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From: Büttner, Markus (Markus.Buettner_at_uni-bayreuth.de)
Date: Wed Jan 13 2021 - 02:47:21 CST

I am referring to the Analysis of Ubiquitin in Non-equlibrium: http://www.ks.uiuc.edu/Training/Tutorials/namd/namd-tutorial-unix-html/node15.html, the section on Heat diffusion.

In the first box, the average temperature of the system is given by the infinite series, with summands of the form 1/n^2 * exp(-(n pi / R)^2 Dt).

Later on, when we're doing the curve fittingfor the parameter D, the formula in the tutorial is given by

y = 200 + 66.87*(exp(-0.0146*a0*x) +0.25*exp(-0.25*0.0146*a0*x)
+1/9*exp(-1/9*0.0146*a0*x) +1/16*exp(-1/16*0.0146*a0*x)
+1/25*exp(-1/25*0.0146*a0*x) +1/36*exp(-1/36*0.0146*a0*x)
+1/49*exp(-1/49*0.0146*a0*x) +1/64*exp(-1/64*0.0146*a0*x)
+1/81*exp(-1/81*0.0146*a0*x) +1/100*exp(-1/100*0.0146*a0*x)).

Since we're approximating the infinite series of the average temperature by the partial sum, shouldn't we have exp(4 * 0.146 * a0 * x) instead of exp(-0.25 * 0.0146 * a0 * x)?

If I'm fitting the curve with the "1/" removed from the exponential terms, I get a much better fit to the numerical data and a diffusion constant much closer to the diffusion constant of water.

Perhaps I'm missing something here, I would be happy if someone could clarify why there are 1/n^2 terms in the exponential terms.

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