• 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.


PO is now at the Physics Department of Bielefeld University!

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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,734 comments
1,470 users with positive rep
818 active unimported users
More ...

  Local Density of the Medium for Classical 1D Harmonic Chain

+ 3 like - 0 dislike

In the first chapter of Altland and Simons (2nd Edition), pg. 34, there is the following exercise:

Consider the one-dimensional elastic chain discussed in Section 1.1. Convince yourself that the continuity equation of the system assumes the form $\partial_t \int_{x_1}^{x_2}dx \rho(x,t) = \partial_t(\phi(x_2,t)-\phi(x_1,t))$, where $\rho(x,t)$ is the local density of the medium. Use this result to show that $\rho = \partial_x\phi$. Show that the momentum $\int dx (\mathrm{particle~density})\times(\mathrm{velocity}) = \int dx\rho\partial_t\phi$ carried by the system coincides with the Noether momentum.

Rather than clutter this question with several additional lines, the 1D harmonic chain of section 1.1 is defined as in this question: Taylor expansion in classical 1D harmonic chain (classical field theory)

Now, my question is, why does $\rho = \partial_x\phi$ and not $\rho = -\partial_x\phi$?

Equivalently, why, in the first equation of the exercise, is the right hand side $\partial_t(\phi(x_2,t)-\phi(x_1,t))$ and not $\partial_t(\phi(x_1,t)-\phi(x_2,t))$?

Given that $\phi(x,t)$ is a displacement field, I would think that the particle density in the region $[x_1,x_2]$ should go up as the particles in the chain are displaced toward the region rather than away from it, i.e. density would rise for $\partial_t \phi(x_1,t) > 0$ (the opposite of what is stated in the exercise).

I have looked online, but found no one else who has talked about the particle density of the 1D harmonic chain, despite the number of books/lecture notes on the 1D harmonic chain.

Also, I have the rest of the question figured out; I just don't see how Altland and Simons got the sign that they did. Every way I think about it, I get the opposite sign from theirs.


This post imported from StackExchange Physics at 2018-03-02 22:41 (UTC), posted by SE-user JTC
asked Feb 28, 2018 in Theoretical Physics by JTC (15 points) [ no revision ]
retagged Mar 2, 2018

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:
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).
Please complete the anti-spam verification

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

Your rights