Scalar probability density function mixing models need not comply with the linearity and independence hypothesis

Abstract

In a mixture of scalar fields undergoing diffusive processes governed by Fick's law, the concentration at each point evolves linearly in the concentrations at all points and independently from the other concentrations, when one considers a finite differences integration of their evolution equations. However, these properties must not necessarily be enforced in probability density function models, since they are relaxed when conditional expected values are taken.