Flory–Schulz distribution

Flory–Schulz distribution
Probability mass function
Parameters	0 < a < 1 (real)
Support	k ∈ { 1, 2, 3, ... }
PMF	$a^{2}k(1-a)^{k-1}$ {\displaystyle a^{2}k(1-a)^{k-1}}
CDF	$1-(1-a)^{k}(1+ak)$ {\displaystyle 1-(1-a)^{k}(1+ak)}
Mean	${\frac {2}{a}}-1$ {\displaystyle {\frac {2}{a}}-1}
Median	${\frac {W\left({\frac {(1-a)^{\frac {1}{a}}\log(1-a)}{2a}}\right)}{\log(1-a)}}-{\frac {1}{a}}$ {\displaystyle {\frac {W\left({\frac {(1-a)^{\frac {1}{a}}\log(1-a)}{2a}}\right)}{\log(1-a)}}-{\frac {1}{a}}}
Mode	$-{\frac {1}{\log(1-a)}}$ {\displaystyle -{\frac {1}{\log(1-a)}}}
Variance	${\frac {2-2a}{a^{2}}}$ {\displaystyle {\frac {2-2a}{a^{2}}}}
Skewness	${\frac {2-a}{\sqrt {2-2a}}}$ {\displaystyle {\frac {2-a}{\sqrt {2-2a}}}}
Excess kurtosis	${\frac {(a-6)a+6}{2-2a}}$ {\displaystyle {\frac {(a-6)a+6}{2-2a}}}
MGF	${\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}$ {\displaystyle {\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}}
CF	${\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}$ {\displaystyle {\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}}
PGF	${\frac {a^{2}z}{((a-1)z+1)^{2}}}$ {\displaystyle {\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 of chains of length $k$ {\displaystyle k} is: $w_{a}(k)=a^{2}k(1-a)^{k-1}{\text{.}}$ {\displaystyle w_{a}(k)=a^{2}k(1-a)^{k-1}{\text{.}}}

In this equation, k is the number of monomers in the chain,^[1] and 0<a<1 is an empirically determined constant related to the fraction of unreacted monomer remaining.^[2]

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

References

[edit ]

^ Flory, Paul J. (October 1936). "Molecular Size Distribution in Linear Condensation Polymers". Journal of the American Chemical Society. 58 (10): 1877–1885. doi:10.1021/ja01301a016. ISSN 0002-7863.
^ IUPAC, Compendium of Chemical Terminology , 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "most probable distribution". doi:10.1351/goldbook.M04035

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Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
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supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda