This is a followup to this post.
In this post, I thought I'd illustrate how a particular distribution looks like when seen through different lenses: N(M), W(M), and w(log M). For concreteness, let us pick a particular generalized exponential function (GEX), which can model a wide variety of empirical distributions as W(M):
In this post, I thought I'd illustrate how a particular distribution looks like when seen through different lenses: N(M), W(M), and w(log M). For concreteness, let us pick a particular generalized exponential function (GEX), which can model a wide variety of empirical distributions as W(M):
\[ W(M) = \frac{m t^{\frac{k+1}{m}}}{\Gamma \left(\frac{k+1}{m}\right)} M^k e^{-t M^m}\]
with m = 0.5, k = 1.5, and t = 0.005.
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| Blue Curve is W(M) and Red Curve is N(M) |
I can get N(M) by considering N(M) = W(M)/M and figuring out the normalization constant. Here, it turns out to be:
\[ N(M) = \frac{m t^{k/m}}{\Gamma \left(\frac{k}{m}\right)} M^{k-1}e^{-t M^m} \]
It is biased towards smaller molecular weights. In this particular example, the number-averaged and weight-averaged molecular weights turn out to be 480K and 1.2M, respectively. Also note that the numbers on the y-axis are "smallish". This is forced by the normalization constraint,
\[ \int_{0}^{\infty} N(M) dM = \int_{0}^{\infty} W(M) dM = 1,\]
since the numbers on the x-axis are largish.
We take the W(M) curve and multiply it by 2.303 * M to get w(log M). On the x-axis we plot log M, as:
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