 Flory-Schulz Distribution
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Flory%E2%80%93Schulz Distribution
Parameters 0 < a < 1 (real) k ? { 1, 2, 3, ... } $a^{2}k(1-a)^{k-1}$ $1-(1-a)^{k}(1+ak)$ ${\frac {2}{a}}-1$ ${\frac {W\left({\frac {(1-a)^{\frac {1}{a}}\log(1-a)}{2a}}\right)}{\log(1-a)}}-{\frac {1}{a}}$ $-{\frac {1}{\log(1-a)}}$ ${\frac {2-2a}{a^{2}}}$ ${\frac {2-a}{\sqrt {2-2a}}}$ ${\frac {(a-6)a+6}{2-2a}}$ ${\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}$ ${\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}$ ${\frac {a^{2}z}{((a-1)z+1)^{2}}}$ 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 (chemistry) of chains of length $k$ is:

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

In this equation, k is the numer 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, 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:

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