What is meant by ##\rho_{xx}##?

[aliens123]

I was wondering what was meant by $$\rho_{xx}$$ in the following paper: https://arxiv.org/abs/1703.07325

I would guess that $$\rho$$ is the density, however I don't see why $$\rho$$ would be treated like a scalar. Unless what they mean is derivatives... Thank you!

 

[Ibix]

It's defined in equation 3.

 

[Delta2]

Normally it should be $$\rho_{xx}=\frac{\partial^2\rho}{\partial x^2}$$

Usually density is a scalar, at least that's how I know it when studying most of physics.

 

[Ibix]

Usually density is a scalar

Yes, but $\rho_{xx}$ has nothing to do with density - see how it's defined in equation 3 in the paper.

 

[Delta2]

Yes, but $\rho_{xx}$ has nothing to do with density - see how it's defined in equation 3 in the paper.

I am having big trouble following the paper, but i think equation 3 is not an equation of definition but an equation of calculation.

Moreover i think the paper tries to study the electron flow as the electrons being some sort of continuum fluid and by applying fluid dynamics equations. Most likely $\rho$ is the density of the "electron continuum fluid"

 

[Ibix]

Most likely $\rho$ is the density of the "electron continuum fluid"

No - check the units. It's a component of the resistivity tensor.

 

[Delta2]

No - check the units. It's a component of the resistivity tensor.

Hmmm, maybe you are right, it seems that they are tensor components cause it also refers to $\eta_{xx}$ and $\eta_{xy}$ as components of the kinematic viscosity tensor.

 

[Ibix]

Hmmm, maybe you are right

Furthermore, there is an explicit statement in the paragraph between equations 11 and 12 that equation 3 is the resistivity tensor.

 

[vanhees71]

Yes, it's the inverse conductivity tensor and thus the resistivity tensor. It's an example, how not to write a paper and that referees are sometimes not picky enough .

 

[robphy]

Side note:

Normally it should be $$\rho_{xx}=\frac{\partial^2\rho}{\partial x^2}$$

Usually density is a scalar, at least that's how I know it when studying most of physics.

When I taught Griffith's level E&M the first time,
I realized another example of notational differences between mathematicians and physicists.
To physicists and advanced physics students, $E_x$ is the x-component of the electric field vector: $\quad$ $\hat x \cdot \vec E$.
To mathematicians and advanced mathematics students, $E_x$ is the partial-derivative of $E$ with respect to $x$: $\quad$ $\displaystyle\frac{\partial E}{\partial x}$.
(I'd be curious to how $\vec\nabla\cdot \vec E$ would be expanded out... presumably, one would define $\vec E= F(x,y,z) \hat x+ G(x,y,z) \hat y + H(x,y,z) \hat z$)

(Long ago, I was a double major in physics and math, but I guess I probably compartmentalized notations.)

Possibly interesting reading:
http://sites.science.oregonstate.edu/math/bridge/papers/bridge.pdf
"Bridging the Gap between Mathematics and the Physical Sciences"
Tevian Dray and Corinne A. Manogue

http://sites.science.oregonstate.edu/physics/bridge/papers/CMJspherical.pdf
"Spherical Coordinates"
Tevian Dray and Corinne A. Manogue

 

[vanhees71]

Never ever use the Notation $E_x$ for the partial derivative of a function $E$ with respect to $x$. You'll run into a lot of trouble doing so. The mathematicians turn to make their lives very difficult by not using a thoughtful notation. This I learned when I attended a pure mathematicians' lecture on functional analysis. The mathematicians never use ornaments to clearly specify what's a vector, a vector component, a scalar etc. They were very confused when it came to solve problems. Usually I first translated the problem into physicists's notation and solved it in this notation. After that I translated it to the mathematicians's notation, because the mathematicians never accept physicists's notation (particularly they don't like the Dirac bra-ket notation for Hilbert spaces and make the scalar product semilinear in the 2nd argument instead in the first just to make their lives more difficult; nor do they appreciate the Ricci calculus when they deal with vector/tensor algebra and calculus, again to make their lives more difficult).

The physicists had to learn the much more powerful and error-minimizing notation themselves. If you look at Maxwell's treatise or even the early writings by Einstein and Minkowski you get into a crisis too!

 

[DrClaude]

Possibly interesting reading:
http://sites.science.oregonstate.edu/math/bridge/papers/bridge.pdf
"Bridging the Gap between Mathematics and the Physical Sciences"
Tevian Dray and Corinne A. Manogue

http://sites.science.oregonstate.edu/physics/bridge/papers/CMJspherical.pdf
"Spherical Coordinates"
Tevian Dray and Corinne A. Manogue

Very nice read. Thanks for recommending,