The Hydrodynamic Limit of the Freezing Model

Abstract

In this thesis the hydrodynamic limit of
the Freezing Model is studied. The model consists of an integer line on which
particles can get frozen to different degrees, analogous to jumping to another integer
line, with certain rates and can get unfrozen with certain rates per frozen
layer. The main result of the thesis is a proof that the hydrodynamic limit for
the Freezing model converges to a system of PDE’s describing the particle
density for each layer, either the ground layer or a frozen one. Firstly, it is
proven that the position of a random walker in the Freezing Model,
appropriately scaled, converges to a so-called Switching Brownian Motion. This
together with duality is used in the rest of the proof. Secondly, it is proven
that the expectation of the empirical field densities of the layers converges
towards the solutions of the aforementioned PDE’s. Lastly, it is proven that
the variance of the empirical field density converges to 0. Under an additional
diffusive scaling of the system of PDE’s for the particle densities, a
condition for diffusive behaviour is set up involving the ratio of the freezing
and unfreezing rates. It is shown that the PDE’s collapse into the heat
equation if this condition is satisfied. Finally, the case where the condition
is not satisfied is investigated. The model is then no longer memoryless like a
Markov process and shows sub-diffusive behaviour. The rescaled position of a
single particle then no longer converges to Brownian motion, but to Brownian
motion on the time scale on which the particle occupies the ground layer. This
time scale is t β−1 with 1 < β < 2. This grows slower than t for large enough
t. Additionally, simulations in Python were built to show that the model
exhibits diffusive or nondiffusive behaviour depending on the jump rates of the
process. Before everything, some mathematical preliminaries about probability
theory, Markov theory, random walks and duality are given.