Statistical theory of dislocation configurations in a random array of point obstacles

The stable configurations of a dislocation in an infinite random array of point obstacles are analyzed using the mathematical methods of statistical mechanics. The theory provides exact distribution functions of the forces on pinning points and of the link lengths between points on the line. The expected number of stable configurations is a function of the applied stress. This number drops to zero at the critical stress. Due to a degeneracy problem in the line count, the value of the flow stress cannot be determined rigorously, but we can give a good approximation that is very close to the empirical value