Let us define a non-dimensional scalar variable or progress variable  such that  at the unburnt mixture and  at the burnt gas side. For example, if  is the unburnt gas temperature and  is the burnt gas temperature, then the non-dimensional temperature can be defined as
The progress variable could be any scalar, i.e., we could have chosen the concentration of a reactant as a progress variable. Since the reaction sheet is infinitely thin, at any point in the flow field, we can find the value of  to be either unity or zero. The transition from zero to unity occurs instantaneously at the reaction sheet. Therefore, the probability density function for the progress variable is given by
where  and  are the probability of finding unburnt and burnt mixture, respectively and  is the Dirac delta function. By definition, the normalization condition leads to
It can be seen that the mean progress variable,
is nothing but the probability of finding burnt gas at location  and at the time . The density function is completely described by the mean progress variable, as we can write (suppressing the variables )
Assuming constant pressure and constant molecular weight, ideal gas law can be shown to reduce to
where  is the heat release parameter. Using the above relation, the mean density can be calculated as follows
The Favre averaging of the progress variable is given by
Combining the two expressions, we find
and hence
The density average is