1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Brownian motion ish

  1. Sep 13, 2012 #1
    Suppose I have a large particle of mass [itex]M[/itex] that is randomly emitting small particles. The magnitude of the momenta of the small particles is [itex]\delta p[/itex] (and it is equal for all of them. Each particle is launched in a random direction (in 3 spatial dimensions--although we can work with 1 dimension if it's much easier). Assume also that these particles are emitted at a uniform rate with time [itex]\delta t[/itex] between emissions.

    So here's my issue. It seems to me that this is a random walk in momentum space. What I would like to know is how to estimate the displacement of the particle after [itex]N[/itex] particles are pooped out. Thus, I need some way to "integrate the velocity".

    However, I want to stress that I only care about an order of magnitude estimate of the displacement here. Has anyone dealt with this kind of a situation?

    I appreciate any help greatly!
  2. jcsd
  3. Sep 13, 2012 #2

    We have the sum of N independent identically distributed random variables so this is going to converge to a Gaussian very quickly, that is with N>30 or so. The momentum will follow 3-D Gaussian with mean of zero, that has got to be available somewhere. (A 2D Gaussian is called a Rayleigh distribution.)

    The 1D case will be a binomial distribution that converges to a Gaussian.
  4. Sep 13, 2012 #3

    Thanks for your help.

    I am aware that the momentum distribution will converge to a Gaussian of width [itex]\sim \sqrt{N} \delta p[/itex]. However, do you know what this will mean for the position distribution? In other words, I am really interested in the distribution of the quantity [itex]\sum_{i} p(t_{i}) [/itex] where the sum is taken over time steps for the random walk.

    My concern is that even though [itex] p [/itex] is expected to be [itex]\sim \sqrt{N} \delta p[/itex] at the end of the walk, I think that the sum may "accelerate" away from the origin because [itex] p [/itex] drifts from its origin.
  5. Sep 14, 2012 #4


    User Avatar
    2017 Award

    Staff: Mentor

    From a dimensional analysis: ##\overline{|x|}=c~ \delta t~\delta v~ N^\alpha##
    A quick simulation indicates ##\alpha \approx 1.5## and ##c\approx 1/2## in the 1-dimensional case. In 3 dimensions, c might be different, while alpha should stay the same.
  6. Sep 14, 2012 #5
    Thanks for the help. In fact, your estimation of [itex]\alpha = \frac{3}{2}[/itex] is the same thing I estimated with the following sketchy method:

    Let [itex]n(t) = \frac{t}{\delta t}[/itex] be the number of particles emitted after time [itex]t[/itex]. Then, the speed of the large particle at time [itex]t[/itex] can be estimated as [itex]\frac{\delta p \sqrt{n(t)}}{M} = \frac{\delta p }{M} \sqrt{\frac{t}{\delta t}} [/itex].

    Then [itex] \left| x(t) \right| \sim \int_{0}^{t} \left| v(t) \right| dt \sim \delta t \delta v \left(\frac{t}{\delta t}\right)^{3/2} [/itex].

    I feel that this estimate is probably an overestimate which is where your [itex] c \sim 1/2 [/itex] may come from.

    Thanks again.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook