Regarding the Entropy of Distinguishable Particles

Explanation, Prediction, and Confirmation, 2011

The Gibbs paradox has frequently been interpreted as a sign that particles of the same kind are fundamentally indistinguishable; and that quantum mechanics, with its identical fermions and bosons, is indispensable for making sense of this. In this article we shall argue, on the contrary, that analysis of the paradox supports the idea that classical particles are always distinguishable. Perhaps surprisingly, this analysis extends to quantum mechanics: even according to quantum mechanics there can be distinguishable particles of the same kind. Our most important general conclusion will accordingly be that the universally accepted notion that quantum particles of the same kind are necessarily indistinguishable rests on a confusion about how particles are represented in quantum theory.

American Journal of Physics, 2011

Discussions of the foundations of statistical mechanics, how they lead to thermodynamics, and the appropriate definition of entropy have occasioned many disagreements. I believe that some or all of these disagreements arise from differing, but unstated assumptions, which can make opposing opinions difficult to reconcile. To make these assumptions explicit, I discuss the principles that have guided my own thinking about the foundations of statistical mechanics, the microscopic origins of thermodynamics, and the definition of entropy. The purpose of this paper will be fulfilled if it paves the way to a final consensus, whether or not that consensus agrees with my point of view.

2002

The properties of classical models of distinguishable particles are shown to be identical to those of a corresponding system of indistinguishable particles without the need for ad hoc corrections. An alternative to the usual definition of the entropy is proposed. The new definition in terms of the logarithm of the probability distribution of the thermodynamic variables is shown to be consistent with all desired properties of the entropy and the physical properties of thermodynamic systems. The factor of 1/N! in the entropy connected with Gibbs' Paradox is shown to arise naturally for both distinguishable and indistinguishable particles. These results have direct application to computer simulations of classical systems, which always use distinguishable particles. Such simulations should be compared directly to experiment (in the classical regime) without ''correcting'' them to account for indistinguishability.

2016

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.

Entropy, 2008

Gibbs' Paradox is shown to arise from an incorrect traditional definition of the entropy that has unfortunately become entrenched in physics textbooks. Among its flaws, the traditional definition predicts a violation of the second law of thermodynamics when applied to colloids. By adopting Boltzmann's definition of the entropy, the violation of the second law is eliminated, the properties of colloids are correctly predicted, and Gibbs' Paradox vanishes.

Foundations of Physics, 2012

Disagreements over the meaning of the thermodynamic entropy and how it should be defined in statistical mechanics have endured for well over a century. In an earlier paper, I showed that there were at least nine essential properties of entropy that are still under dispute among experts. In this paper, I examine the consequences of differing definitions of the thermodynamic entropy of macroscopic systems.

Entropy, 2018

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs' notion of 'generic phase'). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. 'Distinguishability', in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

Journal of Statistical Physics, 2004

In a recent paper, Nagle criticized the new definition of entropy that I had proposed in an earlier work. In the examples for which Nagle claims my definition fails, he took a formula that I had derived for one set of experiments and used it to represent my definition for other experiments. However, the formulas obtained from my definition depend on the specific experimental observables. If my definition is correctly applied to Nagle's experiments, no contradictions remain.

The notion that particles can be indistinguishable in a fundamental sense, which poses a challenge to traditional notions of individuality and identity, has first come up in the context of classical statistical mechanics. In particular, the Gibbs paradox has been interpreted as a sign of the untenability of the classical concept of a particle and as a premonition that quantum theory is needed to make sense of the situation. These ideas have been criticized in the literature, and completely classical solutions of the Gibbs paradox have been proposed. We shall argue, however, that although the criticism was justified, the proposed solutions have not gone to the heart of the matter. As we shall show, the solution of the Gibbs paradox in classical physics is in fact unrelated to fundamental distinguishability issues; only distinguishability in a pragmatic sense plays a role (in this we develop ideas of van Kampen [10]). With regard to quantum mechanics we shall show that the paradox survives in basically the same form even here, in spite of the quantum mechanical (anti-)symmetrization postulates.

2002

The "Gibbs Paradox" refers to several related questions concerning entropy in thermodynamics and statistical mechanics: whether it is an extensive quantity or not, how it changes when identical particles are mixed, and the proper way to count states in systems of identical particles. Several authors have recognized that the paradox is resolved once it is realized that there is no such thing as the entropy of a system, that there are many entropies, and that the choice between treating particles as being distinguishable or not depends on the resolution of the experiment. The purpose of this note is essentially pedagogical; we add to their analysis by examining the paradox from the point of view of information theory. Our argument is based on that 'grouping' property of entropy that Shannon recognized, by including it among his axioms, as an essential requirement on any measure of information. Not only does it provide the right connection between different entropies but, in addition, it draws our attention to the obvious fact that addressing issues of distinguishability and of counting states requires a clear idea about what precisely do we mean by a state.