The effects of local perturbations on conservative particle systems

Abstract

In this thesis, research was done in the area of interacting particle systems. Especially, the symmetric exclusion process with local perturbations was investigated. These perturbations, were in the form of sinks and sources, which add or take away particles at certain rates. Moreover, simulations were done for the asymmetric exclusion process. This process took place on a ring, with the addition of a source. For the symmetric exclusion process with sinks and sources, certain expressions were proven for the expected occupancy of a site. For the simulations, the main goal was to find out how the jumping rates and starting density, influenced the time to get to the fully occupied state, at different source rates. The first proof was for a source at an arbitrary site. From this expression, one could see that if the rates were recurrent, the system converged to the fully occupied state. If, however, the rates were transient, the system had a limiting density. Thereafter, it was shown that if a sink and source are placed in the same arbitrary site, the system always converged to a density, which under the Bernoulli measure, was not equal to the fully occupied state. The fact that the sink and source were in the same site, was an indispensable condition. Subsequently, the case of countably many sources was investigated. For which it was also shown that recurrent rates always yield a fully occupied state, as time tends to infinity. Whereas transient rates, once again, caused a limiting density. Moreover, the special case of a simple random walk in three dimensions or higher was investigated. If, for distances far away from the origin, the source rates could be bounded above by a certain function, then the system would not converge to the fully occupied state. Also, another proof showed that a recurrent set of source sites would always let the process converge to a fully occupied state. Lastly, similar conditions for one time dependent source at the origin were proven. Namely, it was shown for recurrent rates, that if the source dies out quick enough, as t tends to infinity, the system did not converge to the fully occupied state. For the simulations, it was found that the expected time to fill up was influenced by the starting density, both at small and large source rates, in a linear way. This conclusion could not be drawn for the rate difference: p-q. Due to the large error of the fit, a linear relation was not clearly seen.  Finally, some concluding remarks were made on both aspects of the thesis, along with some recommendations for future projects related to this topic.