Annoying things in statistical mechanics

[Kcant]

I've been refreshing myself on some of the statistical mechanics I learned a couple years ago, using Kittel and Kroemer as a guide. However, I've come across a couple things that bother me:

1. When the Boltzmann distribution is derived, no real physics enters the picture. Essentially, the Taylor expansion of the entropy function is used to find the relative probability that two states are occupied. But entropy is just the logarithm of the degeneracy function, so why not just Taylor-expand the degeneracy function itself? What makes entropy special? I know that the degeneracy function is generally a fast-varying function of energy, and that entropy varies much more smoothly, but how can you know this a priori? How do you know that entropy varies sufficiently slowly to accurately approximate probabilities, and that you don't need some higher-order logarithm?

2. When deriving the Fermi-Dirac statistics, why can the energy of an unoccupied state be taken to be zero? Don't you really need some kind of field theory to know that?

 

[Monocles]

1. Entropy is defined as the logarithm of the multiplicity function specifically because it makes the math much easier to work with. If you wanted to, you could define the multiplicity function as entropy, and while the math would change, the physical results would be identical. So, it's basically a matter of convenience. Because the mapping between the multiplicity function and the logarithm of the multiplicity function is bijective, you aren't changing the physics any - just the ease of doing the math.

2. I don't know the answer to this question :( (I'm taking stat mech this semester and we haven't gotten to that yet)

 

[astrokreng]

I understand your frustrations with these concepts in statistical mechanics. It can be frustrating when certain assumptions or simplifications are made without a clear explanation or justification. However, it is important to keep in mind that statistical mechanics is a mathematical framework used to describe the behavior of large systems of particles, and as such, it relies on certain assumptions and approximations to make the calculations tractable.

In regards to your first point, the use of the Taylor expansion of the entropy function is a common approach in statistical mechanics, but it is not the only one. Some textbooks may choose to use the degeneracy function instead, and both approaches are valid. The choice of which function to use often depends on the context and the specific problem being solved. It is also worth noting that the degeneracy function and entropy function are mathematically related, so using one or the other will ultimately lead to the same results.

Furthermore, the assumption that entropy varies smoothly is often justified by experimental evidence. In most cases, the behavior of a system can be accurately described using the first-order term in the Taylor expansion, and higher-order terms do not significantly affect the results. This is why the first-order approximation is commonly used in statistical mechanics.

Regarding your second point, the assumption of an unoccupied state having zero energy is a simplification that is often made in the derivation of Fermi-Dirac statistics. While it may not be strictly accurate, it is a reasonable approximation in many cases and can lead to useful insights about the behavior of systems. In more complex situations, a field theory approach may be necessary, but for many systems, the assumption of zero energy for unoccupied states is a valid and useful simplification.

In conclusion, while it can be frustrating to encounter certain assumptions and simplifications in statistical mechanics, it is important to understand that they are often necessary for practical and mathematical reasons. As scientists, we must continually question and evaluate these assumptions, but also recognize their value in helping us understand and describe the behavior of complex systems.