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Extensivity of entropy and modern form of Gibbs paradox
Abstract
The extensivity property of entropy is clarified in the light of a critical examination of the entropy formula based on quantum
statistics and the relevant thermodynamic requirement. The modern form of the Gibbs paradox, related to the discontinuous
jump in entropy due to identity or non-identity of particles, is critically investigated. Qualitative framework of a new resolution
of this paradox, which analyses the general effect of distinction mark on the Hamiltonian of a system of identical particles,
is outlined.
... In physics, there are two distinct paradoxes, which are both known as the " Gibbs paradox " [2, 3, 20] and are often confused with each other. In the following, the false increase in entropy, which is calculated from the process of combining two gases of the same kind consisting of distinguishable particles, will be referred to as the Gibbs paradox of the first kind (GP1) [16, 21–23, 30, 38, 41, 44]. ...
In physics, there are two distinct paradoxes, which are both known as the “Gibbs paradox”. This article is concerned with
only one of them: the false increase in entropy, which is calculated from the process of combining two gases of the same kind
consisting of distinguishable particles. In the following, this paradox will be referred to as the Gibbs paradox of the first
kind (GP1). (Two particles are said to be distinguishable if they are either non-identical, that is, if they have different
properties, or if they are identical and there are microstates which change under transposition of the two particles.) The
GP1 is demonstrated and subsequently analyzed. The analysis shows that, for (quantum or classical) systems of distinguishable
particles, it is generally uncertain of which particles they consist. The neglect of this uncertainty is the root of the GP1.
For the statistical description of a system of distinguishable particles, an underlying set of particles, containing all particles
that in principle qualify for being part of the system, is assumed to be known. Of which elements of this underlying particle
set the system is composed, differs from microstate to microstate. Thus, the system is described by an ensemble of possible
particle compositions. The uncertainty about the particle composition contributes to the entropy of the system. Systems for
which all possible particle compositions are equiprobable will be called harmonic. Classical systems of distinguishable identical particles are harmonic as a matter of principle; quantum or classical systems
of non-identical particles are not necessarily harmonic, since for them the composition probabilities depend individually
on the preparation of the system. Harmonic systems with the same underlying particle set are always correlated; hence, for
harmonic systems, the entropy is no longer additive and loses its thermodynamic meaning. A quantity derived from entropy is
introduced, the reduced entropy, which, for harmonic systems, replaces the entropy as thermodynamic potential. For identical classical particles, the equivalence
(in particular with respect to the second law of thermodynamics) between distinguishability and indistinguishability is proved.
The resolution of the GP1 is demonstrated applying the previously found results.
KeywordsEntropy-Distinguishable particles-Gibbs paradox
The development of the concept of indistinguishability in the writings of Gibbs is traced, leading from the "Gibbs paradox" to his definition of generic phase.
The traditional resolution of Gibbs paradox seems to give rise to a further paradox. A resolution of this paradox is proposed.
This note deals with the Gibbs paradox for the following case. An ideal gas A is allowed to mix with an ideal gas A* whose atoms are all A atoms in an excited metastable state. It is shown that after a time long compared to the lifetime of the metastable state there has been an entropy increase larger than the mixing entropy which “disappears.”
A presentation is given of the form taken by the thermodynamic theory of homogeneous chemical equilibrium when transcribed into Landsberg's (1961) formulation of thermodynamics in terms of set theory. Certain sets of points can usefully be defined, in order to deal separately and explicitly with metastable equilibrium states, and with open systems as such, without having simultaneously to consider non-equilibrium states. 'Narrower' and 'wider' thermodynamic principles can be distinguished in terms of what is the set for which some proposition is asserted to apply, and Landsberg's formulation makes possible greater clarity in the enunciation of the special axioms of chemical thermodynamics. In particular, the axiom of the ideal behaviour of dilute systems takes a very clear-cut significance. When arguments of the type introduced by Landsberg are used, the assumptions involved in a formal development of the thermodynamic theory of homogeneous chemical equilibrium are clearly apparent. Metastable equilibrium states, and non-equilibrium states, each have their own distinct place in the theory. Some comments are made in relation to the usual presentation.
The sudden drop in the isothermal entropy of mixing when two ideal gases
are made identical is analyzed for ideal quantum gases. For fermions of
large reduced chemical potentials μ/kT this drop can be made
arbitrarily small and this leads to a new resolution of the paradox.
The identification of the phase space ofN classical identical particles with the equivalence class of points is of crucial importance for statistical mechanics. We show that the refined phase space leads to the correct statistical mechanics for an ideal gas; moreover, Gibbs's paradox is resolved and the Third Law of Thermodynamics is recovered. The presence of both induced and stimulated transitions is shown as a consequence of the identity of the particles. Other results are the quantum contribution to the second virial coefficient and the Bose-Einstein condensation. Photon bunching and Hanbury Brown-Twiss effect are also seen to follow from the classical model. The only element of quantum theory involved is the notion of phase cells necessary to make the entropy dimensionless. Assuming the existence of the light quantum or the phonon hypothesis we could derive the Planck distribution law for blackbody radiation or the Debye formula for specific heats respectively.
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