  A systematic coarse graining of microscopic atomic displacements generates a
local elastic deformation tensor ${\mathsf D}$ as well as a positive definite
scalar $\chi$ measuring non-affinity, i.e. the extent to which the
displacements are not representable as affine deformations of a reference
crystal. We perform an exact calculation of the statistics of $\chi$ and
${\mathsf D}$ and their spatial correlations for solids at low temperatures,
within a harmonic approximation and in one and two dimensions. We obtain the
joint distribution $P(\chi, {\mathsf D})$ and the two point spatial correlation
functions for $\chi$ and ${\mathsf D}$. We show that non-affine and affine
deformations are coupled even in a harmonic solid, with a strength that depends
on the size of the coarse graining volume $\Omega$ and dimensionality. As a
corollary to our work, we identify the field, $h_{\chi}$, conjugate to $\chi$
and show that this field may be tuned to produce a transition to a state where
the ensemble average, $<\chi>$, and the correlation length of $\chi$ diverge.
Our work should be useful as a template for understanding non-affine
displacements in realistic systems with or without disorder and as a means for
developing computational tools for studying the effects of non-affine
displacements in melting, plastic flow and the glass transition.
