Duality and Macroscopic Fields
摘要
In this chapter we study consequences of duality in the study of hydrodynamic limits, and more generally macroscopic fluctuation fields. The approach is in the spirit of DeMasi and Presutti (Mathematical Methods for Hydrodynamic Limits. Springer, Berlin, 2006) where the macroscopic limits are studied via duality functions, but we focus on simple applications, mostly emphasizing what extra information one can obtain via duality, such as the deviation from local equilibrium. First we show that the scaling limit of a single dual particle determines the macrosopic equation for the particle density. Then, starting with independent walkers we study the time dependent variance of the density field via two dual particles and compare with the solution of the limiting Ornstein Uhlenbeck process. We show how the propagation of local equilibrium is related to the scaling behavior of an arbitrary number of dual particles. We then turn to the interacting case, where we essentially show the same results, using coupling with independent particles. Finally we consider higher order macrosopic fields, and provide a new application of orthogonal polynomial duality, namely a quantitative version of the Boltzmann-Gibbs principle.