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I have physical data and I wanted to use a logistic function to fit them. In this context, I came across the so-called Boltzmann-fit which fits nicely to the data but since I wonder where does it come from:

boltz1

boltz2

I struggle to separate (if even possible) that Boltzmann-fit from the Maxwell-Boltzmann distribution curve and the Boltzmann equation. I'm also not able to find that much about this fit which makes me wonder as Boltzmann contributed probably among few others some of the most crucial findings.

In short: Where does this "fit" come from? Is it the Boltzmann equation? Is it cconnected to the Maxwell-Boltzmann distribution? Is it something else?

edit: Some "official" information:

boltzmann

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Most physicists would call this fitting by a Fermi function, since Fermi function $$ f(E)=\frac{1}{1+e^{\beta(E-\mu)}}\approx_{\beta\rightarrow 0} e^{\beta(\mu-E)}, $$ which gives you Boltzmann distribution only in a high temperature limit (second equality). While it is not clear where the term Boltzmann-fit comes from, fitting by this curve is the essence of logistic regression (although sometimes hidden in mathematical details), as it approximates a probability distribution where the data can take one of two values.

Another place where it frequently arises is when solving ordinary differential equations with two stable states (again, often well beyond physics).

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  • $\begingroup$ Thank you! I've also wondered about the Fermi term as well.. But why/how are Fermi/Boltzmann functions subject to a probability distribution with two values? Isn't the Fermi function providing continuous values? $\endgroup$
    – Ben
    Dec 7, 2020 at 12:32
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    $\begingroup$ Fermi function is continuous, but it can be interpreted as the probability that the state of energy $E$ is filled or empty. $\endgroup$ Dec 7, 2020 at 12:36
  • $\begingroup$ Got it, thank you! jFYI, I added a plot from an international organisation where you can see that they named this fit Boltzmann. In case it matters somehow.. $\endgroup$
    – Ben
    Dec 7, 2020 at 12:58
  • $\begingroup$ It is quite possible that in some domains it is called this way. $\endgroup$ Dec 7, 2020 at 13:00
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    $\begingroup$ I gave the approximation only to note a possible link to the Boltzmann distribution in statistical physics. It has no relation to the logistic regression. $\endgroup$ Dec 8, 2020 at 9:26

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