An equation in 6N dimensions

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In summary: The same conservation principles apply, that points in phase space can only move around, they never get created or destroyed.
  • #1
Kashmir
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We are in phase space of ##6N## dimensions. Each point ##\mathbf r## in this space has ##6N## coordinates.

IMG_20220404_130052.JPG

Pathria writes "Consider an arbitrary "volume" ##\omega## in the relevant region of the phase space and let the "surface" enclosing this volume be denoted by ##\sigma## then the net rate at which the representative points "flow" out of ##\omega## (across the bounding surface ##\sigma## ) is given by
##
\int \rho \boldsymbol{v} \cdot \hat{\boldsymbol{n}} d \sigma
##"

Where ##\boldsymbol{v}## is velocity and ##\rho## is number density function

I can understand why the equation is true in 3D however in higher dimensions I'm not sure why it holds . Please help me
 
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  • #2
Could you see why it would hold in 2D, where the two dimensions would be ##p_x## and ##x##?

(Note that ##\mathbf{v}## here is a generalization of velocity, not the actual velocity of a particle.)
 
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  • #3
DrClaude said:
Could you see why it would hold in 2D, where the two dimensions would be ##p_x## and ##x##?

(Note that ##\mathbf{v}## here is a generalization of velocity, not the actual velocity of a particle.)
No sir, I cannot.
 
  • #4
But you can see how it works in 3D?
 
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  • #5
DrClaude said:
But you can see how it works in 3D?
I had forgotten that we were talking about genralised velocities. I had in mind a 3D spatial velocity vector.
 
  • #6
Kashmir said:
I had forgotten that we were talking about genralised velocities. I had in mind a 3D spatial velocity vector.
You have to generalize to the time derivative of the coordinates, but the principle is the same.
 
  • #7
DrClaude said:
You have to generalize to the time derivative of the coordinates, but the principle is the same.
I understand that we have a generalized velocity vector now, however I don't get how the same principle would apply?

Is it that it's proof is advanced so the author has skipped it?
 
  • #8
Kashmir said:
I understand that we have a generalized velocity vector now, however I don't get how the same principle would apply?
The same conservation principles apply, that points in phase space can only move around, they never get created or destroyed.

Kashmir said:
Is it that it's proof is advanced so the author has skipped it?
Don't take this the wrong way, but I would say the author doesn't give a proof because they consider this trivial. Again, just think in terms of basic conservation principles.
 
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1. What is an equation in 6N dimensions?

An equation in 6N dimensions refers to a mathematical expression that involves 6 variables, each representing a different dimension. This type of equation is commonly used in physics and other scientific fields to model complex systems.

2. How is an equation in 6N dimensions different from a regular equation?

An equation in 6N dimensions is different from a regular equation in that it has 6 variables, each representing a different dimension, whereas a regular equation typically has only 2 or 3 variables. This allows for a more comprehensive and accurate representation of a system.

3. What are some real-life applications of equations in 6N dimensions?

Equations in 6N dimensions have many real-life applications, such as modeling the behavior of particles in quantum mechanics, predicting the movement of planets in astrophysics, and analyzing complex chemical reactions in chemistry.

4. Are there any limitations to using equations in 6N dimensions?

One limitation of using equations in 6N dimensions is that they can become very complex and difficult to solve, especially as the number of dimensions increases. Additionally, these equations may not always accurately represent real-life systems due to simplifications and assumptions made during the modeling process.

5. How do scientists use equations in 6N dimensions to make predictions?

Scientists use equations in 6N dimensions to make predictions by plugging in values for the variables and solving the equation to determine the outcome. These predictions can then be compared to real-world observations to test the accuracy of the equation and make any necessary adjustments.

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