I'm trying to fit the data for the thermal conductivity of Gallium Arsenide from [1]. Although there are theoretical models that explain this phenomenon, they are too complicated for my purposes, so I just want to find an appropriate fitting curve.
Note the log-log scale of the plot. The blue curve is defined as: $$f(x)=A\exp(B\ln(x)^2)\exp(C\ln x)$$ I got the above by assuming that it is a quadratic function on a log-log plot. It is not backed up by theory, but works well for me.
The red curve is: $$g(x) = (D\cdot x\exp(-E/x))^{-1}$$ which is justified by the theory: it shows the exponential behaviour at first and then, for large x, falls as 1/x. It is a very poor fit, to only two data points: the 2nd and 3rd from the right. If I add one more point to the fit though, the plot looks like this:
which, in x=150-200 range, is a worse fit than the first one. What could I do to "inflect" the blue curve so it comes closer to the 3rd data point from the right?
Here is the raw data:
T k
3.079 1.399
3.700 2.570
4.470 4.860
6.480 11.47
8.740 19.20
11.06 33.07
18.08 42.24
24.00 40.24
29.20 29.00
40.00 15.73
42.60 13.00
67.40 6.460
95.14 2.970
310.0 0.570
If I use the data for $24 \leq T \leq 310$ and the model $$y=a\frac{ e^{b/x}}{x}$$ you gave, a quick and dirty nonlinear curve fit leads to $$y=300.453\frac{ e^{28.5042/x}}{x}$$ which does not look too bad (at least to me). The adjusted $R^2=0.997$.
For sure, if you are concerned by high temperatures, it would be better to minimize the sum of squares of relative errors.