Difference in random walk description

[jk22]

We studied random walk starting from a probability conservation : p(x,t)=p(x+dx,t+dt)+p(x-dx,t+dt)

Which means the particle can go left or right by dx in time dt.

The solution of the differential equation starting from a delta is a gaussian, which means the particle could go apparently at different speed than dx in dt.

Where does this difference comes from ?

 

[mathman]

Gaussian results from the fact that you adding the results of an "infinite" number of "infinitesimal" steps, which are independent and where each step can be + or -. Essentially it is the result of the central limit theorem applied to the limit of a binomial distribution.

 

[jk22]

So after a time dt it could be that we added an infinite number of + hence reaching infinity ?

What could we do if we want a maximal speed ? So that it becomes compatible with relativity.

 

[mathman]

So after a time dt it could be that we added an infinite number of + hence reaching infinity ?

What could we do if we want a maximal speed ? So that it becomes compatible with relativity.

Each step (+ or -dx) takes time dt. The central limit theorem describes the result of integrating over time. The distance traveled in time T has a Gaussian distribution with variance proportional to T.

I don't understand what you are trying to with infinity. If all the steps are in the same direction, it is not a random walk.

 

[jk22]

Adding only plus is a limit case, when the time is small paths going around the origin are preferred but then this difference gets smaller. but what i mean is that the speed at time dt can reach infinity. And i look for a method to avoid this.

 

[mathman]

Adding only plus is a limit case, when the time is small paths going around the origin are preferred but then this difference gets smaller. but what i mean is that the speed at time dt can reach infinity. And i look for a method to avoid this.

What do you mean by "small paths going around the origin"? I thought you were describing a 1 dimensional case.

 

[jk22]

It is rather through or at the origin.

 

[mathman]

The basic point is that each tiny step can be + or -. To get infinite speed you need a preponderance of one or the other.