Abstract
The suggestion that particles of the same kind may be indistinguishable in a fundamental sense, even so that challenges to traditional notions of individuality and identity may arise, has first come up in the context of classical statistical mechanics. In particular, the Gibbs paradox has sometimes been interpreted as a sign of the untenability of the classical concept of a particle and as a premonition that quantum theory is needed. This idea of a ‘quantum connection’ stubbornly persists in the literature, even though it has also been criticized frequently. Here we shall argue that although this criticism is justified, the proposed alternative solutions have often been wrong and have not put the paradox in its right perspective. In fact, the Gibbs paradox is unrelated to fundamental issues of particle identity; only distinguishability in a pragmatic sense plays a role (in this we develop ideas of van Kampen [11]), and in principle the paradox always is there as long as the concept of a particle applies at all. In line with this we show that the paradox survives even in quantum mechanics, in spite of the quantum mechanical (anti-)symmetrization postulates.
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Notes
See ([12] for a discussion of the consequences of choosing conventions that do not make the entropy extensive.
In the later literature quantum mechanics and the symmetrization postulates for states of ‘identical quantum particles’ are frequently invoked as the final justification.
Actually, we should in principle divide by the number of all particles of the same kind in the whole universe—this does not make a difference for our point about the irrelevance of this division made in the text. Compare also the later discussion of \(1/N!\) in quantum mechanics.
Saunders ([8]) says that both the permutability argument and the Ehrenfest-Trkal argument provide a solution for the classical Gibbs paradox, and that quantum indistinguishability does the same for quantum particles. Our conclusion here will be that all three claims are wrong.
Swendsen ([9, 10]) attempts to avoid this conclusion by an appeal to counterfactual reasoning: if the total system were to enter into contact with another system, its number of particles would become variable and subject to a probability distribution. But such counterfactual reasoning has no explanatory force for the actual case of the Gibbs paradox, in which the total system is closed.
This is not to deny that the Ehrenfest-Trkal approach constitutes an essential step forward in the context of problems in which particle numbers can change, for example the study of dissociation equilibria!
The factors appearing in quantum statistics are actually a bit more complicated. Suppose we are considering a system of \(N\) particles, with \(X\) one-particle states available to them. Classical counting leads to \(X^N\) possible states for the \(N\)-particle system, if we assume that each one-particle state can be occupied by more than one particle. The number of quantum states for bosons is \((N + X - 1)!/N!(X-1)!\), which reduces to \(X^N/N!\) if \(X \gg N\) but in any case is a constant as long as \(N\) does not change—this is the essential point for the argument in the text. In the case of fermions the number of states is \(X!/N!(X-N)!\), which also reduces to \(X^N/N!\) if \(X \gg N\) and is also constant when \(N\) does not vary.
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Dedicated to the memory of N.G. van Kampen (1921–2013)
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Dieks, D. The Logic of Identity: Distinguishability and Indistinguishability in Classical and Quantum Physics. Found Phys 44, 1302–1316 (2014). https://doi.org/10.1007/s10701-014-9814-0
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DOI: https://doi.org/10.1007/s10701-014-9814-0