Boltzmann-Charge transport for Traveling-wave

[Mi1z]

Hello all,

I have a sort of fundamental question regarding charge transport problems. In general, the transport problems treated semi-classically using the Boltzmann Transport Equation and coupled with Maxwell's equations.

Specifically, I am contemplating a traveling wave problem. where the excitation force (E-field) is a traveling wave and thus the response (distribution function) is probably also a traveling wave.

Is it possible to solve such a problem analytically ? ie. no numerical or brute force techniques ? and most importantly, why ?

thanks in advance for your help,
cheers!

 

[eljose79]

Hello there,

Thank you for bringing up this interesting question about charge transport problems. I can say that it is indeed possible to solve such a problem analytically without using numerical or brute force techniques.

Firstly, let's start with the basics. The Boltzmann Transport Equation (BTE) is a semi-classical equation that describes the evolution of a distribution function in phase space, which represents the probability of finding a particle at a given position and momentum. This equation is coupled with Maxwell's equations, which describe the behavior of electromagnetic fields.

Now, in the case of a traveling wave problem, the excitation force (E-field) is a traveling wave and the response (distribution function) is also expected to be a traveling wave. This means that the distribution function will have a spatial and temporal dependence, and can be expressed as a function of the wave vector and frequency of the excitation force.

In order to solve this problem analytically, we can use a technique called the method of characteristics. This method involves transforming the BTE into a set of ordinary differential equations, which can then be solved analytically using standard techniques.

So why is it possible to solve this problem analytically? The answer lies in the nature of the BTE and Maxwell's equations. These equations are highly mathematical and well understood, and have been extensively studied by scientists over the years. This has led to the development of analytical methods that can be used to solve complex problems involving these equations.

In conclusion, it is indeed possible to solve charge transport problems analytically, even in the case of traveling wave problems. However, it requires a deep understanding of the underlying equations and the use of advanced mathematical techniques. I hope this helps answer your question. If you have any further queries, please do not hesitate to ask.