  Deriving the time evolution of a distribution of probability (or a
probability density matrix) is a problem encountered frequently in a variety of
situations: for physical time, it could be a kinetic reaction study, while
identifying time with the number of computational steps gives a typical picture
of algorithms routinely used in quantum impurity solvers, density functional
theory, etc. Using a truncation scheme for the expansion of the exact quantity
is necessary due to constraints of the numerical implementation. However, this
leads in turn to serious complications such as the Fermion Sign Problem
(essentially, density or weights will become negative). By integrating angular
degrees of freedom and reducing the dynamics to the radial component, the time
evolution is reformulated as a nonlinear integral transform of the distribution
function. A canonical decomposition into orthogonal polynomials leads back to
the original sign problem, but using a characteristic-function representation
allows to extract the asymptotic behavior, and gives an exact large-time limit,
for many initial conditions, with guaranteed positivity.
