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Séminaire du LJLL
Amit Einav (Université de Durham)
Systems that involve many elements are ubiquitous in our day to day lives. Their investigation, however, is usually hindered by their complexity. The late 50’s have 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 many element systems and relies on two ingredients: an average model of the system and an asymptotic correlation relation that expresses the emerging phenomena we expect to get. Mean field limits of average models have permeated to settings beyond their original conception 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 physical situations, this assumption doesn’t seem reasonable in models that pertain to biological and societal phenomena. In our talk we will give an overview to mean field limits and the notion of chaos, and proceed to 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.