Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles

This review is devoted to the problem of thermalization in a small isolated conglomerate of interacting constituents. A variety of physically important systems of intensive current interest belong to this category: complex atoms, molecules (including biological molecules), nuclei, small devices of condensed matter and quantum optics on nano- and micro-scale, cold atoms in optical lattices, ion traps. Physical implementations of quantum computers, where there are many interacting qubits, also fall into this group. Statistical regularities come into play through inter-particle interactions, which have two fundamental components: mean field, that along with external conditions, forms the regular component of the dynamics, and residual interactions responsible for the complex structure of the actual stationary states. At sufficiently high level density, the stationary states become exceedingly complicated superpositions of simple quasiparticle excitations. At this stage, regularities typical of quantum chaos emerge and bring in signatures of thermalization. We describe all the stages and the results of the processes leading to thermalization, using analytical and massive numerical examples for realistic atomic, nuclear, and spin systems, as well as for models with random parameters. The structure of stationary states, strength functions of simple configurations, and concepts of entropy and temperature in application to isolated mesoscopic systems are discussed in detail. We conclude with a schematic discussion of the time evolution of such systems to equilibrium.Comment: 69 pages, 31 figure

The herd moves? Emergence and self-organization in collective actors?

The puzzle about collective actors is in the focus of this contribution. The first section enters into the question of the adequateness and inadequateness of reductionist explanations for the description of entities. The considerations in this part do not draw on systems and hence not on principles of self-organisation, because this concept necessitates a systemic view. In other words, the first section discusses reductionism and holism on a very general level. The scope of these arguments goes far beyond self-organising systems. Pragmatically, these arguments will be discussed within the domain of corporative actors. Emergence is a concept embedded in system theory. Therefore, in the second part the previous general considerations about holism are integrated with respect to the concept “emergence”. In order to close the argument by exactly characterising self-organising systems and giving the conceptual link between self-organisation and emergence – which is done in the section four – the third section generally conceptualises systems. This conceptualisation is independent of whether these systems are self-organising or not. Feedback loops are specified as an essential component of systems. They establish the essential precondition of system-theoretic models where causes may also be effects and vice versa. System-theory is essential for dynamic models like ecological models and network thinking. In the fourth part mathematical chaos-theory bridges the gap between the presentation of systems in general and the constricted consideration of self-organising systems. The capability to behave or react chaotically is a necessary precondition of self-organisation. Nevertheless, there are striking differences in the answers given from theories of self-organisation in biology, economics or sociology on the question “What makes the whole more than the sum of its parts?” The fracture seems particularly salient at the borderline between formal-mathematical sciences like natural sciences including economy and other social sciences like sociology, for instance in the understanding and conceptualisation of “chaos” or “complexity”. Sometimes it creates the impression that originally well defined concepts from mathematics and natural science are metaphorically used in social sciences. This is a further reason why this paper concentrates on conceptualisations of self-organisation from natural sciences. The fifth part integrates the arguments from a system-theoretic point of view given in the three previous sections with respect to collective and corporative actors. Due to his prominence all five sections sometimes deal with the sociological system theory by Niklas Luhmann, especially in those parts with rigorous and important differences between his conception and the view given in this text. Despite Luhmann’s undoubted prominence in sociology, the present text strives for a more analytical and formal understanding of social systems and tries to find a base for another methodological approach.