# Relation between occupation probability and first passage probability

• WiFO215
In summary, P(r,t) is the occupation probability, which measures the likelihood of a particle in a random walk being at position r at time t if it starts from the origin at time zero. F(r,t) is the first passage probability, which measures the likelihood of a particle in a random walk reaching position r for the first time at time t. The book explains that there is a relationship between these two probabilities - for a random walk to be at position r at time t, it must first reach r at some earlier time step t' and then return to r after t-t' additional steps. This relationship can be expressed by the equation P(r,t) = \delta_{r0} \delta_{t0} + \
WiFO215
Let P(r,t) define the occupation probability, the probability that a particle emulating a random walk will find itself at position r at time t if starts from the origin at time zero.

Let F(r,t) define the first passage probability, the probability that a particle emulating a random walk will find itself at position r at time t FOR THE FIRST TIME if starts from the origin at time zero.

I was reading a book which says this :
"For a random walk to be at position r at time t, the walk must first reach r at some earlier time step t' and then return to r after t-t' additional steps. This connection between F(r,t) and P(r,t) can thus be expressed by the equation

$$P(r,t) = \delta_{r0} \delta_{t0} + \sum_{t'\leqt}F(r,t')P(0,t-t')$$

"

Can someone please explain how this is possible?

For some reason, I cannot edit the above. It is supposed to read t' $$\leq$$ t below the summation sign.

Can someone move this to the Classical Physics subforum? I might probably get more replies there.

## 1. What is the definition of occupation probability?

Occupation probability is the probability that a system, such as a particle in a potential well, will be found at a specific position in space at a given time.

## 2. How is occupation probability related to first passage probability?

First passage probability is the probability that a system will first reach a specific position in space at a given time. Occupation probability and first passage probability are related because they both involve the likelihood of a system being in a certain position at a given time.

## 3. How is the relation between occupation probability and first passage probability used in scientific research?

The relation between occupation probability and first passage probability is often used in research involving stochastic processes, such as diffusion or random walks. It helps to understand the behavior and dynamics of these systems, as well as to make predictions about their future states.

## 4. What factors can affect the relation between occupation probability and first passage probability?

The relation between occupation probability and first passage probability can be affected by various factors, such as the shape of the potential well, the temperature of the system, and the presence of external forces or barriers.

## 5. How can the relation between occupation probability and first passage probability be experimentally determined?

The relation between occupation probability and first passage probability can be experimentally determined by measuring the occupation time, or the time a system spends in a specific position, and the first passage time, or the time it takes for a system to reach a specific position for the first time. These measurements can then be used to calculate the corresponding probabilities.

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