D.57 The particle energy distributions

This note derives the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein energy distributions of weakly interacting particles for a system for which the net energy is precisely known.

The objective is to find the shelf numbers

It will be assumed, following derivation {N.24}, that if the maximum value is found among all shelf occupation numbers, whole numbers or not, it suffices. More daringly, errors less than a particle are not going to be taken seriously.

In finding the maximum of

Mathematicians call this a constrained maximization problem.

According to calculus, without the constraints, you can just put the
derivatives of penalty terms

that correct for any going
out of bounds, {D.48}, and the correct function whose
derivatives must be zero is

where the constants

At the shelf numbers for which the number of eigenfunctions is
largest, the derivatives

It is a much better idea to approximate the differential quotient by a difference quotient, as in

This approximation is very minor, since according to the so-called mean value theorem of mathematics, the location where

!.

Now consider first distinguishable particles. The function

For any value of the shelf number

and

The correctness of the final half particle is clearly doubtful within the made approximations. In fact, it is best ignored since it only makes a difference at high energies where the number of particles per shelf becomes small, and surely, the correct probability of finding a particle must go to zero at infinite energies, not to minus half a particle! Therefore, the best estimate

The case of identical fermions is next. The function to differentiate is now

This time

which can be solved to give

The final term, less than half a particle, is again best left away, to ensure that 0

Finally, the case of identical bosons, is, once more, the tricky one. The function to differentiate is now

For now, assume that

which can be solved to give

The final half particle is again best ignored to get the number of particles to become zero at large energies. Then, if it is assumed that the number

Before addressing these nasty problems, first the physical meaning of
the Lagrangian multiplier

Note the constraints: the number of particles in system

It follows that two systems that have the same value of

Returning now to the nasty problems of the distribution for bosons,
first assume that every shelf has at least two states, and that

implies that every shelf particle number increases when

Note that under reasonable assumptions, it will only be the ground state shelf that ever acquires a finite fraction of the particles. For, assume the contrary, that shelf 2 also holds a finite fraction of the particles. Using Taylor series expansion of the exponential for small values of its argument, the shelf occupation numbers are

For

What happens during condensation is that

The other problem with the analysis of the occupation numbers for
bosons is that the number of single-particle states on the shelves had
to be at least two. There is no reason why a system of
weakly-interacting spinless bosons could not have a unique
single-particle ground state. And combining the ground state with the
next one on a single shelf is surely not an acceptable approximation
in the presence of potential Bose-Einstein condensation. Fortunately,
the mathematics still partly works:

implies that

That then is the condensed state. Without a chemical potential that
can be adjusted, for any given temperature the states above the ground
state contain a number of particles that is completely unrelated to
the actual number of particles that is present. Whatever is left can
be dumped into the ground state, since there is no constraint on

Condensation stops when the number of particles in the states above
the ground state wants to become larger than the actual number of
particles present. Now the mathematics changes, because nature says
“Wait a minute, there is no such thing as a negative number of
particles in the ground state!” Nature now adds the constraint
that

A system of weakly interacting helium atoms, spinless bosons, would have a unique single-particle ground state like this. Since below the condensation temperature, the elevated energy states have no clue about an impending lack of particles actually present, physical properties such as the specific heat stay analytical until condensation ends.

It may be noted that above the condensation temperature it is only the most probable set of the occupation numbers that have exactly zero particles in the unique ground state. The expectation value of the number in the ground state will include neighboring sets of occupation numbers to the most probable one, and the number has nowhere to go but up, compare {D.61}.