, P. and Dunkel, J. (2014) Thermodynamics Law in Isolates Systems.
arXiv:1408.5382v1 [cond-mat.stat.mech]
[15] Niven, R.K. (2005) Physics Letters A, 342, 286-293.
[16] Niven, R.K. (2006) Physica A, 365, 142-149.
[17] Elmaghraby, E.K. (2011) Journal of Modern Physics, 2, 1242-1246.
[18] Miranda, E.N. and Bertoldi, D.S. (2013) European Journal of Physics, 34, 1075.

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