Schulz–Zimm distribution

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]

Schulz–Zimm
Probability density function
Parameters	${\displaystyle k}$ (shape parameter)
Support	${\displaystyle x\in \mathbb {R} _{>0}}$
PDF	${\displaystyle {\frac {k^{k}x^{k-1}e^{-kx}}{\Gamma (k)}}}$
Mean	${\displaystyle 1}$
Variance	${\displaystyle {\frac {1}{k}}}$

This distribution has only a shape parameter k, the scale being fixed at θ=1/k. Accordingly, the probability density function is $${\displaystyle f(x)={\frac {k^{k}x^{k-1}e^{-kx}}{\Gamma (k)}}.}$$

When applied to polymers, the variable x is the relative mass or chain length ${\displaystyle x=M/M_{n}}$. Accordingly, the mass distribution ${\displaystyle f(M)}$ is just a gamma distribution with scale parameter ${\displaystyle \theta =M_{n}/k}$. 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 ${\displaystyle \langle x^{2}\rangle /\langle x\rangle =1+1/k}$.

For large k the Schulz–Zimm distribution approaches a Gaussian distribution. In algorithms where one needs to draw samples ${\displaystyle x\geq 0}$, 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