  The entropic order parameters measure in a universal geometric way the
statistics of non-local operators responsible for generalized symmetries. In
this article, we compute entropic order parameters in weakly coupled gauge
theories. To perform this computation, the natural route of evaluating
expectation values of physical (smeared) non-local operators is prevented by
known difficulties in constructing suitable smeared Wilson loops. We circumvent
this problem by studying the smeared non-local class operators in the enlarged
non-gauge invariant Hilbert space. This provides a generic approach for smeared
operators in gauge theories and explicit formulas at weak coupling. In this
approach, the Wilson and 't Hooft loops are labeled by the full weight and
co-weight lattices respectively. We study generic Lie groups and discuss
couplings with matter fields. Smeared magnetic operators, as opposed to the
usual infinitely thin ones, have expectation values that approach one at weak
coupling. The corresponding entropic order parameter saturates to its maximum
topological value, except for an exponentially small correction, which we
compute. On the other hand, smeared 't Hooft loops and their entropic disorder
parameter are exponentially small. We verify that both behaviors match the
certainty relation for the relative entropies. In particular, we find upper and
lower bounds (that differ by a factor of 2) for the exact coefficient of the
linear perimeter law for thin loops at weak coupling. This coefficient is
unphysical/non-universal for line operators. We end with some comments
regarding the RG flows of entropic parameters through perturbative beta
functions.
