Maxwellian velocity distribution vs. speed distribution

In summary: So, in summary, the Maxwellian velocity distribution is obtained by $$g(v_x)\propto e^{-mv_x^2/2k_B T}$$ and when extended to 3 dimensions the distribution becomes $$\propto e^{-mv_x^2/2k_B T}e^{-mv_y^2/2k_B T}e^{-mv_z^2/2k_B T} = e^{-mv^2/2k_B T}$$
  • #1
weezy
92
5
Maxwellian velocity distribution is obtained by $$g(v_x)\propto e^{-mv_x^2/2k_B T}$$ and when extended to 3 dimensions the distribution becomes: $$\propto e^{-mv_x^2/2k_B T}e^{-mv_y^2/2k_B T}e^{-mv_z^2/2k_B T} = e^{-mv^2/2k_B T}$$
Now looking at the speed distribution we take a spherical shell in phase space between ##v , v+dv## and obtain: ##f(v)dv \propto 4\pi v^2 dv e^{-mv^2/2k_B T}##

My question is that are the two distributions equivalent since ##f(v)## has a extra ##v^2## term but logically the two seem equivalent?
 
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  • #2
weezy said:
Maxwellian velocity distribution is obtained by $$g(v_x)\propto e^{-mv_x^2/2k_B T}$$ and when extended to 3 dimensions the distribution becomes: $$\propto e^{-mv_x^2/2k_B T}e^{-mv_y^2/2k_B T}e^{-mv_z^2/2k_B T} = e^{-mv^2/2k_B T}$$
Now looking at the speed distribution we take a spherical shell in phase space between ##v , v+dv## and obtain: ##f(v)dv \propto 4\pi v^2 dv e^{-mv^2/2k_B T}##

My question is that are the two distributions equivalent since ##f(v)## has a extra ##v^2## term but logically the two seem equivalent?
The velocity distribution goes from minus infinity to plus infinity for each direction and peaks and has mean at v=0. The speed distribution goes from zero to plus infinity and has value zero for speed less than or equal to zero. (It is really undefined for negative speeds.) The mean value of the speed for the Maxwellian is ## v_{mean}=\sqrt{(8/ \pi) k T/m} ##. For the speed, ## v_{rms}=\sqrt{3 k T/m} ## and most probable speed is ## v_{mp}=\sqrt{2 k T/m} ##. Perhaps the rms speed is the place where your statement that the two seem to be equivalent is most applicable. It is necessary to compute the precise form of the speed distribution, (with the spherical shell), in order to compute ## v_{mean} ## and ## v_{mp} ##.
 
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  • #3
The difference reflects the fact that there are two different coordinate systems for the integration. In the first case the integral is over three dimensions with [itex]dv_xdv_ydv_z[/itex]. The the second case the integral is based on spherical coordinates [itex]v^2dvsin\phi d\phi d\theta[/itex] and then the angular variables integrate out to [itex]4\pi[/itex].
 
  • #4
Well the name says it all: You have distribution! So to transform from one distribution to the distribution of a function of the independent variables you have to take into account the corresponding Jacobian.

In your case you have given the velocity distribution, i.e., the number of particles in an infinitesimal cube ##\mathrm{d}^3 \vec{v}## in velocity space around ##\vec{v}## is
$$\mathrm{d} N = \mathrm{d}^3 \vec{v} f(\vec{v}).$$
Now you have ##v=|\vec{v}|##. It's clear that here the case is pretty simple, because you can just use spherical coordinates, in which
$$\mathrm{d} N = \mathrm{d} v \mathrm{d} \vartheta \mathrm{d} \varphi v^2 \sin \vartheta f(\vec{v}).$$
Now you want to "forget" about the angular distribution. So you have to take the integral over ##\vartheta## and ##\varphi##, i.e.,
$$\tilde{f}(v)=\int_0^{\pi} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi v^2 \sin \vartheta f(\vec{v}).$$
Since in your case ##f(\vec{v})=f(v)##, you can do the angle integrals
$$\tilde{f}(v)=v^2 f(v) \int_0^{\pi} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi \sin \vartheta=4 \pi v^2 f(v).$$
 
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1. What is the difference between Maxwellian velocity distribution and speed distribution?

The main difference between these two distributions is that Maxwellian velocity distribution describes the distribution of velocities of a group of particles in a gas, while speed distribution describes the distribution of speeds of a group of particles in a gas.

2. How are Maxwellian velocity distribution and speed distribution related?

Maxwellian velocity distribution is directly related to speed distribution, as it is the probability distribution of velocities multiplied by the velocity itself. In other words, it is the speed distribution multiplied by the velocity.

3. What factors affect the shape of the Maxwellian velocity distribution curve?

The shape of the Maxwellian velocity distribution curve is affected by factors such as temperature, molecular mass, and the number of particles in the gas. Higher temperatures and lighter molecules result in broader, higher curves, while a larger number of particles results in a sharper and narrower curve.

4. How does the Maxwellian velocity distribution help us understand the behavior of gases?

The Maxwellian velocity distribution helps us understand the behavior of gases by providing a statistical description of the velocities of particles in a gas. This information is important for studying phenomena such as diffusion, heat transfer, and viscosity in gases.

5. Can the Maxwellian velocity distribution be applied to other systems besides gases?

Yes, the Maxwellian velocity distribution can be applied to other systems besides gases, such as plasmas, liquids, and even solid materials. However, the assumptions and equations used may differ depending on the system being studied.

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