Map a one dimensional random walk to a two-state paramagnet

[Kitty123]

1. The question asks us to map a one dimensional random walk to a two state paramagnet and then write an expression for the number of journeys of N steps which end up at r=Rdelta.

Then we are asked to find an expression for the probability that N steps will end up at r.

2. N!/((N-Up)!(N-down)!)
The average is N/2
e^(-2x^2/N)
Probability= multiplicity(N)/multiplicity(all)

3. Other than saying that x=r=Rdelta and substituting that into my Gaussian I am really unsure of how to even begin this.

I attached a picture of the question. It’s #4

 

[haruspex]

unsure of how to even begin this.

The first thing it asks for is a verbal description of the mapping. Given a random walk, how might you map that to a microstate?

 

[Kitty123]

I understand the first part. A random walk is like a two state paramagnet because for every spin up or spin down you could go left or right.

After looking back over my notes I think I plug-in delta^2/(2tau)*t for x^2... also in light of the second part of the question plugging this in for x^2 would allow me to show that r has a proportional dependence on sqrt(Dt) where D=delta^2/tau.

I have no idea if this is correct... I think I’m on the right track !

 

[haruspex]

After looking back over my notes I think I plug-in delta^2/(2tau)*t for x^2... also in light of the second part of the question plugging this in for x^2 would allow me to show that r has a proportional dependence on sqrt(Dt) where D=delta^2/tau.

Since the question asks for an expression in terms of N and Rδ, and I do not know how you are relating those to the variables you mention above, I am unable to comment on that.
I believe it is asking you to use N and Rδ somehow to replace N-up, N-down in the equation you quoted:

N!/((N-Up)!(N-down)!)

 

[Kitty123]

Since the question asks for an expression in terms of N and Rδ, and I do not know how you are relating those to the variables you mention above, I am unable to comment on that.
I believe it is asking you to use N and Rδ somehow to replace N-up, N-down in the equation you quoted:

I am suppose to express the variables from the random walk in terms of the variables for the paramagnet. I know that r= total distance traveled, delta= the change in placement, and R must be the total steps taken over N/2 since total steps over N/2* delta would give me a final distance traveled. Some how I am suppose to be able to replace the x in my Gaussian with these other variables.

 

[haruspex]

I know that r= total distance traveled

No, it is clear that r is the finishing position of the walk, i.e. the displacement. The total distance walked is the number of steps, N.
I am unclear what R and δ are, but we are given r=Rδ, so maybe I don't need to know.
It does not seem to me that this part, an expression for the number of journeys, is asking for a Gaussian. It does not mention approximation. I would answer using factorials.

 

[Kitty123]

No, it is clear that r is the finishing position of the walk, i.e. the displacement. The total distance walked is the number of steps, N.
I am unclear what R and δ are, but we are given r=Rδ, so maybe I don't need to know.
It does not seem to me that this part, an expression for the number of journeys, is asking for a Gaussian. It does not mention approximation. I would answer using factorials.

Ok! That makes sense. The second part asks us to use the given Gaussian to calculate the probability, and I was assuming I needed to do that for part a... if I am just plugging things into a multiplicity function for part a that makes sense! Thank you.

 

[Kitty123]

OK. So if I am supposed to use a multiplicity like you said... but I am also supposed to map this onto a two state paramagnet which I solved in the previous problem... then I can use the factorial equation that I gave. In the state paramagnet that I solved I found that N-spin up = (N/2+x) where X is the excess over N/2. This x, for this walk, is the excess of steps over N/2. This means that for a random walk I have to take into account two times step length and position... which means x= r/2l. I can plug that expression into the factorial and I can use that to replace the x in the Gaussian to solve part b!

 

[haruspex]

OK. So if I am supposed to use a multiplicity like you said... but I am also supposed to map this onto a two state paramagnet which I solved in the previous problem... then I can use the factorial equation that I gave. In the state paramagnet that I solved I found that N-spin up = (N/2+x) where X is the excess over N/2. This x, for this walk, is the excess of steps over N/2. This means that for a random walk I have to take into account two times step length and position... which means x= r/2l. I can plug that expression into the factorial and I can use that to replace the x in the Gaussian to solve part b!

Looks right.

 

[Kitty123]

Looks right.

 

[Kitty123]

Ok... so now I need to use the Gaussian to write the probability of arriving at r. Probability is multiplicity(n)/multiplicity(all). I am assuming my Gaussian e^-r^2/2Nl (the Gaussian with x^2 replaced) is the numerator in my probability since this is the multiplicity function for arriving at r. The multiplicity of arriving anywhere should be the original Gaussian e^-2x/N, right?

 

[haruspex]

Ok... so now I need to use the Gaussian to write the probability of arriving at r. Probability is multiplicity(n)/multiplicity(all). I am assuming my Gaussian e^-r^2/2Nl (the Gaussian with x^2 replaced) is the numerator in my probability since this is the multiplicity function for arriving at r. The multiplicity of arriving anywhere should be the original Gaussian e^-2x/N, right?

That's not quite what I get. It should be x², certainly, and I also get a factor ∝ 1/√N.
Please post your steps.
As a check, what should the integral over all x be?

 

[Kitty123]

That's not quite what I get. It should be x², certainly, and I also get the factor as ∝ 1/√N.
Please post your steps.
As a check, what should the integral over all x be?

The Gaussian was derived in a previous question. If the distance traveled over N

That's not quite what I get. It should be x², certainly, and I also get the factor as ∝ 1/√N.
Please post your steps.
As a check, what should the integral over all x be?

Assuming that x is the distance over N/2 steps I can say that x=r/2l where r is position, l is step length, and the factor of 2 comes from having to account for the absolute value of step length since I do not want to end at 0 and I assume that for every step left there is an equal step right. Plugging x= r/2l into the Gaussian that was previously derived I get
e^(-2(r/2l)^2)/N
= e^-2(r^2/4l^2)/N
= e^-r^2/2Nl^2. This is the Gaussian for a random walk that ends at position r.

Since probability is multiplicity(r)/multiplicity(all) I assumed that multiplicity(all) would be my original e^(-2x^2/N) since this is all the possibilities, not just those that end at r.

 

[haruspex]

The Gaussian was derived in a previous question.

Maybe, but it is incomplete. There should be a factor ##\frac 1{\sqrt{2\pi N}}## outside the exponential.
I am guessing the "x" you had in post #11 instead of "x²" was just a typo.

 

[Kitty123]

That's not quite what I get. It should be x², certainly, and I also get the factor as ∝ 1/√N.
Please post your steps.
As a check, what should the integral over all x be?

The Gaussian was derived in a previous question. If the

Maybe, but it is incomplete. There should be a factor ##\frac 1{\sqrt{2\pi N}}## outside the exponential.
I am guessing the "x" you had in post #11 instead of "x²" was just a typo.

Yes. That was a typo. The full Gaussian has a (2^N)*sqrt(2/pi*N) in front of the exponential... so there is a factor of 1/sqrt(N).

Am I correct in thinking that the multiplicity(all) in the probability should be the original Gaussian without the r^2/2l adjustment?

 

[haruspex]

Am I correct in thinking that the multiplicity(all) in the probability should be the original Gaussian without the r^2/2l adjustment?

A multiplicity is an integer, a Gaussian is a probability distribution.
I think you are just being a bit sloppy with your usage of the terms, but that makes it very hard to answer your questions.

 

[Kitty123]

A Gaussian was never explained to us as probability distribution. That makes this simpler. Thank you for your help.

 

[haruspex]

A Gaussian was never explained to us as probability distribution. That makes this simpler. Thank you for your help.

I may have been a bit too specific there. Seems "Gaussian" is a bit more general in that its integral along the real line need not be 1. When normalized to 1 by a suitable constant factor it is a probability distribution.
See https://en.m.wikipedia.org/wiki/Gaussian_function.

 

[Kitty123]

Thinking about the Gaussian here as a probability makes sense (it also makes the second part of the problem much easier!)

I will check out the link you sent after my kiddos go to bed! Thanks for all of your help!