  This paper focuses on the connections between four stochastic and
deterministic models for the motion of straight screw dislocations. Starting
from a description of screw dislocation motion as interacting random walks on a
lattice, we prove explicit estimates of the distance between solutions of this
model, an SDE system for the dislocation positions, and two deterministic
mean-field models describing the dislocation density. The proof of these
estimates uses a collection of various techniques in analysis and probability
theory, including a novel approach to establish propagation-of-chaos on a
spatially discrete model. The estimates are non-asymptotic and explicit in
terms of four parameters: the lattice spacing, the number of dislocations, the
dislocation core size, and the temperature. This work is a first step in
exploring this parameter space with the ultimate aim to connect and quantify
the relationships between the many different dislocation models present in the
literature.
