About chaos and order in many element systemsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Systems that involve many elements, be it a gas of particles or a herd of animals, are ubiquitous in our day to day lives. Their investigation, however, is hindered by the complexity of such systems and the amount of (usually coupled) equations that are needed to be solved. The late 50’s has seen the birth of the so-called mean field limit approach as an attempt to circumvent some of the difficulties arising in treating such systems. Conceived by Kac as a way to give justification to the validity of the Boltzmann equation, the mean field limit approach attempts to find the behaviour of a limiting “average” element in a many element system and relies on two ingredients: an average model of the system (i.e. an evolution equation for the probability density of the ensemble), and an asymptotic correlation relation that expresses the emerging phenomena we expect to get as the number of elements goes to infinity. Mean field limits of average models, originally applied to particle models, have permeated to fields beyond mathematical physics in recent decades. Examples include models that pertain to biological, chemical, and even societal phenomena. However, to date we use only one asymptotic correlation relation – chaos, the idea that the elements become more and more independent. While suitable when considering particles in a certain cavity, this assumption doesn’t seem reasonable in models that pertain to biological and societal phenomena. In our talk we will introduce Kac’s particle model and the notions of chaos and mean field limits. We will discuss the problem of having chaos as the sole asymptotic correlation relation and define a new asymptotic relation of order. We show that this is the right relation for a recent animal based model suggested by Carlen, Degond, and Wennberg, and highlight the importance of appropriate scaling in its investigation.