Simulating Random Walk: Calculating Diffusion Length

[AndersonMD]

I am trying to do simulations of a random walk, I get out a normal distribution in 1D how do I get the "diffusion length" from the gaussian fit?

 

[mathman]

From wikipedia

Gaussian random walk

A random walk having a step size that varies according to a normal distribution is used as a model for real-world time series data such as financial markets. The Black-Scholes formula for modeling equity option prices, for example, uses a gaussian random walk as an underlying assumption.

Here, the step size is the inverse cumulative normal distribution Φ − 1(z,μ,σ) where 0 ≤ z ≤ 1 is a uniformly distributed random number, and μ and σ are the mean and standard deviations of the normal distribution, respectively.

For steps distributed according to any distribution with a finite variance (not necessarily just a normal distribution), the root mean squared expected translation distance after n steps is

E|S_n| = σ√n.

 

[AndersonMD]

So, if I am looking for the diffusion length of an exciton with lifetime $$\tau$$, where $$l_{D}=\sqrt{D_{X}\tau}$$, and I want to find out what the equivalent diffusion length in my simulation is where I am using random steps of length dx, I can fit the gaussian and find the E mentioned above?

 

[mathman]

Your original question and your comment are confusing me. Are you talking about a random walk with steps of fixed length (random direction) or are the step lengths distributed normally? Also, how many dimensions is your walk? I am not familiar with the physics notion (exciton) and the diffusion length (?) formula.

 

[AndersonMD]

I think I figured it out.
In general (1D) you can solve for:

$\frac{\partial n_{x}}{\partial t} = D_{x}\frac{\partial^{2} n_{x}}{\partial x}-\frac{n_{x}}{\tau} + I(x,t)$

This can be solved with a Gaussian and $\sigma^{2} = 4D_{x}t$. What I was trying to do was using a random step MATLAB simulation with a time step, lifetime, and spatial step figure out what the equivalent diffusion length was.