Knot Invariants and Cellular Automata

摘要

The goal of this project was to build on an understanding of the connectionsbetween knot invariants, exactly solvable statistical mechanics models and discrete dynamical systems toward an answer to the question of how early and robust thermodynamic behavior appears in lattice gas automata. Preliminary work focussed on developing an understanding of state models in knot theory and their relation to statistical mechanics models. Rather than considering the complicated and somewhat more difficult case of lattice gas hydrodynamics the researcher began with a simpler model: a reversible cellular automaton in one dimension. This particular automaton has an additive conserved quantity which Takesue has used to construct a one dimensional statistical mechanics model. He observes in simulations that the cellular automaton displays thermodynamic behavior which is well described by this one dimensional statistical mechanics model. In previous works it was shown that there is a naturally associated two dimensional statistical mechanics model as well, then examined the properties of this model, and suggested how these properties may relate to the behavior of the dynamical cellular automaton.