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Re: Probability Distribution Function
On 2/26/12 at 4:18 AM, niels.martinsen at gmail.com (Niles) wrote: >I have a probablity distribution (Maxwell-Boltzmann) giving the >probability of a classical particle having some velocity v. No. The Maxwell-Boltzmann distribution is a distribution of particle speeds not particle distributions. That is the Maxwell-Boltzmann distribution is a continuous distribution of a scalar quantity not a distribution of a vector quantity such as velocity. If I seem to be being a bit fussy about the difference between speed (a scalar) and velocity (a vector) it is because of what you post next. >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 regarding how I should do this. There are several difficulties here. The trajectory of a particle will be determined by its velocity. But velocity a vector quantity is not described by a Maxwell-Boltzmann distribution. You could assign a random velocity to a given particle by selecting the speed of the particle as a random deviate from a given Maxwell-Boltzmann distribution and a direction selected from a uniform distribution. But this leads to the next difficulty. It is far from clear what you mean by applying a function to "the whole distribution". What do you mean by the "whole distribution"? >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. This suggests you are thinking of say n particles with different velocities. You could look at the center of mass for your system of particles, rather than each particle individually. Done this way, you need not concern yourself with the behavior of each particle. But if you do want to model an individual particle, you are going to need to consider collisions between particles. It seems to me the way to approach this would be to treat individual particles trajectories as 3D Brownian motion with the trajectory of the center of mass for the system superimposed. >1) My first question is if this is a correct method I am using? I don't think so. But I can't be certain from your post. >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.