Entropy dissipation estimates for the linear Boltzmann operator by José Alfredo Cañizo (University of Birmingham)

We prove a linear inequality between the entropy and entropy dissipation functionals for Boltzmann  linear operator (with an equilibrium background). This provides a positive answer to the analogue of Cercignani’s conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels. For Maxwellian kernels, the proof of such an inequality is surprisingly simple and relies on a general estimate of the entropy of the gain part operator obtained in works by Matthes, Toscani and Villani. For more general kernels, the proof relies on some comparison principle. Finally, we also show how, in the grazing collision limit, our results allow to recover known logarithmic Sobolev inequalities.

A lattice model for gang territoriality by Alethea Barbaro (Case Western Reserve University)

In cities with a lot of gang activity, police and residents can often map out every gang’s territory, down to the block and sometimes even down to which side of a street belongs to which gang.  One activity of the younger members of a gang is to advertise the gang by “tagging”, and there is evidence that this may be one of the ways that a gang claims territory.  In this talk, we discuss a model for territorial formation which is motivated by gang graffiti.  We begin with a particle model on a lattice with agents belonging to two gangs.  These agents then put down graffiti and move preferentially away from sites marked by the other gang. For certain parameters, the agents gradually segregate into distinct territories.  We reframe the model as a spin model with a Hamiltonian which captures the same basic dynamics and prove that a phase transition occurs in two dimensions.

This is joint work with Lincoln Chayes and Maria Rita D’Orsogna.
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