<p>You can't apply the Euler equation in the multi-stream regime, where streams of material moving in different directions cross, because this system isn't a fluid. That is, the density and velocity at each position don't suffice to specify the system. Instead you need to work with distribution functions and the Boltzmann equation. (Although as you note, you can integrate it to get something similar to the Euler equation, where the velocity dispersion acts somewhat like a pressure.)</p>
<p>Within this treatment, if it is necessary to account for collisions, you can add a collision term to the Boltzmann equation. I'm not familiar with a context where it is meaningful to specify a pressure.</p>

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2022-10-03T19:56:28.107Z
2022-10-03T19:56:28.107Z
Sten
<p>i have two questions, and they are related.
In the Euler equation <span class="math-container">$\frac{D\vec{u}}{Dt}=-\vec{\nabla}\Phi-\frac{\vec{\nabla}P}{\rho}$</span>
where <span class="math-container">$\frac{D\vec{u}}{Dt}=\frac{\partial \vec{u}}{\partial t}+(\vec{u}\cdot \vec{\nabla})\vec{u}$</span>. If for example one wants to applie it to a galaxy, in the context of N body problem. What does this pressure means? Is the interstellar medium that gives sense to this term? i think that the answer to the last question is no, one should deal with navier stokes equation and when it says viscocity there you have your ISM. Because i only imagine gravitational interactions (assuming there are no physical collitions)i can't figure it out.
I understand that you can use Bolztman equations without collisions and take the first momentum order of it, and you have something that is like a pressure,like dispersion of velocities. Or maybe using focker-planck equation first momentum order, but that never done it.
The thing is if you have N particles you will have a density <span class="math-container">$\rho (\vec{x},t) = \sum_{i=1}^{N}m_{i}\delta(\vec{x}-\vec{x_{i}}(t))$</span> . In this condition (euler equation) is P=0?</p>

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2022-10-04T20:06:13.290Z
Well now i see that boltzmann equations is for only one particle ):
martín canullán
<stellar-density>
Pressure on euler equation for astrophysical scenario
martín canullán
# Pressure on euler equation for astrophysical scenario

(Asked by: username_0 on 2022-10-03)

i have two questions, and they are related.
In the Euler equation $\frac{D\vec{u}}{Dt}=-\vec{\nabla}\Phi-\frac{\vec{\nabla}P}{\rho}$
where $\frac{D\vec{u}}{Dt}=\frac{\partial \vec{u}}{\partial t}+(\vec{u}\cdot \vec{\nabla})\vec{u}$. If for example one wants to applie it to a galaxy, in the context of N body problem. What does this pressure means? Is the interstellar medium that gives sense to this term? i think that the answer to the last question is no, one should deal with navier stokes equation and when it says viscocity there you have your ISM. Because i only imagine gravitational interactions (assuming there are no physical collitions)i can't figure it out.
I understand that you can use Bolztman equations without collisions and take the first momentum order of it, and you have something that is like a pressure,like dispersion of velocities. Or maybe using focker-planck equation first momentum order, but that never done it.
The thing is if you have N particles you will have a density $\rho (\vec{x},t) = \sum\_{i=1}^{N}m\_{i}\delta(\vec{x}-\vec{x\_{i}}(t))$ . In this condition (euler equation) is P=0?

**Tags:** stellar-density
> **Comments:**
> **username_0**: Well now i see that boltzmann equations is for only one particle ):  

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## 1 answers

### Answer by username_1 on 2022-10-03. Score: 3

You can't apply the Euler equation in the multi-stream regime, where streams of material moving in different directions cross, because this system isn't a fluid. That is, the density and velocity at each position don't suffice to specify the system. Instead you need to work with distribution functions and the Boltzmann equation. (Although as you note, you can integrate it to get something similar to the Euler equation, where the velocity dispersion acts somewhat like a pressure.)

Within this treatment, if it is necessary to account for collisions, you can add a collision term to the Boltzmann equation. I'm not familiar with a context where it is meaningful to specify a pressure.


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