Flory–Schulz distribution

Flory–Schulz distribution

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

The Flory–Schulz distribution is a mathematical function named after Paul Flory and G. V. Schulz that describes the relative ratios of polymers of different length after a polymerization process, based on their relative probabilities of occurrence. The probability mass function (pmf) can take the form of:

$f_a(k) = a^2 k (1-a)^{k-1}$

In this equation, k is a variable characterizing the chain length (e.g. number average molecular weight, degree of polymerization), and a is an empirically-determined constant.

The form of this distribution implies is that shorter polymers are favored over longer ones. Apart from polymerization processes, this distribution is also relevant to the Fischer–Tropsch process that is conceptually related, 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:

$\left\{\begin{array}{l} (a-1) (k+1) f(k)+k f(k+1)=0, \\[10pt] f(0)=0,f(1)=a^2 \end{array}\right\}$