  We formulate the Kohn-Sham density functional theory (KS-DFT) as a
statistical theory in which the electron density is deter-mined from an average
of correlated stochastic densities in a trace formula. The key idea is that it
is sufficient to converge the total energy per electron to within a predefined
statistical error in order to obtain reliable estimates of the electronic band
structure, the forces on nuclei, the density and its moments, etc. The
fluctuations in the total energy per electron are guaranteed to decay to zero
as the system size increases. This facilitates "self-averaging" which leads to
the first ever report of sublinear scaling KS-DFT electronic structure. The
approach sidesteps calculation of the density matrix and thus is insensitive to
its evasive sparseness, as demonstrated here for silicon nanocrystals. The
formalism is not only appealing in terms of its promise to far push the limits
of application of KS-DFT, but also represents a cognitive change in the way we
think of electronic structure calculations as this stochastic theory seamlessly
converges to the thermodynamic limit.
