Random Walk Diffusion: Analytic Expression for Probability

[AndersonMD]

Hello

I am trying to find an analytic expression for the probability that a particle will have passed a position x after a time t.

It is a 1D random walk, the probability distribution after time t that a particle will end at position x is given by a gaussian, but I need to know how many particles passed the position at any point in time.

Cheers

 

[Dickfore]

I am trying to find an analytic expression for the probability that a particle will have passed a position x after a time t.

It is a 1D random walk, the probability distribution after time t that a particle will end at position x is given by a gaussian

so, you answered your own question.

but I need to know how many particles passed the position at any point in time.

A random walker by definition is a single particle. If you have a collection of N, non-interacting particles, then the answer to your other question is just the probability from the above question multiplied by the total number of particles.

 

[AndersonMD]

No, there is a difference between the probability of ending up at a certain location (gaussian), and the probability that it is passed through a position before ending up somewhere else (looks like a triangular function with gaussian wings)

 

[Dickfore]

Oh, I see. You ask for the probability that I will pass through a point with a coordinate $x$ either from the left or from the right within a time interval $[0, t]$. Is this what you mean?

 

[AndersonMD]

Yes, exactly.