Flory–Schulz distribution

The Flory–Schulz distribution is a discrete probability distribution named after Paul Flory and Günter Victor Schulz that describes the relative ratios of polymers of different length that occur in an ideal step-growth polymerization process. The probability mass function (pmf) for the mass fraction of chains of length $k$ is: $$w_{a}(k)=a^{2}k(1-a)^{k-1}{\text{.}}$$

Flory–Schulz distribution

Probability mass function
Parameters	0 < a < 1 (real)
Support	k ∈ { 1, 2, 3, ... }
PMF	$a^{2}k(1-a)^{k-1}$
CDF	$1-(1-a)^{k}(1+ak)$
Mean	${\frac {2}{a}}-1$
Median	${\frac {W\left({\frac {(1-a)^{\frac {1}{a}}\log(1-a)}{2a}}\right)}{\log(1-a)}}-{\frac {1}{a}}$
Mode	$-{\frac {1}{\log(1-a)}}$
Variance	${\frac {2-2a}{a^{2}}}$
Skewness	${\frac {2-a}{\sqrt {2-2a}}}$
Excess kurtosis	${\frac {(a-6)a+6}{2-2a}}$
MGF	${\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}$
CF	${\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}$
PGF	${\frac {a^{2}z}{((a-1)z+1)^{2}}}$

In this equation, k is the number of monomers in the chain, and 0<a<1 is an empirically determined constant related to the fraction of unreacted monomer remaining.

The form of this distribution implies is that shorter polymers are favored over longer ones -the chain length is geometrically distributed. Apart from polymerization processes, this distribution is also relevant to the Fischer–Tropsch process that is conceptually related, where it's known as Anderson-Schulz-Flory (ASF) distribution, in that lighter hydrocarbons are converted to heavier hydrocarbons that are desirable as a liquid fuel.

The pmf of this distribution is a solution of the following equation: $$\left\{{\begin{array}{l}(a-1)(k+1)w_{a}(k)+kw_{a}(k+1)=0{\text{,}}\\[10pt]w_{a}(0)=0{\text{,}}w_{a}(1)=a^{2}{\text{.}}\end{array}}\right\}$$