From solution to mother equation

[naviakam]

Hi
I have got two solutions to unknown equations:
1. power equation in the form y=k x(-n) where k is coefficient and n is the power
1. exponential equation of the form: y=y0+k exp(-(x-x0)/t)

Is it possible to find the original equation/s leading to these solutions (kind a reverse engineering!)?

Best

 

[fresh_42]

Do you look for one solution for both equations with different initial conditions, or are these two different problems? If so, why don't you differentiate to find the "mother equation"?

 

[naviakam]

Do you look for one solution for both equations with different initial conditions, or are these two different problems? If so, why don't you differentiate to find the "mother equation"?

There must be one solution to both. By the way, how to differentiate in order to find the mother equation?

 

[fresh_42]

There must be one solution to both. By the way, how to differentiate in order to find the mother equation?

Your solutions can be viewed as flow through a vector field. Since one is exponential and the other one polynomial, they correspond to two different initial values, or there is a point where the behavior substantially changes, i.e. a composite function.

Is your $x$ the variable or is it itself a function of time $t$: $x=x(t)$?

 

[naviakam]

It's a variable, refers to the particle energy.

 

[martinbn]

Differentiate and substitute, for example the first one after taking derivatives is $y'=-knx^{-n-1}$. Multiply by $x$, you get $xy'=-knx^{-n}$, use the solution to find $xy'=-ny$

 

[fresh_42]

Then $y'=-knx^{-n-1}$ and $y'\cdot x +ny=0$ in the first case, and similar for the second. Could it be, that the first equation is the asymptotic behavior of the second? And is $t$ really a constant?

 

[martinbn]

Could it be, that the first equation is the asymptotic behavior of the second? And is $t$ really a constant?

One is polynomial, the other exponential!

 

[fresh_42]

One is polynomial, the other exponential!

I know.

Your solutions can be viewed as flow through a vector field. Since one is exponential and the other one polynomial, they correspond to two different initial values, or there is a point where the behavior substantially changes, i.e. a composite function.

I thought of a collection of exp curves getting flatter and flatter with $y=0$ as asymptote. Something similar to this: $y=\exp(-tx)$ with $x\to 0$ or $x\to \infty$.

 

[naviakam]

Thanks.

Ions are accelerated in a potential, the graph for number of ions versus ion energy plotted in origin and fitted. There are two plots: one is fitted with the power and the other with exponential decay in the form mentioned above (y is the number and x is the energy). We've been asked to guess the original function leading to these spectra. Amazingly, both equations seem originating from a single function!

 

[pasmith]

What form of ODE are you looking for? You can easily take ODEs satisfied by kx^n and y_0 + ke^{x/t} respectively in the form f(y,y',x) = 0 and mulitply them together to get \left(y'' - \frac{y'}{t}\right)\left(y' + \frac{ny}{x}\right) = 0 but that may not be the form of ODE you are looking for.

 

[naviakam]

What form of ODE are you looking for? You can easily take ODEs satisfied by kx^n and y_0 + ke^{x/t} respectively in the form f(y,y',x) = 0 and mulitply them together to get \left(y'' - \frac{y'}{t}\right)\left(y' + \frac{ny}{x}\right) = 0 but that may not be the form of ODE you are looking for.

1. The second term is fine but how the first term was derived?

Please show that.

2. As mentioned, capacitor discharge in a gas accelerates ions and ion spectra fitted in Origin software. The original equation must be obtained (from power and exp fits) and related to the accelerating potential.

 

[martinbn]

1. The second term is fine but how the first term was derived?
View attachment 275367
Please show that.

2. As mentioned, capacitor discharge in a gas accelerates ions and ion spectra fitted in Origin software. The original equation must be obtained (from power and exp fits) and related to the accelerating potential.

Is this a homework question?

 

[naviakam]

Is this a homework question?

Lab

 

[berkeman]

Is this a homework question?

Lab

Then it should have been posted in the Homework section of PF. I will move your thread there now. Also keep in mind that for schoolwork problems/projects, we require that you do the bulk of the work. We do not do your homework for you here.

 

[naviakam]

Then it should have been posted in the Homework section of PF. I will move your thread there now. Also keep in mind that for schoolwork problems/projects, we require that you do the bulk of the work. We do not do your homework for you here.

How this equation:

is related to the capacitor potential?

 

[haruspex]

1. The second term is fine but how the first term was derived?
View attachment 275367
Please show that.

2. As mentioned, capacitor discharge in a gas accelerates ions and ion spectra fitted in Origin software. The original equation must be obtained (from power and exp fits) and related to the accelerating potential.

It might help if you were to explain what y, x and t represent in that set-up.
In particular, is it perhaps y=y(x,t), or y=y(x(t)), or t is constant, or...?

 

[naviakam]

It might help if you were to explain what y, x and t represent in that set-up.
In particular, is it perhaps y=y(x,t), or y=y(x(t)), or t is constant, or...?

y is the number of ions, x is the energy. but t: I just fitted the curve and exponential decay fit was the best. t was given by the fit in the origin software.

 

[naviakam]

Differentiate and substitute, for example the first one after taking derivatives is $y'=-knx^{-n-1}$. Multiply by $x$, you get $xy'=-knx^{-n}$, use the solution to find $xy'=-ny$

how the equation containing y0 is differentiated?

 

[naviakam]

What form of ODE are you looking for? You can easily take ODEs satisfied by kx^n and y_0 + ke^{x/t} respectively in the form f(y,y',x) = 0 and mulitply them together to get \left(y'' - \frac{y'}{t}\right)\left(y' + \frac{ny}{x}\right) = 0 but that may not be the form of ODE you are looking for.

first term is :
y"+(1/t)Y'=0
right?

 

[haruspex]

y is the number of ions, x is the energy. but t: I just fitted the curve and exponential decay fit was the best. t was given by the fit in the origin software.

So t is an arbitrary constant? Then why doesn't the equation reduce to $y=y_0+k'e^{-\frac xt}$, where $k'=ke^\frac{x_0}t$?

 

[naviakam]

So t is an arbitrary constant? Then why doesn't the equation reduce to $y=y_0+k'e^{-\frac xt}$, where $k'=ke^\frac{x_0}t$?

then what is the next step?

 

[haruspex]

then what is the next step?

I see nothing better than has already been suggested, @pasmith's post #11, with your sign correction in post #20. But it does look very artificial and unconvincing.

I'm not sure I understand your situation. If these equations come from curve fitting, how can you be sure they really are of the form you specify?
What happens if you insist on one form, either power or exponential, and fit both sets of data to that?

I am also surprised that whatever software you used to produce the exponential fit generated three constants when two would have sufficed.

 

[naviakam]

I see nothing better than has already been suggested, @pasmith's post #11, with your sign correction in post #20. But it does look very artificial and unconvincing.

I'm not sure I understand your situation. If these equations come from curve fitting, how can you be sure they really are of the form you specify?
What happens if you insist on one form, either power or exponential, and fit both sets of data to that?

I am also surprised that whatever software you used to produce the exponential fit generated three constants when two would have sufficed.

Then regarding to post #11: how to correlate this equation with the potential? I want to find type of potential that generated this equation!

 

[naviakam]

in other word possible to draw ion spectra (of the form above) from potential?

 

[pasmith]

I assume the physics will tell you which functions can be potential functions, and what parameters they have which you can vary to fit your data. You should not be trying to fit your data to exponential or power laws unless the physics tells you to expect those (because a potential of a particular form will give you a curve which looks like that).

You say that this is lab work. Does your assignment include instructions on how to go about analysing your results to obtain the potential? Has this been covered in lectures? What does your textbook have to say on the matter?

 

[naviakam]

this is the lab for a plasma course. we discharge capacitor in a gas and measure the ion spectrum. then we have been asked to correlate the spectra with governing potential!

 

[naviakam]

I assume the physics will tell you which functions can be potential functions, and what parameters they have which you can vary to fit your data. You should not be trying to fit your data to exponential or power laws unless the physics tells you to expect those (because a potential of a particular form will give you a curve which looks like that).

You say that this is lab work. Does your assignment include instructions on how to go about analysing your results to obtain the potential? Has this been covered in lectures? What does your textbook have to say on the matter?

is it possible at all to correlate this equation with the potential function?

 

[Fred Wright]

First, there is no "mother equation". I am going to make some assumptions about your experiment.
1. You have a Langmoir probe inserted in the plasma connected to a power supply to bias it to negative/positive voltages.
2. You obtain a current-voltage plot.
3. The plot has three regions:
Region I (Townsend discharge region):
In this region the probe is negatively biased; electrons are rejected from the probe and ions are collected. The ion's current that passes area $A_p$ in the plasma is linked to both ion density and velocity,
$$
J_p =\frac{I_p}{A_p}=\frac{n_iev_{th}}{4}=\frac{n_ie}{4}\left (\frac{2K_B T}{m_i} \right )^{\frac{1}{2}}
$$
$J_p$ is ion current density, $I_p$ is plasma current, $n_i$ is ion density, $v_{th}$ is thermal velocity, $e$ is electric charge, $K_B$ is Boltzmann's constant, $m_i$ is ion mass, and $T$ is absolute temperature. Does this equation's form look familiar to you?

Region II (glow discharge region):
Rendering the value of V less negative will compel the probe to collect both ions and electrons (high thermal energy). As $V$ is made more positive, the collected ions and electrons are expected to cancel (or balance) each other out. This probe-plasma potential, $V_f$ is called the floating potential. In the event of thermalized plasma, this voltage is $\frac{1}{K_BT}$ (expressed in eV). When $V$ is increased $V_f$ electron current increases.The resulting current is exponentially related to $V$. In region II, the electron current are defined by:
$$
J_e=\frac{I_e}{A_p}=\frac{n_eev_{th}}{4}e^{\frac{-eV}{K_BT_e}}
$$
$T_e$ is the electron temperature. Does this equation look familiar to you?

Region III
In this region the plasma current saturates at a plasma space potential value $\psi_p$, caused by the space charge limitation during current collection.

 

[naviakam]

First, there is no "mother equation". I am going to make some assumptions about your experiment.
1. You have a Langmoir probe inserted in the plasma connected to a power supply to bias it to negative/positive voltages.
2. You obtain a current-voltage plot.
3. The plot has three regions:
Region I (Townsend discharge region):
In this region the probe is negatively biased; electrons are rejected from the probe and ions are collected. The ion's current that passes area $A_p$ in the plasma is linked to both ion density and velocity,
$$
J_p =\frac{I_p}{A_p}=\frac{n_iev_{th}}{4}=\frac{n_ie}{4}\left (\frac{2K_B T}{m_i} \right )^{\frac{1}{2}}
$$
$J_p$ is ion current density, $I_p$ is plasma current, $n_i$ is ion density, $v_{th}$ is thermal velocity, $e$ is electric charge, $K_B$ is Boltzmann's constant, $m_i$ is ion mass, and $T$ is absolute temperature. Does this equation's form look familiar to you?

Region II (glow discharge region):
Rendering the value of V less negative will compel the probe to collect both ions and electrons (high thermal energy). As $V$ is made more positive, the collected ions and electrons are expected to cancel (or balance) each other out. This probe-plasma potential, $V_f$ is called the floating potential. In the event of thermalized plasma, this voltage is $\frac{1}{K_BT}$ (expressed in eV). When $V$ is increased $V_f$ electron current increases.The resulting current is exponentially related to $V$. In region II, the electron current are defined by:
$$
J_e=\frac{I_e}{A_p}=\frac{n_eev_{th}}{4}e^{\frac{-eV}{K_BT_e}}
$$
$T_e$ is the electron temperature. Does this equation look familiar to you?

Region III
In this region the plasma current saturates at a plasma space potential value $\psi_p$, caused by the space charge limitation during current collection.

diagnostic is a plastic plate (CR-39) placed above a cylindrical magnet. ions deflected passing through the magnet based on their energy and form tracks on the plate.

 

[naviakam]

What would be type of the potential responsible for the following equation:

 

[haruspex]

What would be type of the potential responsible for the following equation:
View attachment 275499

In post #18 you let slip the crucial fact that your equations in post #1 come from attempts to fit data to readings. From then it was clear that attempting to find an equation that merged them over the whole range of x, as in posts up to that point, were a wild goose chase. Ignore all posts before #18.

Presumably what you have is a plotted curve that looks exponential over one part of the range and polynomial over another part. If so, there is probably no way to merge them into a single analytic expression.
E.g. consider a velocity/time graph for an object falling into water. There is no way to merge the equation for before hitting the water with the the one for after into a single analytic equation.

If my presumption is incorrect then please explain in detail what the situation is. Your response to @Fred Wright in post #30 sheds very little light on how you got your data.

 

[naviakam]

In post #18 you let slip the crucial fact that your equations in post #1 come from attempts to fit data to readings. From then it was clear that attempting to find an equation that merged them over the whole range of x, as in posts up to that point, were a wild goose chase. Ignore all posts before #18.

Presumably what you have is a plotted curve that looks exponential over one part of the range and polynomial over another part. If so, there is probably no way to merge them into a single analytic expression.
E.g. consider a velocity/time graph for an object falling into water. There is no way to merge the equation for before hitting the water with the the one for after into a single analytic equation.

If my presumption is incorrect then please explain in detail what the situation is. Your response to @Fred Wright in post #30 sheds very little light on how you got your data.

capacitor discharge in a gas produces ions. these ions are measured by magnetic spectrometry in which particles pass through the magnet and curve according to their energy and form tracks on a plastic plate.
from magnetic spectrometry two types of ion spectra are obtained (not from a single discharge but from two separate discharges). the initial condition is similar for both discharges but two different spectrum are produced: one fits with power and the other with exponential fit.
what I am looking for is to guess the corresponding potential responsible for such spectra!

 

[haruspex]

the initial condition is similar for both

What is the difference?

You wrote that y is a number of ions and x is an energy. Does that mean y of the ions have energy x? If so, is that as a histogram, i.e. y is a count of ions in some energy range around x?

one fits with power and the other with exponential fit.

Just how good is the fit? How many datapoints? What value of n do you get?
Can you post the data in machine-readable form?

 

[naviakam]

What is the difference?

You wrote that y is a number of ions and x is an energy. Does that mean y of the ions have energy x? If so, is that as a histogram, i.e. y is a count of ions in some energy range around x?

Just how good is the fit? How many datapoints? What value of n do you get?
Can you post the data in machine-readable form?

 

[haruspex]

View attachment 275538

Ok, so y is the log of the intensity?
As I asked, what is the power (n) in that fit, and what is the difference between the two runs?

 

[naviakam]

Ok, so y is the log of the intensity?
As I asked, what is the power (n) in that fit, and what is the difference between the two runs?

1. yes y is the number of ions.

2. n=4

3. the difference is the gas pressure in which the capacitor discharges:
at 6mbar the spectrum fits with exp
at 10mbar it's a power fit

 

[haruspex]

yes y is the number of ions.

The "power fit" graph you posted is log-linear, so is y in the power equation the number of ions (intensity?) or the log of the number?

And does this mean your exponential graph is also after taking logs, so you really have log(intensity) is negative exponential?!

n=4

Exactly?

the difference is the gas pressure in which the capacitor discharges:
at 6mbar the spectrum fits with exp
at 10mbar it's a power fit

So the full story is y=y(x,p), where p is the gas pressure.
Extremely artificially, you could write
$y=P(p)y_1(x)+(1-P(p))y_2(x)$, where P(6mbar)=0 and P(10mbar)=1.
I'm not seriously suggesting that would be valid, but you get the idea.

 

[naviakam]

The "power fit" graph you posted is log-linear, so is y in the power equation the number of ions (intensity?) or the log of the number? If you were to plot x^-4 on log-linear I would expect a straight line.

And does this mean your exponential graph is also after taking logs, so you really have log(intensity) is negative exponential?!

Exactly?

So the full story is y=y(x,p), where p is the gas pressure.
Extremely artificially, you could write
$y=P(p)y_1(x)+(1-P(p))y_2(x)$, where P(6mbar)=0 and P(10mbar)=1.
I'm not seriously suggesting that would be valid, but you get the idea.

1. it's been plotted in log format, y is number only not log

2. 3.8<n<4.1

3. then what is the estimated potential leading to each term in the equation $y=P(p)y_1(x)+(1-P(p))y_2(x)$?

 

[haruspex]

it's been plotted in log format, y is number only not log

Why would you plot it log-linear if trying to demonstrate a power law fit? Plot y-y₀ log-log against x, or plot y-y₀ against $x^{-4}$.

3. then what is the estimated potential leading to each term in the equation $y=P(p)y_1(x)+(1-P(p))y_2(x)$?

As I wrote, I am not seriously suggesting that is the correct form. I was just showing you how, given a function of two variables, the dependence on one might look very different for a different value of the other. To get any reliable sense of what is going on you would need results for several intermediate values of the pressure.

Anyway, what is this potential you are asking about? The voltage on the capacitor? Some potential function?

 

[naviakam]

Why would you plot it log-linear if trying to demonstrate a power law fit? Plot y-y₀ log-log against x, or plot y-y₀ against $x^{-4}$.

As I wrote, I am not seriously suggesting that is the correct form. I was just showing you how, given a function of two variables, the dependence on one might look very different for a different value of the other. To get any reliable sense of what is going on you would need results for several intermediate values of the pressure.

Anyway, what is this potential you are asking about? The voltage on the capacitor? Some potential function?

I was wondering if from the spectra we can guess a general potential function for each of the following equations:

 

[haruspex]

I was wondering if from the spectra we can guess a general potential function for each of the following equations:
View attachment 275586
View attachment 275587

As I indicated, I don't know what you mean by a "potential function" in this context.
Perhaps you could illustrate by starting from the other end: pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to.

Also, please post your "power fit" data as a graph of ln(y-y₀) against ln(x).

 

[naviakam]

As I indicated, I don't know what you mean by a "potential function" in this context.
Perhaps you could illustrate by starting from the other end: pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to.

Also, please post your "power fit" data as a graph of ln(y-y₀) against ln(x).

1. the capacitor discharge voltage is 10 kV, but ions are accelerated to MeV by an unknown potential. Then we try to guess this accelerating potential (starting from the other side: from spectra to potential!).

2. " pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to. "
Would you please help me how to do this.

3. here is the raw data:

1026.35 2.12E+10
973.90 2.28E+10
925.86 2.53E+10
881.75 2.73E+10
841.16 3.43E+10
803.73 4.18E+10
769.15 5.28E+10
737.15 6.57E+10
707.47 8.61E+10
679.92 1.15E+11
654.29 1.42E+11
630.42 1.89E+11
608.16 2.57E+11
587.38 3.53E+11
567.95 4.66E+11
549.77 6.18E+11
532.74 8.32E+11
516.77 1.10E+12
501.78 1.52E+12
487.70 1.90E+12
474.47 2.60E+12
462.02 3.33E+12

 

[haruspex]

1. the capacitor discharge voltage is 10 kV, but ions are accelerated to MeV by an unknown potential. Then we try to guess this accelerating potential (starting from the other side: from spectra to potential!).

2. " pick some example of such a function and show what y=f(x,p), or differential equation, it would lead to. "
Would you please help me how to do this.

3. here is the raw data:

1026.35 2.12E+10
973.90 2.28E+10
925.86 2.53E+10
881.75 2.73E+10
841.16 3.43E+10
803.73 4.18E+10
769.15 5.28E+10
737.15 6.57E+10
707.47 8.61E+10
679.92 1.15E+11
654.29 1.42E+11
630.42 1.89E+11
608.16 2.57E+11
587.38 3.53E+11
567.95 4.66E+11
549.77 6.18E+11
532.74 8.32E+11
516.77 1.10E+12
501.78 1.52E+12
487.70 1.90E+12
474.47 2.60E+12
462.02 3.33E+12

Are these the data that you claim a power law for?
For those data I get a decent straight line fit between ln(I) and V. Just a bit of a curve at one end. Looks like I is proportional to exp(-0.02V).

If it is not the "power law" data, please post that too.

 

[naviakam]

Are these the data that you claim a power law for?
For those data I get a decent straight line fit between ln(I) and V. Just a bit of a curve at one end. Looks like I is proportional to exp(-0.02V).

If it is not the "power law" data, please post that too.

This data and many others fit well with power law without any curve at the end.

However what I want to know is that if we have power law and exponential equations (as stated in post #39) how we can assign a potential function to them? This is what we need to present to the class! Please help on this.

Thanks

 

[haruspex]

This data and many others fit well with power law without any curve at the end.

Do you mean a power law or a polynomial?

how we can assign a potential function to them?

As I keep telling you, I don't even understand that question. Presumably you do have some idea of how a potential function would lead to an intensity versus voltage function, so give me an example. Pick some arbitrary potential function and show what I=f(V) it would lead to.

 

[naviakam]

Do you mean a power law or a polynomial?

As I keep telling you, I don't even understand that question. Presumably you do have some idea of how a potential function would lead to an intensity versus voltage function, so give me an example. Pick some arbitrary potential function and show what I=f(V) it would lead to.

If we have $Y=KX^{-n}$ where Y is number of particles and X is their energy.
Then we have a potential like $V=CI^{4}$ where V is the potential, C is a constant and I is the current.
The potential lead to the particles to be accelerated!

How this two equations could be related mathematically?

 

[haruspex]

If we have $Y=KX^{-n}$ where Y is number of particles and X is their energy.
Then we have a potential like $V=CI^{4}$ where V is the potential, C is a constant and I is the current.
The potential lead to the particles to be accelerated!

How this two equations could be related mathematically?

What current is this? Is it the current carried by the flow of ions?
If so, seems to me that you are not looking for a "potential function" but rather two functions of potential: how the current (rate of ions) and the energy per ion depend on an applied potential.
I.e. I=I(V) and E=E(V) lead to an observed relationship between I and E, and you want to find out what I(V) and E(V) are.
Am I close?

 

[naviakam]

What current is this? Is it the current carried by the flow of ions?
If so, seems to me that you are not looking for a "potential function" but rather two functions of potential: how the current (rate of ions) and the energy per ion depend on an applied potential.
I.e. I=I(V) and E=E(V) lead to an observed relationship between I and E, and you want to find out what I(V) and E(V) are.
Am I close?

Yes you are right, then how it is formulated mathematically?

 

[haruspex]

Yes you are right, then how it is formulated mathematically?

There is clearly no unambiguous way to extract two relationships from the one, but an obvious guess is that in a vacuum the energy would be directly proportional to the voltage, leaving you with a simple way to extract the relationship between voltage and current from the observed data.
But you got different results at some other pressure, so this is not a vacuum. Presumably the gas present saps some of the KE, but should not affect the current.