Sensitivity Analysis

Coarse-grained V representability

“…Standard tools of convex analysis will be used, without much comment apart from reminders of definitions. The reader for whom the material is unfamiliar can find it in a number of textbooks, for example, chapter I of Ekeland and Témam [9], or van Tiel [11]. Rockafellar [12] treats the finite-dimensional theory exhaustively.…”

“…This condition rearranges to the statement that the hyperplane ρ → F L (ρ 0 )+ v(ρ−ρ 0 ) dx lies below the graph of F L and touches it at ρ 0 . That is, v is a tangent functional to F L at ρ 0 and, since it is continuous, a subgradient [9,11,12] at ρ 0 . Recall that the set of all subgradients at…”

“…Combining this theorem with the cg-Hohenberg-Kohn Theorem gives Gâteaux differentiability [9,11,13] of the restriction of F L to X N .…”

Methods for calculating nonconcave entropies

“…The latter question naturally arises when studying long-range systems. If s(u) is nonconcave, then the Legendre transform of ϕ(β) cannot yield s(u) simply because Legendre transforms only yield concave functions [24,25]. By reading a bit about convex analysis, one learns in fact that the Legendre transform of ϕ(β), defined as…”

“…These mathematical results are illustrated in the next example using a simple spin model which will stay with us for the rest of the paper. For proofs of these results, the reader should consult the classical book of Rockafellar [24] or the more readable treatise of van Tiel [25].…”