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Schulz–Zimm distribution

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Schulz–Zimm
Probability density function
Parameters (shape parameter)
Support
PDF
Mean
Variance

The Schulz–Zimm distribution is a special case of the gamma distribution. It is widely used to model the polydispersity of polymers. In this context it has been introduced in 1939 by Günter Victor Schulz[1] and in 1948 by Bruno H. Zimm.[2]

This distribution has only a shape parameter k, the scale being fixed at θ=1/k. Accordingly, the probability density function is

When applied to polymers, the variable x is the relative mass or chain length . Accordingly, the mass distribution is just a gamma distribution with scale parameter . This explains why the Schulz–Zimm distribution is unheard of outside its conventional application domain.

The distribution has mean 1 and variance 1/k. The polymer dispersity is .

For large k the Schulz–Zimm distribution approaches a Gaussian distribution. In algorithms where one needs to draw samples , the Schulz–Zimm distribution is to be preferred over a Gaussian because the latter requires an arbitrary cut-off to prevent negative x.

Schulz–Zimm distribution with k=100, and Gaussian distribution with same mean and variance

References

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  1. ^ G V Schulz (1939), Z. Phys. Chem. 43B, 25-46. - Eq (27a) with -ln(a), k+1 in place of our x,k.
  2. ^ B H Zimm (1948), J. Chem. Phys. 16, 1099. - Proposes a two-parameter variant of Eq (13) without derivation and without reference to Schulz or whomsoever. One of the two parameters can be eliminated by the requirement <n>=1.