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Questions tagged [boltzmann-equation]

DO NOT USE THIS TAG for Boltzmann's constant, Maxwell-Boltzmann distribution, Stefan-Boltzmann law & Boltzmann brains!

This tag is for the Boltzmann equation. DO NOT USE THIS TAG for Boltzmann's constant, Maxwell-Boltzmann distribution, Stefan-Boltzmann law & Boltzmann brains!

77 questions
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Relaxation of the Boltzmann transport equation

My professor in kinetic gas theory said that when considering the Boltzmann Transport Equation (BCE) $$ \partial_tf + \frac{\vec{p}}{m}\cdot\nabla_{\vec{q}}f + \vec{F}\cdot\nabla_{\vec{p}}f = (\partial_tf)_{Coll} $$ Over long periods of time, the…
Tomas Noguera
  • 370
7
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What is the difference between Liouville's theorem and the Boltzmann transport equation?

From what I understand, Liouville's theorem is about the probability density $\rho$ of an ensemble existing in a differential volume in phase space $d\mathbf{r}d\mathbf{p}$. So the statement for Liouville's theorem is \begin{align*} \frac{d}{dt}…
megamence
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Derivation of the Drude model conductivity from the Boltzmann transport equation

The electric conductivity in the Drude model according to Wikipedia is \begin{align*} \sigma&=\frac{n q^2\tau}{m}. \end{align*} I'm trying to derive this from the Boltzmann equation in the relaxation time…
MrPillow
  • 191
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1 answer

Acoustic finite-size effects of simulated fluids under periodic boundary conditions

Consider a fluid simulated in a finite box of specific size. An impulse to the fluid element at the center in a given direction is physically expected to propagate at the speed of sound and attenuate during propagation. If the impulse is attenuated…
YoussefMabrouk
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5
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1 answer

Hydrodynamic equation to Boltzmann's equation

How to get the four-velocity of a fluid in terms of its microscopic distribution function $f(x^{i},\vec{p})$? For the sake of initial simplicity, the fluid can be thought of to be single component. The distribution function $f(x^{i},\vec{p})$…
FieldTheorist
  • 1,173
5
votes
2 answers

Boltzmann equation for Photons

I am computing the Boltzmann equation for Photons from the book "Modern Cosmology" by Scott Dodelson. This is the colission term from Compton scattering Then, the Dirac delta is expanded I am trying to understand how to expand the Dirac delta…
Cristhian Calderon
  • 101
4
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6 answers

Is the intensity of light ONLY dependent on the number of photons, and nothing else?

Recently, my teacher just told us that intensity is not linearly dependent on temperature and that it's ONLY dependent on photons. But then, what about Boltzmann's law? Isn't intensity dependent on the fourth power of absolute temperature? Even if…
ihateelectricalphysics
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Time scales in the BBGKY Hierarchy

In Kardar's or D. Tong's lectures on the BBGKY hierarchy, it is argued that the equation for the two-particle distribution function $f_2$, $$ \left( \frac{\partial}{\partial t} + \frac{\vec{p}_1}{m} \cdot \frac{\partial}{\partial \vec{q}_1} +…
Tristan
  • 31
3
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1 answer

Time derivative expansion in the Knudsen number for the Chapman-Enskog expansion

For the Chapman-Enskog expansion we expand the distribution function as a series in the Knudsen number $\epsilon$ as (I am following this paper): $$f = f^{(0)} +\epsilon f^{(1)} + ...$$ and it is also necessary to do this to do a similar expansion…
realanswers
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A question on relaxation time approximation

I was learned that the form of collision term in relaxation time approximation is set to be: $$\left (\frac{\partial f}{\partial t}\right)_c=-\frac{f-f^0}{\tau}$$ in with $f^0$ is local equilibrium distribution function. Many places use the simplest…
Haiqin Tang
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3
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1 answer

Issue deriving Fokker-Planck equation starting from Boltzmann's equation

I was trying to derive the Fokker-Planck equation starting from the Boltzmann's equation and I run into some issue while trying to do so. Starting from Boltzmann and using the notation $f \equiv f(x, v, t)$ the normalized density: $$\dfrac{\partial…
Syrocco
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How to find plasmon from Landau-Silin equation?

In David Pine's Theory Of Quantum Liquids: Normal Fermi Liquids, it's said that we can find charged Fermi liquid has plasmon modes easily from Eq. (3.40), replicated as follows: $$ (\boldsymbol{q} \cdot \boldsymbol{v}_p - \omega) \delta…
jywu
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Electron transport in fluids VS solids ("structure factor" VS "phonons")

Assume to have a neutral plasma consisting of some ions $X^+$ in a neutralizing gas of electrons $e^-$. Imagine that the $X^+$ can be in a gaseous (G), liquid (L) or solid (S) phase. We want to calculate the conductivity (electrical and/or thermal…
Quillo
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Does the Boltzmann equation reduce to Navier-Stokes equation?

Since the derivation of the Boltzmann Equation uses the molecular chaos assumption, it seems to me that it should not be valid for dense systems such as fluids. Now, according to Chapman-Enskog theory, the Boltzmann equation can be represented by a…
Roshan Tom
  • 31
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How to reduce an integral in phase space to an one-dimensional form?

I've been trying for a very long time to show that the following integral: $$I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow…
RicardoP
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