Particle systems with sources and sinks

Abstract

This project aims to describe crude models that work towards a mathematical model for transcription and translation. Transcription refers to the conversion of genetic code to a readable format for the cell, while translation refers to the conversion of this code into polypeptide chains using ribosomes. The main proposal for such a model is the totally asymmetric exclusion process which has already been applied to the case of translation [1]. In this report two simpler models and its mathematical abstractions are examined. These two are independent random walks on a grid Zd and the totally symmetric exclusion process on Zd where we allow a maximum of one particle per site. Since transcription and translation deal with particles being added and removed, we also consider the two models under which a source is placed in the origin for particles. Starting with independent random walkers, we find a principle called duality. This says that we can describe the independent random walks of a whole configuration by letting a single particle execute a random walk starting from x and evaluate the random walk at a later time t. Since precise quantitative behaviour is rather dificult to extract without further assumptions, we instead turn towards invariant distributions. It is proven that the Poisson measure is invariant on Zd, with and without a source in the origin for independent random walkers.