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Boltzmann Transport Equation
LECTURE 2
• Boltzmann Transport Equation
• Continuity (Balance) Equations
• Method of Moments
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Sec. 5.2
Boltzmann Transport Equation
• This famous equation describes how an electron moves in 6-D phase space.
• Consider 2-D (x and kx) and some force Fx
• Electron moves a distance vx∆t in time ∆t
• Electron changes momentum according to
• i.e., transport between position states is taken to be classical
• Transport between momentum states can be classical and quantum mechanical (scattering)
• Probability of occupancy of the 2 states must be equal
• Do Taylor expansion
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Sec. 5.2
BTE: a balance equation
What is being balanced here?
Make the BTE into a general balance equation
What does each term mean physically?
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Sec. 5.2.1
Method of moments
• The BTE solves for f, the non-equilibrium distribution function
• This is very difficult to do.
• We use the Method of Moments to solve for properties related to f
• Specifically, charge, current density, and KE density
Note the ORDER of
progression of the exponent
of v.
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Sec. 5.2
Method of moments
Multiply each term by Cvn and integrate over k.
For example, zeroth order moment:
This gives
Why zero for R and G?
What is this well-known equation?
How do we find Je in order to evaluate n ?
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Sec. 5.2.2
1st order moment
What is the meaning of div JΦ ?
What are W and <<τM>> ?
We now have an expression for Je
What is the difficulty in evaluating Je ?
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Sec. 5.2.2
Continuity of current density
• Write down the steady-state version of the current continuity equation
• Neglect the gradient in KE/electron
• This gives us the DDE
• and gives definitions for
mobility and diffusivity.