The correct expression for pressure, temperature in Landau's 2-fluid model

Setup: the system of equations

I would like to numerically solve the equations of motion in Landau's [Ref (1)] two-fluid description of superfluid He II, which are

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 \\ \frac{\partial j_i}{\partial t} + \sum_k \frac{\partial \Pi_{ik}}{\partial x_k} = 0 \\ \frac{\partial \mathbf{v}_s}{\partial t} = -\nabla\Big( \phi + \frac{v_s^2}{2} - \frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2\Big) \\ \frac{\partial (\rho S)}{\partial t} + \nabla \cdot (\rho S \mathbf{v}_n ) =0$$ where $$\rho = \rho_s + \rho_n \\ \mathbf{j} = \rho_s \mathbf{v}_s + \rho_n \mathbf{v}_n \\ \Pi_{ik} = p \delta_{ik} + \rho_nv_i^{(n)}v_k^{(n)}\rho_s v_i^{(s)}v_k^{(s)}.$$ We have four differential equations, which I want to solve for the four variables $\mathbf{v}_s$, $\mathbf{v}_n$, $\rho$ and $S$. So far so good!

Note: I write everything as vectors, but initially I will try 1D, and maybe do 2D later.

The problem

However, I would also like to know $\rho_n$ (which gives me $\rho_s=\rho-\rho_n$); in fact, I seem to need this to compute $\mathbf{j}$. I also want to know $T$. I am most interested in seeing second sound, so $T$ is essential. Moreover, I need to know $p$, because it appears in the stress tenser $\Pi_{ik}$. I also need to know $\phi$, which Landau calls the "thermodynamic potential". I don't know what this is, but after reading some other sources I am for now going to replace it with the chemical potential $\mu$ --- but see question (2) below.

To summarise, I believe I need expressions for

• $\rho_n$,
• $T$,
• $p$ and
• $\mu$

in terms of the independent variables $\mathbf{v}_s$, $\mathbf{v}_n$, $\rho$ and $S$. My questions in their broadest form are:

1. What expressions should I use for these?
2. Am I correct to identify $\phi=\mu$? If not, what is $\phi$? I confess that I am by now thoroughly confused by all the thermodynamic variables. I proceed assuming I am correct about this.

Attempt and ideas

Idea one

I could use the various relations from the statisical physics of degenerate Bose gases, given in (e.g.) Refs (2, 3). Ref (3) sections 2.2 and 2.3 for example gives me functions in terms of (ph = phonon, rot = roton contributions)

$$\rho_n = \rho_{n,ph}(T, \mathbf{v_n}-\mathbf{v}_s) + \rho_{n,rot}(T, \mathbf{v_n}-\mathbf{v}_s) \\ s = s_{ph}(T) + s_{rot}(T) \\ p = p_{rot}(T) + p_{ph}(T).$$

I have nothing for the superfluid contribution to $p$ (can I ignore this?)... But it otherwise looks like I have $p$ and $T$, which means I can use the Duhem-Gibbi relation for $\nabla \mu = \frac{m}{\rho}\nabla p - sm \nabla T$.

Idea two

I find some relation between $s$ and $T$, and use something like (in 1D)

$$s = \frac{const}{T} \int d\epsilon \frac{ \sqrt{\epsilon}}{ \exp(\frac{1}{kT}(\epsilon - \mu)) -1}$$ to get $\mu$. But I'm still missing the relation between $s$ and $T$, and an expression for $p$, so maybe this is not as well formed as idea one.

More questions

1. None of these agree with, e.g., Tisza [ref (4)] who uses $s=s_0\Big(\frac{T}{T_{\lambda}}\Big)^{5.5}$ and $\rho_n = \rho \Big( \frac{T}{T_{\lambda}} \Big)^{5.5}$. Am I muddling the interacting and non-interacting Bose gas pictures? Tisza's expressions are empirical, for interacting He II -- is this why they are so different?
2. I am also worried that Tong says (p89): "However, in He-4, the interactions between atoms are strong and the system cannot be described using the simple techniques developed above"
3. Shouldn't the pressure depend on the density? The first sound speed is $c_1=\frac{\partial p}{\partial \rho}$ - have I just killed this?

Apologies for the long and muddled post, I have made a real effort but feel quite fundamentally confused. If anyone can shed light in some of the gaps, it would be much appreciated!

References

1. Landau, L., "Theory of the Superfluidity of He II" (1941) Physical Review 60, 356.

2. Tong, David, "Lectures on Statistical Physics", (2012) Ch 3: Quantum gases - link

3. Schmitt, Andreas, "Introduction to Superfluidity" (2014), Ch 2: Superfluid helium -- link

4. Tisza, Laszlo, "The Theory of Liquid Helium" (1947), Physical Review 72(9), 838.