Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations

For describing the dislocation pattern formation, two different approaches are applied. In one of them the collective behavior of many individual dislocations is investigated by analytical or numerical methods, in the other one the dislocation system is described by continuous functions of the space coordinates. A method is proposed for establishing the link between the two length scales. Starting from the exact evaluation equation of the $N$-dislocation distribution function, a hierarchy of equations is derived for the one-, two-, three-, etc., particle distribution functions. Although the dislocations form a nonconservative system, the applied method is similar to the derivation of the so called BBGKY hierarchy which is frequently used in plasma physics. By assuming that the three-particle correlation function is negligible a closed set of equations can be obtained for the dislocation density and the two-particle correlation functions. The possible origin of appearance of dislocation patterns is also investigated.