  We numerically investigate the distribution of Drude weights $D$ of many-body
states in disordered one-dimensional interacting electron systems across the
transition to a many-body localized phase. Drude weights are proportional to
the spectral curvatures induced by magnetic fluxes in mesoscopic rings. They
offer a method to relate the transition to the many-body localized phase to
transport properties. In the delocalized regime, we find that the Drude weight
distribution at a fixed disorder configuration agrees well with the
random-matrix-theory prediction $P(D) \propto (\gamma^2+D^2)^{-3/2}$, although
the distribution width $\gamma$ strongly fluctuates between disorder
realizations. A crossover is observed towards a distribution with different
large-$D$ asymptotics deep in the many-body localized phase, which however
differs from the commonly expected Cauchy distribution. We show that the
average distribution width $\langle \gamma\rangle $, rescaled by $L\Delta$,
$\Delta$ being the average level spacing in the middle of the spectrum and $L$
the systems size, is an efficient probe of the many-body localization
transition, as it increases/vanishes exponentially in the delocalized/localized
phase.
