Classical position/velocity probability

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Discussion Overview

The discussion revolves around the probability of finding a particle in certain regions of space based on its velocity, particularly focusing on two different velocity functions. Participants explore how the time spent by a particle at various velocities affects the probability distribution of those velocities.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the probability of finding a particle in a region is inversely proportional to its velocity in that region, proposing two cases of velocity functions.
  • Another participant agrees that the probability for lower velocities should be greater but questions the reasoning behind the equality of probabilities for certain velocity ranges.
  • A different participant argues that the probability in low velocity regions should depend on the difference between parameters v0 and v1, suggesting that the closer they are, the longer the particle spends in low velocity regions.
  • One participant points out that while the cosine function is symmetric, the overall velocity function is not symmetric due to the presence of the parameter v0, which affects the likelihood of finding the particle in specific regions.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between velocity and probability, with some agreeing on the general trend that lower velocities correspond to higher probabilities, while others challenge specific reasoning and conclusions drawn from the velocity functions. The discussion remains unresolved regarding the exact nature of these probabilities.

Contextual Notes

Participants have not fully clarified the assumptions regarding the definitions of probability in this context, nor have they resolved the mathematical implications of the velocity functions discussed.

j1m1
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Probability to find a particle in some region of space is inversely proportional to velocity particle has in that region of space.
Let's say we have two cases: one particle has velocity given by v(t)=v0*Cos(w*t), and other by v(t)=v0-v1*Cos(w*t), (v0>v1).
Since particle spends more time in regions of low velocity this should imply that probability to find a particle with low velocity is bigger than to find it with high velocity . For the first case probability to find a particle with velocity around v0 should be equal to probability to find a particle with velocity around -v0. In the second case the probability to find a particle with velocity around v0-v1 should be much bigger that to find it with velocity v0+v1, but on the other hand v(t) is distribution of velocity of a particle in time, and from this it looks that probability to find a particle with velocity around v0-v1 should be equal to probability to find a particle with velocity around v0+v1.
All opinions appreciated.
 
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j1m1 said:
In the second case the probability to find a particle with velocity around v0-v1 should be much bigger that to find it with velocity v0+v1
It will be bigger, but not necessarily much bigger.

but on the other hand v(t) is distribution of velocity of a particle in time, and from this it looks that probability to find a particle with velocity around v0-v1 should be equal to probability to find a particle with velocity around v0+v1.
I don't follow your reasoning here. Can you try to explain in more detail why you think this?
 
I think probability in low velocity region should depend on v0-v1 (the closer they are together, the longer the time particle spends in low velocity region).
My reasoning is: if you look at v(t)=v0-v1*Cos(w*t), particle's velocity oscillates around v0, and Cos function is symmetric in t, meaning in one full period it will be as much positive as it will be negative,and in the same way. If I were to choose random time t, just by looking at this velocity distribution, I would have equal probability that particle has velocity around v0+v1 as v0-v1.
 
The cos function is symmetric, but the function ##t\mapsto v(t)## is not symmetric because of the asymmetric velocity v0, and it is that function that determines the likelihood of finding a point in a region, not just the Cos function.
 
Ok I think I understand now. Thank you for clarifying.
 

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