Getting to the Boltzmann Equation from a Nonequilibrium Distribution Function

Boltzmann equation. This will give us an expression for the relaxation time, which is defined as:\frac{1}{\tau \left( \mathbf{k} \right)}=\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}Once we have an expression for the relaxation time, we can then substitute it back into the nonequilibrium distribution function and simplify to arrive at the Boltzmann equation.
  • #1
Denver Dang
148
1

Homework Statement


Hi.

I have a course where I am supposed to show how to get to the Boltzmann equation from a nonequilibrium distrubution function.
At the moment I'm kinda lost to how this is done, so hopefully a hint or two would be of some help :)

Homework Equations


The nonequilibrium distribution function:

[tex]g\left( \mathbf{k},t \right)={{g}^{0}}\left( \mathbf{k} \right)+\int_{-\infty }^{t}{dt'}{{e}^{-\left( t-t' \right)\,/\tau \left( \mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right) \right)}}\left( -\frac{\partial f}{\partial \mathrm{ }\varepsilon\text{ }} \right)\times \mathbf{v}\left( \mathbf{k}\left( t' \right) \right)\cdot \left[ -e\mathbf{E}\left( t' \right)-\nabla \mu \left( t' \right)-\frac{\mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right)-\mu }{T}\nabla T\left( t' \right) \right][/tex]
where g0 is the local equilibrium disibution function and f being the Fermi function.

The Boltzmann equation is given by:

[tex]\frac{\partial g}{\partial t}+\mathbf{v}\cdot \frac{\partial g}{\partial \mathbf{r}}+\mathbf{F}\cdot \frac{1}{\hbar }\frac{\partial g}{\partial \mathbf{k}}={{\left( \frac{\partial g}{\partial t} \right)}_{coll}}[/tex]

The Attempt at a Solution


As far as I have been told, I should be able to come to the Boltzmann equation from the first equation with the use of these equations:

[tex]{{\left( \frac{dg\left( \mathbf{k} \right)}{dt} \right)}_{\mathrm{coll}}}=-\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}[/tex]

and:

[tex]\int{\frac{d\mathbf{k}'}{{{\left( 2\mathrm{ }\pi\text{ } \right)}^{3}}}{{W}_{\mathrm{k},\mathrm{k}'}}\left[ g\left( \mathbf{k} \right)-g\left( \mathbf{k}' \right) \right]}=\frac{1}{\tau \left( \mathbf{k} \right)}\left[ g\left( \mathbf{k} \right)-{{g}^{0}}\left( \mathbf{k} \right) \right][/tex]

These last two equations seems to be somewhat related to the relaxtion-time-approximation.
But how I get from the first equation on the very top, to the Boltzmann equation, well, there I'm kinda lost tbh :/

So anyone with an idea?

It's from the book: "Solid state physics by Ashcroft and Mermin" if that helps.
 
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  • #2


Hi there,

I can give you some hints on how to derive the Boltzmann equation from the nonequilibrium distribution function. First, let's start with the definition of the nonequilibrium distribution function:

g\left( \mathbf{k},t \right)={{g}^{0}}\left( \mathbf{k} \right)+\int_{-\infty }^{t}{dt'}{{e}^{-\left( t-t' \right)\,/\tau \left( \mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right) \right)}}\left( -\frac{\partial f}{\partial \mathrm{ }\varepsilon\text{ }} \right)\times \mathbf{v}\left( \mathbf{k}\left( t' \right) \right)\cdot \left[ -e\mathbf{E}\left( t' \right)-\nabla \mu \left( t' \right)-\frac{\mathrm{ }\varepsilon\text{ }\left( \mathbf{k} \right)-\mu }{T}\nabla T\left( t' \right) \right]

This equation describes the evolution of the distribution function in time, where the first term on the right-hand side is the equilibrium distribution function and the second term accounts for deviations from equilibrium due to external forces, such as an electric field or a temperature gradient.

Now, let's consider the left-hand side of the Boltzmann equation:

\frac{\partial g}{\partial t}+\mathbf{v}\cdot \frac{\partial g}{\partial \mathbf{r}}+\mathbf{F}\cdot \frac{1}{\hbar }\frac{\partial g}{\partial \mathbf{k}}

The first term represents the time evolution of the distribution function, which is also described by the first term on the right-hand side of the nonequilibrium distribution function. The second term accounts for the spatial variation of the distribution function, and the third term accounts for the effect of external forces on the distribution function.

To derive the Boltzmann equation, we need to equate the two expressions for the time evolution of the distribution function. This means that we need to set the first term on the right-hand side of the nonequilibrium distribution function equal to the first term on the left-hand side of
 

FAQ: Getting to the Boltzmann Equation from a Nonequilibrium Distribution Function

1. What is the Boltzmann Equation?

The Boltzmann Equation is a mathematical equation that describes the statistical behavior of a system of particles in thermodynamic equilibrium. It is used to predict the distribution of particles in a system based on their energy levels and the temperature of the system.

2. How is the Boltzmann Equation derived from a nonequilibrium distribution function?

The Boltzmann Equation can be derived from a nonequilibrium distribution function by taking the limit of the distribution function as the system approaches equilibrium. This involves making simplifying assumptions and performing mathematical manipulations to arrive at the final form of the Boltzmann Equation.

3. What is the importance of the Boltzmann Equation in physics?

The Boltzmann Equation is a fundamental equation in statistical mechanics and plays a crucial role in understanding the behavior of gases, liquids, and solids. It is also used in many fields of physics, such as thermodynamics, kinetic theory, and quantum mechanics.

4. Can the Boltzmann Equation be applied to all systems?

The Boltzmann Equation is most commonly used for systems in thermal equilibrium, where the particles have reached a steady state. However, it can also be applied to nonequilibrium systems with some modifications. Additionally, it is only valid for systems that can be described by classical mechanics.

5. How is the Boltzmann Equation related to entropy?

The Boltzmann Equation is closely related to entropy, which is a measure of the disorder or randomness of a system. The equation shows that the entropy of a system is directly proportional to the natural logarithm of the number of possible microstates of the system, indicating a higher entropy for more disordered systems.

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