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There is an ambiguity. Although I did not understand your analysis of the problem completely, charge carriers certainly can run against the (averaged) electric force due to difference in available bands and other particle statistics effects. The gauge freedom is irrelevant. There are two cases for the “ubiquity”. First, these non-Maxwellian deviations ...


I think someone made a mistake on the indices of the summations. The first summation is the elastic deformation energy, using the applied stress tensor $\sigma_{ij}$ and resulting strains $\Delta\epsilon_{ij}$. This applies across the entire volume that your strain field covers. This applies to the volume $V$ which is area I. The $i$ and $j$ in this term ...

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