Understanding the Boltzmann Equation Derivation in Electric and Magnetic Fields

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SUMMARY

The discussion focuses on the derivation of the Boltzmann equation in the context of electric and magnetic fields, referencing the work of Ashcroft and Mermin. Participants emphasize the importance of applying the chain rule to derive the necessary derivatives, specifically df0/dE and df0/dr. The relationship dk/dE is identified as proportional to 1/Vk, necessitating the inclusion of mass constants and hbar in the calculations. This highlights the mathematical intricacies involved in the derivation process.

PREREQUISITES
  • Understanding of the Boltzmann equation
  • Familiarity with electric and magnetic field concepts
  • Knowledge of chain rule in calculus
  • Basic principles of quantum mechanics, including constants like hbar
NEXT STEPS
  • Study the derivation of the Boltzmann equation in detail
  • Learn about the application of the chain rule in physics
  • Explore the relationship between energy and wave vector (dk/dE)
  • Investigate the role of mass constants in quantum mechanics
USEFUL FOR

Physicists, graduate students in theoretical physics, and researchers focusing on statistical mechanics and quantum field theory will benefit from this discussion.

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Below is part of derivation of the Boltzmann equation in an electric and magnetic field.
I don't understand how to arrive at the bottom equation though. It is known that the dependence of the original distribution function is the given. My idea is to use chain rule but I don't see how to get a derivative of only ε.
 

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ashcroft and mermin? The chain rule sounds like the right idea. Multiply df0/dk by dk/dE to get df0/dE. I believe the dk/dE is proportional to 1/Vk; it needs a mass constant and hbar somewhere. Similar procedure goes for df0/dr.
 

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