In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.

Is there any interpretation of these operators in terms of Grothendieck (or other) style geometric ideas such as thickenings, formal schemes, lambda rings, or crystalline cohomology?

A related and possibly equivalent question is whether there is any precise sense in which the "extra" material in divided power algebras (compared to polynomial or power-series rings) is dual to the "extra" differential operators in positive characteristic(s).