4. i) Show that the classical limit of the Fermi-Dirac...

4. i) Show that the classical limit of the Fermi-Dirac distribution function makes the collision integral of the Boltzmann equation (14.3.8) vanish. Assume the chemical potential is not affected by the collisions. ii) Show also that the left side of (14.3.8) vanishes when f depends only on the energy $\varepsilon = mv^2/2 + U(\mathbf{r})$, $\frac{\partial f}{\partial t} + \mathbf{v} \cdot \frac{\partial f}{\partial \mathbf{r}} + \frac{\mathbf{F}}{m} \cdot \frac{\partial f}{\partial \mathbf{v}} = \int_{\mathbf{v}_1} \int_{\Omega'} (f'f_1' - ff_1) \mathbf{V} \sigma \,d\Omega' \,d^3v_1$ (14.3.8) $\sigma = \sigma(V')$.

Transcript

00:01 In this question we are asked about two different equations that describe electronic oscillators.

00:07 So we are asked to compare equation 1 and equation 2 and show that equation 2 reduces to equation 1 as the frequency approaches 0.

00:18 We're going to use a simplification for exponentials.

00:22 If you expand out an exponential in terms of polynomials, we can say that e to the x is 1 plus x plus x squared over 2 factorial.

00:32 Etc.

00:33 We can keep adding additional terms.

00:36 If x is small, then we can rewrite this as e to the x equals 1 plus x, ignoring the latter terms.

00:47 And that is because if you have an x that is small and then you square it, it becomes even smaller.

00:56 So this is only true in the case of a small x.

01:04 Now let us look down here at our equations.

01:08 Let's write out the two equations.

01:10 There's similar theme between the two.

01:14 They have many of the same components.

01:18 The difference is equation 2 contains an exponential.

01:23 This is where the exponential simplification is going to come into play.

01:30 So we can see here that equation 2 states the density of states, which is dependent on frequency and temperature, is equal to 8 pi h over c cubed, h being planks constant, c being the speed of light cubed, times frequency cubed, dv, we're looking at the change of frequencies, divided by the exponential portion minus 1.

01:57 Now let us look at equation 1.

02:00 In equation 1, we see that the density of states, also dependent on frequency and temperature, is equal to 8 pi, frequency squared, rather than a cube.

02:15 Kb, now kb is the bolt's most constant times t change and frequency, all divided by the speed of light cubed...

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