Relation between occupation probability and first passage probability

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    Probability Relation
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Discussion Overview

The discussion revolves around the relationship between occupation probability and first passage probability in the context of particle dynamics. Participants explore the implications of certain expressions and assumptions related to these probabilities, focusing on the mathematical formulation and underlying concepts.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant links to a previous post in a different sub-forum, indicating a lack of responses there and seeking insights here.
  • Another participant explains that the expression suggests a particle arriving at a position r at time t must have first arrived at that position at an earlier time t', with the delta function indicating the particle's initial position was at 0.
  • A follow-up comment raises a question about the assumption that the probability distribution P remains unchanged once the particle reaches position r, noting that this assumption is not explicitly stated.
  • Another participant points out an implicit assumption of stationarity, suggesting that the distribution is dependent only on the time difference rather than on absolute time.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions underlying the probability expressions, particularly regarding the constancy of the probability distribution and the implications of stationarity. No consensus is reached on these points.

Contextual Notes

The discussion highlights potential limitations in the assumptions made about the probability distributions, particularly concerning their dependence on time and the implications of stationarity. These aspects remain unresolved.

WiFO215
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All the expression is saying is that a particle arriving at r at time t will have had a first arrival at r at time t' (which may = t) ahd then arrive at a relative position of 0 in time t-t'. The delta function term simply says that P(0,0) reflects the fact that it was at 0 to start with.
 
mathman said:
All the expression is saying is that a particle arriving at r at time t will have had a first arrival at r at time t' (which may = t) ahd then arrive at a relative position of 0 in time t-t'. The delta function term simply says that P(0,0) reflects the fact that it was at 0 to start with.

He is describing relative position zero using the same probability distribution P? So he is making the assumption there that P doesn't change once the thing reaches r? He hasn't mentioned this explicitly.
 
There is an implicit assumption of stationarity, i.e. the distribution depends only on time difference, not absolute time.
 
Okay! Thanks.
 

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