Abstract
Statistics of distinguishable particles has become relevant in systems of colloidal particles and in the context of applications of statistical mechanics to complex networks. In this paper, we present evidence that a commonly used expression for the partition function of a system of distinguishable particles leads to huge fluctuations of the number of particles in the grand canonical ensemble and, consequently, to nonequivalence of statistical ensembles. We will show that the alternative definition of the partition function including, naturally, Boltzmann’s correct counting factor for distinguishable particles solves the problem and restores ensemble equivalence. Finally, we also show that this choice for the partition function does not produce any inconsistency for a system of distinguishable localized particles, where the monoparticular partition function is not extensive.
Highlights
The Gibbs paradox, namely, the entropy not being extensive for a classical ideal gas, is commonly solved by adding an ad hoc term to the entropy, −k log( N!) or, using Stirling formula, −kN log( N/e), where N is the number of particles and k is Boltzmann’s constant
We have shown that the common textbook expression of the partition function of a system of distinguishable particles which does not include the N! term, leads to abnormally large fluctuations of the number of particles in the grand canonical ensemble
The large fluctuations go against the postulates of statistical mechanics, which require that the relative fluctuations of the number of particles in the grand canonical ensemble vanish in the thermodynamic limit, such that it is possible to identify the mean value of the number of particles as the physically measurable N and ensure ensemble equivalence
Summary
The Gibbs paradox, namely, the entropy not being extensive for a classical ideal gas, is commonly solved by adding an ad hoc term to the entropy, −k log( N!) or, using Stirling formula, −kN log( N/e), where N is the number of particles and k is Boltzmann’s constant. Another example is that of statistical mechanics of networks [17,18,19], where edges/links of the network are considered as particles, and pairs of vertices/nodes are considered as energy states, establishing a straightforward analogy with quantum physical systems. In this case, links correspond to individual identifiable actions and it seems very forced to regard them as indistinguishable [20].
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