16:00
We are surrounded by systems that involve many elements and the interactions between them: the air we breathe, the galaxies we watch, herds of animals roaming the African planes and even us – trying to decide on whom to vote for.
As common as such systems are, their mathematical investigation is far from simple. Motivated by the realisation that in most cases we are not truly interested in the individual behaviour of each and every element of the system but in the average behaviour of the ensemble and its elements, a new approach emerged in the late 1950s - the so-called mean field limits approach. The idea behind this approach is fairly intuitive: most systems we encounter in real life have some underlying pattern – a correlation relation between its elements. Giving a mathematical interpretation to a given phenomenon and its emerging pattern with an appropriate master/Liouville equation, together with such correlation relation, and taking into account the large number of elements in the system results in a limit equation that describes the behaviour of an average limit element of the system. With such equation, one hopes, we could try and understand better the original ensemble.
In our talk we will give the background to the formation of the ideas governing the mean field limit approach and focus on one of the original models that motivated the birth of the field – Kac’s particle system. We intend to introduce Kac’s model and its associated (asymptotic) correlation relation, chaos, and explore attempts to infer information from it to its mean field limit – The Boltzmann-Kac equation.