What is the number of quantum states compatible with isolated ideal gas macrostate $N,V,U$ and molecular mass $m$?

Question

What is the degeneracy of an energy level $U$ of an ideal gas of $N$ particles with molecular mass $m$ in a volume $V$?

This sounds like a standard textbook problem about the Boltzmann entropy of ideal gas, but I'm actually asking a different thing, and it is more difficult: what is the value of the positive integer number of quantum states $W$ compatible with the macrostate $N,V,U$ of isolated gas of molecular mass $m$. So I'm not asking about the Boltzmann entropy (whose value is arbitrary and that is fine) but about the actual number of Hamiltonian eigenstates compatible with the constraints (which I expect to have definite answer in quantum theory).

The usual way to find this number works with classical statistical physics calculation of phase volume $\Omega_{compatible}$ (compatible with the macrostate and molecular mass $m$); this volume is then usually divided by the "smallest volume" of the "elementary" phase space element $h^{3N}$ (or $h^{3N}N!$) corresponding to how much phase space "fuzzily corresponds" to a single quantum state.

There is a problem with this method; the actual macrostate constraint defines only a hyper-surface in the phase space, which has zero thickness, and thus zero phase space volume. To get a non-zero phase space volume $\Omega$ (to be later divided by the elementary phase space volume), we must introduce some relevant region in the phase space that has non-zero thickness. Thus the hyper-surface is supplanted by a hyper-layer of some non-zero thickness $\Delta P$ (an interval of radial momentum variable $P$), which is related to an interval of allowed energy via $\Delta E = \frac{P\Delta P}{m}$. But $\Delta P$ is completely arbitrary, and thus my question cannot be answered.

Or alternatively, one can often find the compatible phase volume is written down using the delta distribution as

$$ \Omega_{compatible} = E_0 \int d^{3N}q~ d^{3N}p ~\delta (E-H). $$

This number has units of phase space volume, thus when divided by $h^{3N}$ or $h^{3N} N!$ we get a dimensionless number, which seems on the right track to getting the number of quantum states, independent of the choice of units of length, mass and time.

However, it is still not a definite answer to my question, because of the arbitrary value of non-zero constant $E_0$. True, the resulting ratio is dimensionless, but its value is still arbitrary, due to the thing that makes it dimensionless ($E_0$) having an arbitrary value!

Thus both these methods of finding $W$ introduce an arbitrary energy parameter - be it $\Delta E$ or $E_0$ - and this makes $W$ arbitrary, and thus does not answer my question.

Now, it is true that none of this prevents us from defining the Boltzmann entropy via $S =k_B \ln \Omega_{compatible}$ or $S =k_B \ln \frac{\Omega_{compatible}}{h^{3N}N!}$, where $\Omega_{compatible}$ has dimension of $(qp)^{3N}$; thermodynamic entropy value is arbitrary, so the Boltzmann entropy can be arbitrary too, as long as the arbitrary parameter like $E_0$ or $h$ has the same value for all the considered macrostates. This is because difference of entropy between two macrostates is independent of $E_0$ or $h$, and in thermodynamics it is this difference what matters, not value of entropy of single macrostate.

However, this relaxed attitude towards definition entropy (which is justified) does not help us to answer the question. In quantum statistical physics, isolated system in a known macrostate should have a definite number of compatible microstates, not an arbitrary one, because in quantum statistical physics of a system in a box, we have a discrete set of states, not a continuum.

Also, it seems that if we somehow get the unique answer for $W$, we can then infer what the effective value of $\Delta E$ or $E_0$ in the above formulae should be. And thus we can see that in quantum statistical physics, these parameters should not be arbitrary.

The answer most probably depends on whether the gas is boson or fermion, but answer to any of these two cases will be interesting.

Answer 1

As a prolific answerer of your standing, the stuff I am about to suggest, is likely to already be things that you have already seen before. However, it is still worth pointing out explicitly. I do not think that you are asking this question just for your own sake, but rather, I think you are trying to make your own lecture notes and are asking for a clear exposition on the topic.

First of all, a few attempts that you have listed involve phase space volumes, and then dividing by $h^{3N}$, with or without $N!$. This cannot work because phase space itself is at most a semiclassical concept. You can either have position wavefunctions, or work in terms of the momenta. Yes, I am aware that there are things like the density operator cast into the Wigner quasiprobability distribution, but going that direction will not help you get what you want with ease. In particular, a treatment that will be simple enough to be part of an introduction to statistical thermodynamics would be to just follow the standard treatment on the topic, which uses only a counting of the momentum states (really, energy states that way). The positions will simply not come into the picture.

Another issue is that it is rather awkward to fixate yourself on the microcanonical viewpoint. Instead, the natural exposition in the quantum realm is to start with either the canonical or grand canonical ensemble, and then work backwards to deduce what the appropriate microcanonical expression should be. Part of the problem is that the integral over the Dirac delta distribution fixing the energy eigenvalue will not work well in the quantum regime. Semiclassically, we expect this to be a smooth function on phase space, so that the integration will pick up a sensible value. However, it is clear upon thought of the quantum underpinnings that the energy eigenvalues distribution will be spiky, highly fluctuating around the semiclassical approximation. Think of this like "the number of lattice points of fixed radius $R$ away from origin", which will have zeroes, weakly degenerate values, and stronger degenerate values, fluctuating quickly around the continuous approximation. Mind you, the microcanonical viewpoint is physically silly; the canonical ensemble is a much better starting point that is more trustworthy as a physical picture to consider.

Note also that, strictly speaking, the introductory computations ignore interactions. This is necessarily wrong, because we all know how quantum theory likes to split degenerate energy eigenstates into multiple equally spaced energy eigenstates, and that energy eigenvalues tend to repel each other, leading to many closely spaced eigenstates. Still, the introductory computations are already in great agreement with experiment, and the first few correction terms also work really well.

With the above in mind, why do you not take the standard textbook introduction? Start with the treatment of a Bose gas or Fermi gas in a big box, in canonical or grand canonical ensemble, and get its partition function. This is a sum, maybe converted into an integral, in momentum variables, that is really just trying to enumerate the energy eigenvalues yet handling the degeneracy correctly. There is no position sum, but the volume of the system enters because the $\mathrm dk$ momentum increment is reciprocally related to the lengths of the box. This is the part that phase space arguments are getting wrong.

After getting the correct partition function, you can transition to the microcanonical system, and read off the result you want. This will give the correct semiclassical result, for both the Boson gas and the Fermi gas. Then you can take the Maxwell-Boltzmann limit and check that they agree with each other. You can, of course, take the limit earlier.

Answer 2

Preliminaries: Your question touches the connection of statistical mechanics with combinatorics and number theory, see e.g. Ref. 1. I think it is not possible to give a definite answer to your question, i.e. as far as I know there is no closed expression which would answer your question in full generality.

However, below we will also investigate some models to see how your problem can be reduced to a combinatorial question, some of which can be given an implicit solution, e.g. in terms of generating functions. I will also give some references for further reading.

In every case below, we will consider a system of $N$ non-interacting particles, which means that the many-body Hamiltonian takes the form $H=\sum\limits_{i=1}^N h_i$, where $h_i$ is the Hamiltonian of the $i$-th particle.

The basic idea in all cases will be that we will put basis elements into a one-to one correspondence with sequences of numbers, which allows us to get the dimension of the eigenspace corresponding to some eigenvalue by counting the number of sequences which fulfill some constraint.

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A: $N$ distinguishable harmonic oscillators in one-dimension

I think it is instructive to first start with distinguishable particles. The single-particle Hamiltonian admits a complete orthonormal basis, denoted by $\{|n\rangle\}_{n\in \mathbb N_0}$ with corresponding energies $\epsilon_n=n$ (omitting with out loss of generality additive and multiplicative positive constants). The $N$-particle Hilbert space is the $N$-fold tensor product of the single-particle Hilbert space and hence an orthonormal basis for this space is given by $$\{|n_1\rangle\otimes |n_2\rangle\otimes \cdots \otimes|n_N\rangle\}_{n_1,n_2,\ldots,n_N\in \mathbb N_0} \tag 1\quad ,$$

which is also the eigenbasis of the many-particle Hamiltonian $H$. As it is easily shown, every state of the above basis has an energy equal to $$E=\sum\limits_{i=1}^N n_i \tag 2 \quad .$$

You now ask for the dimension of the eigenspace of an arbitrary given eigenvalue $E$, denoted by $\dim E(N)$, which is simply the number of elements of $(1)$ with the same energy. Thus, your question can be reformulated as the number of ways to write a non-negative integer $E$ as a sum of $N$ non-negative numbers. This is the problem of weak-k-composition of a number. The answer, which can readily be obtained e.g. with the method of stars and bars (see also the link), is hence $$\dim E(N)=\binom{E+N-1}{N-1} \quad. \tag 3$$

If you check the above link, you will notice that by similar reasoning you can also obtain the case where all $n_i\in \mathbb N$ instead. Finally, let me remark that compositions treat e.g. $4=3+1$ and $4=1+3$ as different compositions of the number $4$, which naturally reflects the physics of distinguishable particles.

B: $N$ indistinguishable harmonic oscillators in one-dimension

Let us now study the case of identical (spinless) bosons and fermions, while the single-particle setting remains the same. An orthonormal eigenbasis of the many-body Hamiltonian for the bosonic case is given by

$$\{c^B(n_1,n_2,\ldots,n_N)\,S_N|n_1\rangle\otimes |n_2\rangle\otimes\cdots \otimes |n_N\rangle\}_{n_1\leq n_2\leq\ldots \leq n_N} \quad , \tag 4$$

with $n_j\in \mathbb N_0$ for $j=1,2,\ldots, N$, $c^B(n_1,n_2,\ldots,n_N)>0$ a normalization factor and $S_N$ the symmetrization operator of $N$ particles. Each state of this basis has an energy equal to $(1)$. The difference to the previous case is that the basis naturally encapsulates the indistinguishability of the particles, and hence also modifies the corresponding combinatorial problem, which now asks:

Given a non-negative integer $E$, how many sequences of $N$ non-decreasing and non-negative integers do exist such that their sum yields $E$? Equivalently, this is the number of ways to write $E$ as a sum of at most $N$ natural numbers (ignoring order), which is (an instance of) the problem of partition in number theory. This number then is equal to the number of partitions of $E+N$ into exactly $N$ natural numbers. Let $p_N(E)$ denote this number.

For fermions, an orthonormal basis if given by

$$\{c^F(n_1,n_2,\ldots,n_N)\,A_N|n_1\rangle\otimes |n_2\rangle\otimes\cdots \otimes |n_N\rangle\}_{n_1<n_2<\ldots <n_N} \quad , \tag 5$$

with $n_j\in N_0$ for $j=1,2,\ldots, N$, $c^F(n_1,n_2,\ldots,n_N)>0$ a normalization factor and $A_N$ the anti-symmetrization operator of $N$ particles. Each state again has an energy $(1)$ and your question now can be framed as:

Given a non-negative integer $E$, how many sequences of $N$ strictly increasing and non-negative integers do exist such that their sum equals $E$? Equivalently, you can ask for the number of ways (ignoring order) to write $E+N$ as a sum of exactly $N$ natural numbers which are all different, i.e. you ask for the number of k-partitions with the restriction that no element of the partition should repeat.

Note that in contrast to the case of distinguishable particles, partitions (as opposed to compositions) regard $4=3+1$ and $4=1+3$ as the same decomposition.

As a final remark here, let me mention that a generating function for $p_N(E)$ is related to the canonical partition function (Zustandssumme) of $N$ identical bosons in a harmonic oscillator potential, which I've derived some time ago here. This is shown in reference 2, where also an asymptotic relation for $p_N(E)$ is given. For the fermionic case, see references 2 and 5.

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C: $N$ distinguishable particles in a one-dimensional box (ideal gas)

Here, we will distinguish two boundary conditions. First, we consider periodic boundary conditions, where the single-particle energies are given by $\epsilon_k=k^2$ (omitting again irrelevant constants) with $k\in \mathbb Z$. The many-body Hamiltonian admits an orthonormal eigenbasis obtained from the $N$-fold tensor product of the eigenbasis of the single-particle Hamiltonian:

$$ \{|k_1\rangle\otimes |k_2\rangle\otimes \cdots\otimes |k_N\rangle\}_{k_1,k_2,\ldots,k_N \in \mathbb Z} \tag 7 \quad .$$

Each of these states has an energy equal to $$ E=\sum\limits_{i=1}^N k_i^2 \quad , \tag 8$$

and hence your question asks for the number of ways to write a non-negative integer $E$ as a sum of squares of $N$ integer numbers. The answer is the sum of squares function from number theory, see also Wikipedia. It admits a representation in terms of generating functions, which I refrain from reproducing here, as it is given in the link, with some more explicit formulas for certain special cases.

If I had to guess, I'd say that the generalization to three-dimensions should be straightforward, but you should double check.

Next, we can also consider hard walls, i.e. particles in a box with infinite potential at the walls. The single-particle energies read $\epsilon_n=n^2$, for $n\in \mathbb N$. I refrain from repeating basically the same procedure as above and we just note that hence the question now asks for the number of ways to write a natural number $E$ as a sum of squares of $N$ natural numbers.

I don't know if this has a solution (either explicit or implicit in terms of generating functions or similar).

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D: $N$ indistinguishable particles in a one-dimension box (ideal gas)

It should now be evident how to phrase the questions for identical bosons and fermions.

Unfortunately, I don't know of any solution to this problem in a form similar to the above problems. However, reference 3 and 4 discuss some asymptotic (related) cases which might be of interest to you.

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References and further reading:

1. physics- and probability-related approaches to integer partitioning problems. Link.
2. F. C. Auluck and D. S. Kothari. Statistical mechanics and the partitions of numbers. 1946. Link
3. B. K. Agarwala and F. C. Auluck. Statistical mechanics and partitions into non-integral powers of integers. 1951 Link
4. H. N. V. Temperley. Statistical Mechanics and the Partition of Numbers. I. The Transition of Liquid Helium. 1949. Link
5. Kubasiak, A. et al. Fermi-Dirac statistics and the number theory. 2005. Link
6. Zhou, Chi-Chun, and Wu-Sheng Dai. A statistical mechanical approach to restricted integer partition functions. 2018. Link
7. A. Rovenchak. Statistical mechanics approach in the counting of integer partitions. 2016 Link