Maxwell Boltzmann distribution

  • #1
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Summary:

Difference between Maxwell Boltzmann and fermi dirac distribution
In the Aschcroft & Mermin solid state book there is a curve to compare F.D and M.B distribution. I can't understand the concept of M.B curve; what does mean exactlly when x =0? It means the probability of zero energy for particles is most or ...???

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Answers and Replies

  • #2
mjc123
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Figure 2.1 seems odd; it appears to be plotting the distribution of velocity (note, velocity, not speed) versus kinetic energy. The energy distribution does not have a maximum at zero. Look up the MB distribution; observe its forms for the distribution of velocity, speed and energy.

Equation 2.1 is the distribution of vectorial velocity, i.e. with a particular direction as well as magnitude (or alternatively, a particular set of components {vx, vy, vz}). It has a maximum at zero velocity. The distribution of the speed is found by summing Eq. 2.1 over all directions (or all sets of {vx, vy, vz} that give the same magnitude v), and is given by
f(v) = 4πv2f(v)
This has a value of zero at v=0, and a maximum at a finite v. The energy distribution goes as E1/2e-E/kT
 
  • #3
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Thanks so much for your simple description.
Now I can understand the difference between graph for speed, energy and velocity.

But I can't understand "what fig2.1 try to say", it shows if x decreases then the probability increases. The plot is Ae^(-x). The meaning of x=0 is have zero velocity.
Does it mean if T increases then x decreases and the probability for have less velocity increases?
 
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  • #4
mjc123
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Figure 2.1 is a very bad graph. It is plotting the probability of one variable (velocity) against a different variable (energy/kT). All it really shows, qualitatively, is that MB and FD are very different.
If T increases, the distribution broadens, and the probability at low velocities decreases. The x axis in Fig 2.1 is E/kT, and the distribution of (E/kT) is independent of temperature.
 
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  • #5
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Thanks. I thought so but I wasn't sure. You really help me to understand this. Thanks so much.
 
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  • #6
Charles Link
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Usually there is a ## v^2 ## factor on the Maxwell-Boltzmann distribution, and the peak speed (## v_{mp}=##most probable speed) occurs at ## v_{mp}=(2kT/m)^{1/2} ##.
 
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