Probability distributions and particles

[Niles]

Hi

My question is not related to a specific piece of software, but more of a technique. I have a probablity distribution giving the probability of a classical particle having some velocity v. Now, what I have is a function to calculate the trajectory for a particle with some velocity v_i. I need to apply this function to the whole distribution. My question is how I should do this.

Originally what I had thought about doing is to partition the distribution into N small bins, and associate a velocity to each bin. My plan was then to calculate the trajectory for each velocity (=bin), and the "output-velocity" I weigh with the original probability/weight.

1) My first question is if this is a correct method I am using?

2) I have already implemented this is Mathematica. However, for some distinct bins some of the "output"-velocities are the same. So I need to figure out some way to add them up, which I don't find that easy. My problem is to determine how close two data points have to be in order to be binned together. Niles.

 

[mmwave]

Hi Niles,

Thank you for your question. Your method of partitioning the distribution into bins and calculating the trajectory for each velocity is a valid approach. This is commonly known as the Monte Carlo method, where random samples are taken from a probability distribution to approximate a solution. However, there are a few things to consider when using this method.

1) It is important to make sure that the number of bins you choose is large enough to accurately represent the distribution. If the bins are too large, you may not capture the true behavior of the distribution. On the other hand, if the bins are too small, you may end up with a large number of bins with very few data points, which can lead to errors in your calculations.

2) When calculating the trajectory for each velocity, it is important to consider the weight or probability associated with each velocity. This weight should be taken into account when summing up the results for each bin. It is also important to keep track of the number of data points in each bin, as this can affect the accuracy of your results.

Regarding your second question, determining how close two data points have to be to be binned together can be a bit subjective. It ultimately depends on the accuracy you are looking for in your results. One approach could be to use a fixed interval, such as 1 standard deviation from the mean, to determine which data points should be binned together. Another approach could be to use a dynamic interval, where the bin size is adjusted based on the distribution itself. This can be done by dividing the range of the data points by the number of bins you want and using that as the interval size.

I hope this helps answer your questions. Good luck with your calculations!