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American Journal of Physics

The fraction of “all different” combinations: Justifying the semiclassical partition function

Ralph F. Baierlein

Published Apr 1, 1997• 3 citations
Publisher

Abstract

Modern statistical mechanics directly acknowledges the indistinguishability of identical particles, a characteristically quantum property. Under laboratory conditions such that classical physics ought to suffice for most purposes, the indistinguishability is often incorporated by a judicious division by N!. This paper explores a common justification for that procedure—and finds it wanting. Nonetheless, the procedure is valid, but for a different reason.

DOI
10.1119/1.18540

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Modern statistical mechanics directly acknowledges the indistinguishability of identical particles, a characteristically quantum property. Under laboratory conditions such that classical physics ought to suffice for most purposes, the indistinguishability is often incorporated by a judicious division by N!. This paper explores a common justification for that procedure—and finds it wanting. Nonetheless, the procedure is valid, but for a different reason.
It is widely accepted that the papers by Bose in 1924 and by Einstein in 1924-1925 definitely opened the path to the quantum concept of ‘indistinguishability’. Since then, distinction in physics between ‘identity’ and ‘indistinguishability’ of particles has come to be required. However, even before the rise of quantum physics, Maxwell, Boltzmann and Gibbs, among others, had to tackle the difficulties and paradoxes that appear when counting identical particles. Later, after Sackur’s and Tetrode’s first quantum derivation of the canonical partition function of an ideal gas, much confusion arose around its dependence on the number of molecules in the system. Many other physicists, such as Planck, Lorentz or Schrodinger took part in those debates. In 1921, Ehrenfest and Trkal devoted a whole paper to this subject. Clearly, in the days before Bose’s seminal work, embroiling and misunderstanding were commonplace. In considering the emergence of the distinction between ‘identical’ and ‘indistinguishable’ particles, we will go through the so-called statistical route to quantum mechanics. The first developments in statistical mechanics became immediately confused and entangled with the irruption of quantum physics. In the first quarter of the twentieth century the mutual influence between both disciplines was profuse. We will pay special attention to the story of the so-called ‘Gibbs Paradox.’ We will argue that what is now often known by this name in some textbooks of statistical mechanics did not exist as such in those foundational days.
Abstract According to classical physics particles are basic building blocks of the world. These classical particles are distinguishable objects, individuated by unique combinations of physical properties. By contrast, in quantum mechanics the received view is that particles of the same kind (“identical particles”) are physically indistinguishable from each other and lack identity. This doctrine rests on the quantum mechanical (anti)symmetrization postulates together with the “factorist” assumption that each single particle is represented in exactly one factor space of the tensor product Hilbert space of a many-particle system. Even though standard in theoretical physics and the philosophy of physics, the assumption of factorism and the ensuing indistinguishability of particles are problematic. Particle indistinguishability is irreconcilable with the everyday meaning of “particle”, and also with how this term is used in the practice of physics. Moreover, it is a consequence of the standard view that identical quantum particles remain indistinguishable even in the classical limit, which makes a smooth transition to the classical particle concept impossible. Lubberdink (1998; 2009) and Dieks and Lubberdink (2011) have proposed an alternative conception of quantum particles that does not rely on factorism and avoids these difficulties. We further explain and discuss this alternative framework here. One of its key consequences is that particles in quantum theory are not fundamental but emergent; another that once they have emerged, quantum particles are always physically distinguishable and thus possess a physically grounded identity.
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.
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.