Effective models for many particle systems. BCS theory and the Kac model

In the beginning of the 20th century, the dependence of electrical resistance on temperature became a very active field of research in physics. Measurements on the resistance of many different metals at low temperatures, which were carried out by experimental physicists, amongst them James Dewar and John Ambrose Fleming, gave room for the idea that the resistance of metals could vanish completely at very low temperatures. Lord Kelvin [63] in 1902 made the prediction that resistance decreases with falling temperature up to a certain point. After this minimum is reached, he expected the resistance to increase again as temperature decreases further. This behavior turned out to be true for semiconductors. However, in the case of superconductivity things are different. At a certain temperature, the so-called critical temperature, resistance abruptly vanishes and stays zero at all temperatures below the critical temperature. This phenomenon was discovered by Heike Kamerlingh Onnes in 1911 and he was awarded the Nobel Prize for his discovery of superconducting materials in 1913. It took almost 46 years of intensive research until, in 1957, John Bardeen, Leon Neil Cooper and John Robert Schrieffer published their famous paper with the title ''Theory of Superconductivity'' [6]. Their theory of superconductivity, usually referred to as BCS theory, is based on the idea of electron pairing driven by an effective attraction mediated by phonons. This publication was a breakthrough, presenting the first microscopic model for the remarkable effect of vanishing resistance and the three authors were awarded the Nobel prize for their discovery in 1972. Before, Cooper [24] had realized that a very tiny attractive interaction between particles in a Fermi gas suffices to cause pairing between electrons. While this interaction is possible because of interactions through the lattice in the case of metals, it is of local type in other situations, as for example for superfluid cold gases. The arising electron pairs are known as Cooper pairs. Approximately, Cooper pairs behave like Bosons and they form a condensed state which is not identical with a Bose-Einstein condensate, but nonetheless similar. Amongst the famous and typical properties of superconductors that can be explained with the appearance of this strongly correlated quantum state are, for example, infinite conductivity, the Meissner effect, flux quantization and the isotope effect, see the original work of Bardeen, Cooper and Schrieffer [6] and the book of Fetter and Walecka [31]. The second model, which will play a role in this thesis, is a model in kinetic theory and was introduced by Mark Kac in 1956 in his article ''Foundations of kinetic theory'' [61]. The Kac model is a linear, microscopic model to describe a gas of interacting particles in a probabilistic way. This model, and in particular the Kac master equation, due to its simplicity, have a special place among the models describing a large number of interacting particles. One goal, and the main motivation for Kac's work in [61], was to provide a satisfactory derivation of the spatially homogeneous Boltzmann equation. Indeed, Kac in [61] was able to derive the spatially homogeneous, non-linear Kac-Boltzmann equation. It was in this context that Kac introduced his notion of propagation of chaos as well as his definition of chaotic sequences, what he called 'sequences that have the Boltzmann property'. Both these concepts turned out to be very useful tools in his derivation. So far, the - much more difficult - derivation of the Boltzmann equation from the laws of classical mechanics has only been shown for situations with very few collisions, see the work of O. Lanford [66, 67]. In addition, Kac in [61] wished to lay a basis and give a mathematical setting for the study of approach to equilibrium, which he did by presenting his approach to the problem through master equations. These two models, the BCS model and the Kac model, are the effective theories, which will be studied in this thesis.