Calculating the error in <x^2> from the error in <x> (Molecular Dynamics)

In summary: Since the error, ε, will be randomly positive or negative, I think the 2xε term will have an expected value of 0. So that leaves just ε2, which I think many others have already said.
  • #1
Kieran
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Hi, does anyone know of an easy way to calculate the error in <x^2> from the error in <x>? I am running a molecular dynamics simulation and trying to work out the error in the fluctuation of kinetic energy <dEk> = <3/2NT^2> - <3/2NT>^2 from the error in <T>.

Thanks in advance
 
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  • #2
You say you have the error in <T> ? Is'n that the same as ##<\sigma^2> = <(T-<T>)^2> = <T^2> - (<T>)^2 \ \ ## ?
 
  • #3
BvU said:
You say you have the error in <T> ? Is'n that the same as ##<\sigma^2> = <(T-<T>)^2> = <T^2> - (<T>)^2 \ \ ## ?
Yes, I have the standard error in the mean of T but I also need the error in the mean of T^2
 
  • #4
So you don't just want the spread <dEk> but also the error in this spread ? Doesn't that depend on the size of the system ?

[edit] adding: can't you use something like $${\sigma_{T^2}\over T^2 } = 2 {\sigma_T \over T} \ \ \ \rm ?$$
 
  • #5
BvU said:
So you don't just want the spread <dEk> but also the error in this spread ? Doesn't that depend on the size of the system ?

[edit] adding: can't you use something like $${\sigma_{T^2}\over T^2 } = 2 {\sigma_T \over T} \ \ \ \rm ?$$
I think so, this is what I'm having difficulty with. There must be error in <dEk> because there is error in <T>?
 
  • #6
[edit] adding: can't you use something like $${\sigma_{T^2}\over T^2 } = 2 {\sigma_T \over T} \ \ \ \rm ?$$[/QUOTE]
I know of this formula but I wasn't sure if it would work with the error being in the average of T...
 
  • #7
I suppose you are doing something very sophisticated ?
The Maxwell Boltzmann distribution has well-defined characteristics with 'exact' ##\sigma##. It's only when you generate samples, that the error in such ##\sigma## (determined from the sample) comes into the picture ?
 
  • #8
BvU said:
I suppose you are doing something very sophisticated ?
The Maxwell Boltzmann distribution has well-defined characteristics with 'exact' ##\sigma##. It's only when you generate samples, that the error in such ##\sigma## (determined from the sample) comes into the picture ?
Well I'm using a molecular dynamics program to determine a set of T values over time, so I get some fluctuation about the average T and I've worked out the error in the average of T. The problem is that I now need the error in <dEk> which I think needs the error in T but I'm not completely sure.. :confused:
 
  • #10
Dale said:
I think you want to use the propagation of errors formula.

https://en.m.wikipedia.org/wiki/Propagation_of_uncertainty
I've looked at this carefully but I am still unsure and can't seem to find any information of whether these formulas work for errors in the mean of quantities. For example, I'm not sure that

d<T^2>/<T^2> = 2d<T>/<T>
 
  • #11
Hmm, I can't answer for certain, but when I have been in similar situations I would just generate 10000 normally distributed numbers and see if the formula works.

It probably depends on the mean, so you might try mean 100, variance 9 or something similar.
 
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  • #12
Dale said:
Hmm, I can't answer for certain, but when I have been in similar situations I would just generate 10000 normally distributed numbers and see if the formula works.

It probably depends on the mean, so you might try mean 100, variance 9 or something similar.
Ah that sounds like a good plan, thanks for your advice
 
  • #13
You are welcome. Sometimes a few simulations are nearly as good as a proof.
 
  • #14
Kieran said:
Well I'm using a molecular dynamics program to determine a set of T values over time, so I get some fluctuation about the average T and I've worked out the error in the average of T. The problem is that I now need the error in <dEk> which I think needs the error in T but I'm not completely sure.. :confused:
Looks a bit like sample mean sigma ##\sigma_m## is ##\displaystyle \sigma_{\rm\ population}\over \sqrt N## and since you only have the sample it's ##\displaystyle \sigma_m## is ##\displaystyle\sigma_{\rm\ sample}\over \sqrt {n-1}##.

Fill us in on the result of Dale's suggestion ! :smile: My bet is on the simplest guess: double the relative error in T to get the relative error in T2
 
  • #15
I think using (x+ε)2 = x2+2xε+ε2 would give the answer that the error would be 2[itex]\bar{x}[/itex]ε + ε2
But I do agree that a simulation would be the best way to confirm a result one way or another.

EDIT (CORRECTION): Since the error, ε, will be randomly positive or negative, I think the 2xε term will have an expected value of 0. So that leaves just ε2, which I think many others have already said. (But 2xε the term can increase the variance of the error greatly, depending on the magnitude of x.)
 
Last edited:

What is the formula for calculating the error in from the error in ?

The formula for calculating the error in from the error in is: = ( + Δ)^2 = + 2 + (Δ)^2. This formula is derived from the Taylor series expansion of and takes into account the propagation of uncertainty.

What is the significance of calculating the error in from the error in in Molecular Dynamics?

In Molecular Dynamics simulations, represents the average value of a physical quantity, such as the position or velocity of a particle. represents the average squared value of that quantity. By calculating the error in from the error in , we can better understand the reliability and accuracy of our simulation results and make informed decisions about the validity of our data.

How can we reduce the error in from the error in in Molecular Dynamics?

There are several ways to reduce the error in from the error in in Molecular Dynamics. One approach is to increase the number of data points or simulation runs, which can improve the accuracy of the average values. Additionally, using more precise or accurate measurement techniques can also reduce the error in . It is also important to carefully consider and minimize any sources of systematic error in the simulation.

What are the limitations of calculating the error in from the error in in Molecular Dynamics?

One limitation of this calculation is that it assumes a linear relationship between and . In reality, this may not always be the case, especially for highly nonlinear systems. Additionally, the accuracy of the calculation relies on the accuracy of the individual and values, so any errors in those measurements can affect the final result.

Are there any alternative methods for calculating the error in from the error in in Molecular Dynamics?

Yes, there are alternative methods for calculating the error in from the error in in Molecular Dynamics, such as using Monte Carlo simulations or Bayesian inference. These methods may provide more accurate or efficient results in certain cases, but they also have their own limitations and assumptions that should be carefully considered before use.

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