Technical note: General formulation for the distribution problem – prognostic assumed probability density function (PDF) approach based on the maximum-entropy principle and the Liouville equation

Abstract. A general formulation for the distribution problem is presented, which is applicable to frequency distributions of subgrid-scale variables and hydrometeor size distributions, as well as to probability distributions characterizing data uncertainties. The general formulation is presented based upon two well-known basic principles: the maximum-entropy principle and the Liouville equation. The maximum-entropy principle defines the most likely general distribution form if necessary constraints are specified. This paper proposes to specify these constraints as the output variables to be used in a host model. Once a general distribution form is defined, the problem of the temporal evolution of the distribution reduces to that of predicting a small number of parameters characterizing it. This paper derives prognostic equations for these parameters from the Liouville equation. The developed formulation, which is applicable to a wide range of atmospheric modeling problems, is specifically applied to the condensation growth of cloud droplets as a demonstration.