Solved: Is Fermi-Dirac a Solution of Boltzmann Equation?

[dextercioby]

Confused and dizzy...

INTRODUCTION (I'm good at those...)

Studying transport phenomena in solid state physics is a relevant matter which is to be found in most solid-state physics books.Back home,in my third year,i have been taught this chapter by a teacher which followed closely the approach by Kireev in his extraordinary book "Semiconductor's Physics".In that book this simple chapter streches over more than 200 pages,so you can call it a thorough approach.
In the course on "Nonequilibrium Processes" I've been taught that,even until today,a general solution of the Boltzmann equation has not been given.Probably knowing that,Kireev tries to solve this equation perturbatively in a crystal in which is virtually irrelevant who carries energy and electric charge.He searches the sollution in the so-called "relexation time" approximation.He assumes the usual form for the interraction term (that delicious integral with transition amplitudes (the one who probably makes the virtually classical BE semi-classical,meaning it envolves quantum objects,like amplitudes of transition probability...)) and says that the zero-th approximation for the distribution function that he searches is Fermi-Dirac law for fermions (hence holes are to be looked upon as fermions)(sic),and then carries on to find the linear approximation for the distribution function for the free particles in crystal in both electric and magnetic fields acting symultaneously.

POSSIBLY THE QUESTION
This is definitely weird...I mean,the notion of zero-th approximation to an integral/integro-differential equation implies that it satisfies/it's a solution of the damn equation,but with certain restrictions.For exemple,the Maxwell-Boltzmann distribution law is a solution of the Boltzmann integro-differential equation but for time independent virtual statistical ensembles,and in the ugly interraction integral u make the Boltzmann symplifying assumption (that actually closes the equation itself)-the so called "molecular chaos ipothesis".
To resume,in my mind,for everything not to stink,there is only one question(this is the question ):
Is the Fermi-Dirac distribution function a solution of the BE on some (unknown to me) assumptions?
My answer is "no".Then if isn't,it should not be stated as the zero-th approximation of an equation who basically has little to do (or probably nothing) with it.
On the other hand,i'm well aware of the 1926 Sommerfeld model of free electron gas which states that free electrons in a solid obbey FD statistics.So it should be normal that in the absence of any exterior fields the distribution function be the one of FD.But what about (semiclassical) Boltzmann equation and it's devious coonection with FD statistics??

 

[dextercioby]

Nobody willing to give me an explanation??"Rusinica!Rusinica!Rusinicaaaaaa!"

Is it too complicated??I bet other TP matters are much more difficult,than trying to explain to me why the hell the zero-th approximation of an equation has nothing to do with the equation...

P.S.Is there another forum where somebody can give me an answer,preferably correct (or at least documented) one?

 

[R0man]

RESPONSE:

I can understand your confusion and feeling dizzy when trying to understand the connection between Fermi-Dirac distribution and the Boltzmann equation. It is a complex topic and often requires a thorough understanding of both statistical mechanics and solid state physics.

To answer your question, yes, the Fermi-Dirac distribution is a solution of the Boltzmann equation, but under certain assumptions and approximations. As you mentioned, the Maxwell-Boltzmann distribution is also a solution of the Boltzmann equation, but with different assumptions and restrictions. In the case of the Fermi-Dirac distribution, it is a solution of the Boltzmann equation in the limit of low temperatures and high densities, where the interactions between particles become dominant.

The reason why the Fermi-Dirac distribution is often referred to as the "zero-th approximation" is because it is the simplest form of the distribution function that satisfies the Boltzmann equation under these specific conditions. However, it is important to note that it is not a solution for all cases and situations.

Furthermore, the connection between the Boltzmann equation and Fermi-Dirac distribution comes from the fact that the Boltzmann equation is a classical description of a system, while the Fermi-Dirac distribution is a quantum mechanical concept. The 1926 Sommerfeld model you mentioned is a good example of this connection, where the free electron gas is described by the Boltzmann equation, but the particles obey Fermi-Dirac statistics.

In summary, while the Fermi-Dirac distribution is a solution of the Boltzmann equation under certain assumptions, it is important to understand that it is not the only solution and that the connection between the two lies in the specific conditions and approximations used. I hope this helps clarify your confusion and dizziness.