5 - Quantum statistics  pp. 267-334

Quantum statistics

By Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni

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In Chapter 3 we exhibited the limitations of a purely classical approach. For example, if the temperature is below a threshold value, some degrees of freedom become ‘frozen’ and the equipartition theorem is no longer valid for them. The translational degrees of freedom of an ideal gas appear to escape this limitation of the classical (or more precisely, semi-classical) approximation. We shall see in this chapter that, in fact, this is not so: if the temperature continues to decrease below some reference temperature, the classical approximation will deteriorate progressively. However, in this case, the failure of the classical approximation is not related to freezing degrees of freedom but rather to the symmetry properties of the wave function for identical particles imposed by quantum mechanics. A rather spectacular consequence is that the kinetic energy is no longer a measure of the temperature. In a classical gas, even in the presence of interactions, the average kinetic energy is equal to 3kT/2, but this result does not hold when the temperature is low enough, even for an ideal gas. For example, if we consider the conduction electrons in a metal as an ideal gas, we shall show that the average kinetic energy of an electron is not zero even at zero temperature. In addition, this kinetic energy is about 100 times kT at normal temperature. Let us consider another example. In a gaseous mixture of helium-3 and helium-4 at low temperature, the average kinetic energies of the two isotopes are different: the average kinetic energy of helium-3 is larger than 3/2kT while that of helium-4 is smaller.