Fig. 1

An overview of the formation of the line bundle ensemble by a generalized canonical approach. In (a), note that a single parent microstate \(\mathcal {L}_{0}\) is used to generate the mean-field density ρ(r) by convolution with some weight function w_L, i.e. ρ(r):=(ϱ∗w_L)(r). b shows a toy model of the induced probability measure on the space of dislocation configurations. The ensemble average of the discrete density (i.e. integration against this probability measure in the space of line configurations), is constrained to equal the mean field density. By equating the ensemble average to the field generated by means of the convolution operation, we can induce a probability measure by means of some maximum entropy argument. One property of the probability measure, however, is guaranteed by the line bundle constraint: the class of states \(\{\mathcal {L}\perp \boldsymbol {\rho }\}\) (for which the discrete density is of the opposite sign as ρ), is necessarily of null probability (see Appendix B)