The Logic of Identity: Distinguishability and Indistinguishability in Classical and Quantum Physics

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 [10]), 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.

The definition of the thermodynamic entropy in statistical mechanics

“…The entropy defined this way satisfies the postulates for thermodynamics [1][2][3]. I've addressed the objections by Dieks [11,12] and Peters [13,14] to this derivation of the entropy from statistical mechanics and shown that they are not valid.…”

“…The two-step approach can be described as hybrid because it starts with a definition of entropy, notes that the definition fails in some respect, and then corrects it to agree more closely with the thermodynamic properties of entropy. The historical reason for this peculiar question lies in the effort to maintain a definition of entropy in the form of the logarithm of a volume in phase space by modifying it to correct the dependence on particle number [11][12][13][28][29][30]. Since this process usually involves the inclusion of a negative term, −k B ln N !, the result is often called a "reduced entropy.…”

“…I have presented other derivations in the past that are equivalent, though perhaps not as direct [5][6][7][8][9][10]. They have been criticized by Dieks [11,12] and Peters [13,14], whose arguments will be discussed in Sections V and VI. * swendsen@cmu.edu…”

The Gibbs Paradox and Particle Individuality

“…The quantum theory of “identical particles” will turn out to be only marginally relevant: even in the case of quantum systems, the Gibbs paradox persists—it is merely the numerical value of the mixing entropy that is affected by quantum considerations. The present discussion is more comprehensive than in our earlier papers on this subject [ 1 , 2 , 3 , 4 ], adds new considerations both in the classical and quantum contexts, and corrects inaccuracies.…”

Probability, Entropy, and Gibbs’ Paradox(es)

“…When I first wrote about this way of defining the thermodynamic entropy, I only used exchanges between two systems to illustrate the idea [ 25 , 26 , 27 , 28 ]. Objections to this approach and its extension to many systems were raised [ 31 , 32 , 33 ] and answered [ 12 , 29 ]. For completeness, I have included a sketch of the original argument in Appendix A [ 25 ].…”