Empirical PDF

Ideally, one would like to choose Np and M large enough that e is dominated by statistical error (X ), which can then be reduced through the use of multiple independent simulations. In any case, for fixed Np and M, the relative magnitudes of the errors will depend on the method used to estimate the mean fields from the notional-particle data. We will explore this in detail below after introducing the so-called empirical PDF. 

Note that, for a uniform grid, both Vp(.v) and A7P scale as Nv/M. Using these definitions, the empirical PDF is defined as... 

For the moment estimates, we have seen that the composition PDF, /, (delta functions (i.e., the empirical PDF in (6.210)). However, it should be intuitively apparent that this representation is unsatisfactory for understanding the behavior of fyiir) as a function of fj. In practice, the delta-function representation is replaced by a histogram using finite-sized bins in composition space (see Fig. 6.5). The histogram h, (k) for the /ctli cell in composition space is defined by... 

Because all notional particles are assumed to have equal weight,6 the empirical PDF for the / th grid cell is given by7... 

Note that the empirical PDF in (7.2) depends explicitly on the number of notional particles IV/. However, it also depends implicitly on the number of grid cells (M) through the particle transport algorithm described below. The statistical error associated with estimation of the mean composition (< />) using (7.4) will scale as ,2)/N/)1/2. In zones of the flow where the variance (e//2) is small (e.g., in streams with pure fluid (e/> 2) = 0), Ni can be small However, in zones where mixing and reactions are important, Ni = 100-500 is typically required to obtain particle-number-independent estimates. 

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