From solution to mother equation

[haruspex]

Yes.

One thing this thread has shown me is how hard it can be to discriminate between an exponential and a power law just based on the data.
I went back to your data in post #43 and found very good fits for both types. So I plotted the one against the other over the range of the x values, 460 to 1026. I.e. y_exp against y_pow where:
$y_{exp}=e^{-\frac x{50}}$ and $y_{pow}=x^{-10}$.
The graph is an excellent fit to $y_{pow}=2.28*10^{-23}y_{exp}+1.01*10^{-29}$. I found this very hard to believe, but I cannot see any errors in my work.
Of course, it could not remain so straight as x approaches zero since y_pow would shoot off to infinity.

I strongly suspect a general principle here, but I haven't figured out the details. Probably something to do with which terms in the expansion of e^-λx dominate for a given x.From this I suggest that you should treat all sets of data as being of the same type, either power or exp, and see which fits better overall.
I have to agree that for the data in post #57 power law looks more persuasive than exponential, so maybe take all as being power.Edit:
Tried other powers, an x range of 500 to 1000, and observed a general behaviour that $x^{-n}$ v. $e^{-knx}$ is a pretty straight line where k=0.00128 and n ranges from 4 to 12. But if I change the range of x (still over an octave) it breaks down quite fast; probably need k to be a function of a, where the x range is a to 2a.

 

[naviakam]

One thing this thread has shown me is how hard it can be to discriminate between an exponential and a power law just based on the data.
I went back to your data in post #43 and found very good fits for both types. So I plotted the one against the other over the range of the x values, 460 to 1026. I.e. y_exp against y_pow where:
$y_{exp}=e^{-\frac x{50}}$ and $y_{pow}=x^{-10}$.
The graph is an excellent fit to $y_{pow}=2.28*10^{-23}y_{exp}+1.01*10^{-29}$. I found this very hard to believe, but I cannot see any errors in my work.
Of course, it could not remain so straight as x approaches zero since y_pow would shoot off to infinity.
View attachment 276217
I strongly suspect a general principle here, but I haven't figured out the details. Probably something to do with which terms in the expansion of e^-λx dominate for a given x.From this I suggest that you should treat all sets of data as being of the same type, either power or exp, and see which fits better overall.
I have to agree that for the data in post #57 power law looks more persuasive than exponential, so maybe take all as being power.Edit:
Tried other powers, an x range of 500 to 1000, and observed a general behaviour that $x^{-n}$ v. $e^{-knx}$ is a pretty straight line where k=0.00128 and n ranges from 4 to 12. But if I change the range of x (still over an octave) it breaks down quite fast; probably need k to be a function of a, where the x range is a to 2a.

There are many other data sets best fitted with either power law or exponential, therefore $Y=P(p)Y1(E)+(1−P(p))Y2(E)$ still holds.
Considering the potential in post #60, how mathematically Y(V) and E(V) are formulated?

 

[haruspex]

There are many other data sets that are best fitted with either power law or exponential, therefore $Y=P(p)Y1(E)+(1−P(p))Y2(E)$ still holds.
Considering the potential in post #60, how mathematically Y(V) and E(V) are formulated?

Please post a dataset which clearly is closer to exponential than to a power law.

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$ doesn't work because you have various powers and probably various coefficients in the exponentials.

 

[naviakam]

Please post a dataset which clearly is closer to exponential than to a power law.

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$ doesn't work because you have various powers and probably various coefficients in the exponentials.

 

[naviakam]

Please post a dataset which clearly is closer to exponential than to a power law.

$Y=P(p)Y1(E)+(1−P(p))Y2(E)$ doesn't work because you have various powers and probably various coefficients in the exponentials.

This is another data fits well with Exp:
1205.36 1.69E+10
1135.78 1.72E+10
1072.51 1.81E+10
1014.83 1.79E+10
962.07 1.81E+10
913.71 1.79E+10
869.25 1.91E+10
828.31 2.04E+10
790.51 2.09E+10
755.54 2.32E+10
723.14 2.73E+10
693.06 2.95E+10
665.08 3.73E+10
639.02 4.24E+10
614.71 5.28E+10
592.01 6.60E+10
570.77 8.40E+10
550.88 1.11E+11
532.22 1.47E+11
514.71 1.86E+11
498.25 2.40E+11
482.77 3.15E+11
468.19 4.00E+11
454.45 5.07E+11

 

[haruspex]

This is another data fits well with Exp:
1205.36 1.69E+10
1135.78 1.72E+10
1072.51 1.81E+10
1014.83 1.79E+10
962.07 1.81E+10
913.71 1.79E+10
869.25 1.91E+10
828.31 2.04E+10
790.51 2.09E+10
755.54 2.32E+10
723.14 2.73E+10
693.06 2.95E+10
665.08 3.73E+10
639.02 4.24E+10
614.71 5.28E+10
592.01 6.60E+10
570.77 8.40E+10
550.88 1.11E+11
532.22 1.47E+11
514.71 1.86E+11
498.25 2.40E+11
482.77 3.15E+11
468.19 4.00E+11
454.45 5.07E+11

Again, I get a tolerable fit with either exp or power:

I slightly prefer the power fit; it seems to have less of a twist near the origin.

One of the tricky parts about fitting curves to data is that the best fit depends on what you choose to plot against what. E.g. the best least squares fit for $y=Ae^{kx}$ will be different from the best for $\ln(y)=\ln(A)+kx$. The latter model will put more emphasis on the match for the smaller values of y.
To optimise this, it helps if you have some idea how the accuracy of the measurements varies across the range.

Now, this is not an area I know much about, but I took a look at the black body spectrum formula. I note that this mixes power and exponential, so maybe something similar is going on here.

 

[naviakam]

Again, I get a tolerable fit with either exp or power:
View attachment 276269
I slightly prefer the power fit; it seems to have less of a twist near the origin.

One of the tricky parts about fitting curves to data is that the best fit depends on what you choose to plot against what. E.g. the best least squares fit for $y=Ae^{kx}$ will be different from the best for $\ln(y)=\ln(A)+kx$. The latter model will put more emphasis on the match for the smaller values of y.
To optimise this, it helps if you have some idea how the accuracy of the measurements varies across the range.

Now, this is not an area I know much about, but I took a look at the black body spectrum formula. I note that this mixes power and exponential, so maybe something similar is going on here.

Yes black body spectrum consists of a power term multiplied by exponential. It is intensity against wavelength compared with our case which is intensity against energy. In black body the total power is proportional to $T^4$ where T is temperature, and our potential is proportional to $I^4$ where I is current.

Then, how black body spectrum formula and our spectrum could be related?

Or, back to my first question, how our spectrum could be connected to the potential mathematically?

 

[haruspex]

Then, how black body spectrum formula and our spectrum could be related?

Or, back to my first question, how our spectrum could be connected to the potential mathematically?

As I have indicated, I do not know enough about the physics involved to answer questions like that.
Presumably you do have some such understanding, which is why I asked you to show me an example of how some given potential would lead to a particular spectrum. If you cannot do that we're both in the dark.

What I can do is analyse your data and suggest ways to unify the results into a consistent relationship.
How many data sets do you have? If you care to post more I will study them.

 

[naviakam]

As I have indicated, I do not know enough about the physics involved to answer questions like that.
Presumably you do have some such understanding, which is why I asked you to show me an example of how some given potential would lead to a particular spectrum. If you cannot do that we're both in the dark.

What I can do is analyse your data and suggest ways to unify the results into a consistent relationship.
How many data sets do you have? If you care to post more I will study them.

As requested in post #62:
considering intensity $Y=P(p)Y1(E)+(1−P(p))Y2(E)$ (where $Y1$ and $Y2$ are the power and exp functions respectively) and potential $V=YE/It$, how mathematically $Y(V)$ and $E(V)$ are formulated/written?