Equilibrium and non-equilibrium molecular absorption: A study of the Ising Model and the infinite parking limit problem
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Randomness is present in so many everyday systems that we often forget its importance in both mathematical and chemical situations. Chemical reactions depend on random interactions and collisions, the movement of particles is often randomly determined, and randomness plays a role in the way in which diffusing particles interact with a solid surface. Looking specifically into this last situation, we know that there are many ways in which diffusing particles can interact with solids. Particles can diff use through either water or air, and at low concentrations, this process is well-modeled by random processes. When looking at this situation, there are two distinct types of molecular absorption to consider: equilibrium or nonequilibrium absorption. That is, when the diffusing particle comes into contact with the surface, it can either stick exactly where it lands or move around a little to come to a more stable equilibrium arrangement. These processes might look very similar on the macroscopic level, but on the microscopic level, they are studied using very different mathematical techniques. Sometimes the process is modeled by a simpler problem in order to use more rigorous mathematics to address things which are very complicated. Other times, it is more beneficial to solve a problem computationally with programming. These different approaches each have benefits and are both used in comprehensive studies of natural processes.To get a feel for these two different ways of studying this problem, we will examine two different models: an equilibrium model and a non-equilibrium model. One of the most common models used to study equilibrium systems is the Ising model. We start by thinking of the Ising model as a infinite system with boundary conditions, and then we consider the limit as the size of the grid goes to infinity. The value of the spin at each lattice site depends on two different things-the values of the neighboring lattice sites and any applied external magnetic fields. The contributions of each of these pieces depends on the system and is controlled by constants. It was originally proposed by Ising to model the spontaneous magnetization in ferromagnetic substances. The restriction on the spin states and the ordered pattern of the grid points allow this model to be analyzed mathematically, while still giving interesting information about the behavior of many different chemical systems. As the number of points in the grid goes to infinity, we use this model to look at phase transitions, switches between one defined state and another.