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Probability of a single-particle state being occupied

For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is

(a) 1 eV less than mu.

(b) 0.01 eV less than mu.

(c) equal to mu.

(d) 0.01 eV greater than mu.

(e) 1 eV greater than mu.

Solution Preview

Perhaps it's best to start from the beginning. The fundamental assumption in statistical mechanics is that all microstates of an isolated system are equally likely. Suppose we fix some small energy interval which represents our ignorance about the total energy content of a system. Then all the microstates in that energy interval are equally likely. We denote the total number of microstates of the system as a function of the energy by Omega(E), but remember that Omega does also depends on the energy uncertainty.

Omega(E) is proportional to the chosen energy uncertainty dE, some books write Omega(E)dE, instead of Omega(E) for the number of microstates so that Omega(E) becomes independent of dE. However, the entropy is related to the total number of microstates: S = k_{b} Log(Omega) and does depend on dE, i.e. there is a term k Log(dE). Suppose you are dealing with a system containing 10^23 particles with 1 joule of energy and dE = 1 joule*y. Then the entropy is some constant of order unity times 10^23 k_{b} and the k_{b} log(dE) term makes a contribution of k Log(y) which is much smaller than the main contribution unless you take y of order
10^(-23). The spacing between the energy levels is of this order.

It turns out that if you knew exactly in which state the system is then your dE would be this small and the entropy would then become zero! So, the notion of the entropy depends crucially on our ignorance about the system. If our ignorance about a macroscopic system is also macroscopic then you get a macroscopic entropy (i.e. a value for the entropy which has a value within a few orders of magnitude of order unity when expressed in the macroscopic units of joule per kelvin). This entropy then does depend so extremely weakly on dE that you can disregard this dependence.

Except for some simple cases, working with Omega is very inconvenient. It is instead more convenient to work with systems at constant temperature (so-called canonical ensemble) or at constant temperature and chemical potential (so-called grand canonical ensemble).

You can derive the canonical ensemble as follows. The system you are looking at is now not isolated but kept in contact with a heat reservoir at some temperature. The combined system including the heat reservoir is an isolated system. Let's denote the system that is kept ...

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