Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,851 answers , 20,616 comments
1,470 users with positive rep
501 active unimported users
More ...

free fall using stochastic mechanics

+ 3 like - 0 dislike
157 views

I am trying to obtain the standard expressions for free fall in a constant and stationary gravitational field using stochastic mechanics.  When the initial velocity of the particle  is $v_0$ we write the following equation for the subsequent Brownian movement

$${\frac {\partial }{\partial t}}p \left( x,t \right) +v_{{0}}{\frac {
\partial }{\partial x}}p \left( x,t \right) =\eta \left( t \right) {
\frac {\partial ^{2}}{\partial {x}^{2}}}p \left( x,t \right)$$

where $p(x,t)$ is the probability that the particle is at position x at time t; and $\eta \left( t \right) $ is the diffusivity of the particle in the gravitational field at time t.  Assuming that the particle at t = 0 is at x = 0, we have that

$$p \left( x,0 \right) =\delta \left( x \right) $$

where $\delta \left( x \right) $ is the Dirac delta function.

We consider a general form for the diffusivity given by

$$\eta \left( t \right) =\eta\,{t}^{n}$$

where $n$ is a positive integer.

The solution of this problem is obtained with Maple using the Fourier transform and the result is

From this solution the expected position for the particle is

$$E \left( x \left( t \right)  \right) =v_{{0}}t$$

and the corresponding variance for the position at time t is

$${\it Var} \left( x \left( t \right)  \right) =2\,{\frac {\eta\,{t}^{n+
1}}{n+1}}$$

The classical trajectory for the particle is reconstructed according with

$$x \left( t \right) =E \left( x \left( t \right)  \right) +\sqrt {{\it
Var} \left( x \left( t \right)  \right) }$$

Then we have that

$$x \left( t \right) =v_{{0}}t+{\frac {\sqrt {2}\sqrt {\eta}{t}^{n/2+
1/2}}{\sqrt {n+1}}}$$

From this last expression the velocity of the particle at time t is

$$v \left( t \right) =v_{{0}}+(1/2)\,\sqrt {2}\sqrt {\eta}{t}^{n/2-1/2}
\sqrt {n+1}$$

If we attempt to recover the classic expression

$$v \left( t \right) =v_{{0}}+gt$$

we need to choose that

$$n = 3$$

and

$$\eta={g}^{2}/2$$

My questions are:

1. Do you know some reference where all these computations can be found ?.

2. It is possible to use these results with the aim to describe the free fall as a real diffusion in a gravitational field?

asked Jul 12, 2015 in Theoretical Physics by juancho (860 points) [ revision history ]
edited Jul 12, 2015 by juancho

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
$\varnothing\hbar$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...