Understanding the Boltzmann Equation Derivation in Electric and Magnetic Fields

In summary, the conversation involves deriving the Boltzmann equation in an electric and magnetic field. The speaker is having trouble understanding how to arrive at the bottom equation and suggests using the chain rule. It is mentioned that the derivative of only ε is needed and multiplying df0/dk by dk/dE can help obtain df0/dE. The speaker also mentions the need for a mass constant and hbar in the calculation. Similar steps are required for df0/dr.
  • #1
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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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  • #2
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.
 

1. What is the derivation Boltzmann equation?

The derivation Boltzmann equation is a fundamental equation in statistical mechanics that describes the behavior of a system of particles in terms of their individual velocities and interactions. It is derived from the Boltzmann transport equation, which governs the evolution of a system's distribution function.

2. What is the significance of the Boltzmann equation in physics?

The Boltzmann equation is significant because it provides a link between macroscopic observables, such as temperature and pressure, and microscopic quantities, such as particle velocities and interactions. It is also a key tool in understanding the behavior of systems out of equilibrium.

3. How is the Boltzmann equation derived?

The derivation of the Boltzmann equation involves starting from the Boltzmann transport equation and using various mathematical techniques, such as the Chapman-Enskog method and the Grad method, to simplify the equation and make certain assumptions about the system. These assumptions include the dilute gas approximation, which assumes that collisions between particles are rare, and the molecular chaos assumption, which assumes that particles are uncorrelated before and after collisions.

4. What are the assumptions made in the derivation of the Boltzmann equation?

The assumptions made in the derivation of the Boltzmann equation include the dilute gas approximation, which assumes that collisions between particles are rare, and the molecular chaos assumption, which assumes that particles are uncorrelated before and after collisions. Additionally, the derivation also assumes that the system is in a state of local thermodynamic equilibrium, meaning that macroscopic observables, such as temperature and pressure, are uniform throughout the system.

5. What are some applications of the Boltzmann equation?

The Boltzmann equation has a wide range of applications in various fields of physics, including gas dynamics, plasma physics, and kinetic theory. It is also used in the study of non-equilibrium processes, such as chemical reactions and transport phenomena. Additionally, the Boltzmann equation is an essential tool in the development of new materials and technologies, such as semiconductor devices and advanced materials for energy storage and conversion.

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