Mixture of two fluids: Is that EoS possible and how to interpret it?

Question

In the context of cosmology, a perfect fluid is generally described by an equation of state (EoS) of the following form: $$\tag{1} p = w \rho, $$ where $p$ is the fluid's pressure and $\rho$ is its energy density, while $w$ is a parameter (generally a simple constant). Non-linear relations are possible, for example the van der Wals EoS (the state parameter $w$ in (1) could be turned into a function of $\rho$): $$\tag{2} p(\rho) = \frac{w \rho}{1 - \alpha \rho} - \beta \rho^2. $$ A mixture of independant fluids could be defined by adding several densities and pressures: \begin{align} \rho &= \sum_i \rho_i, & p &= \sum_i p_i. \tag{3} \end{align} Now, I'm wondering if it's posible to mix the EoS of several superposed fluids by adding crossed terms. For example: \begin{align} p_1(\rho_1, \rho_2) &= w_1 \rho_1 + \alpha \rho_2^3, \tag{4} \\[2ex] p_2(\rho_1, \rho_2) &= w_2 \rho_2 + \beta \rho_1 + \gamma \rho_1^2. \tag{5} \end{align} I'll take radiation and matter and the following EoS as an example that puzzles me: \begin{align} p_{\text{mat}} &= \frac{\rho_{\text{mat}} \, \rho_{\text{rad}}}{3(\rho_{\text{mat}} + \rho_{\text{rad}})}, \tag{6} \\[1ex] p_{\text{rad}} &= \frac{\rho_{\text{rad}}^2}{3(\rho_{\text{mat}} + \rho_{\text{rad}})}. \tag{7} \end{align} Notice that \begin{align} p_{\text{mat}} + p_{\text{rad}} &= \frac{1}{3} \, \rho_{\text{rad}}, & \frac{p_{\text{rad}}}{p_{\text{mat}}} &= \frac{\rho_{\text{rad}}}{\rho_{\text{mat}}}. \tag{8} \end{align} Usually, radiation propagating in vacuum has the well known EoS $p_{\text{rad}} = \frac{1}{3} \, \rho_{\text{rad}}$, so (7) shouldn't be allowed. But the total energy-momentum tensor of the mixture has the following trace: $$\tag{9} \rho - 3 p = \rho_{\text{mat}} + \rho_{\text{rad}} - 3 p_{\text{mat}} - 3 p_{\text{rad}} = \rho_{\text{mat}}. $$ In this theoretical example, radiation is propagating in a matter fluid (it is superposed to it and is interacting with it!). Matter could react to the presence of radiation by absorbing or emitting some radiation, so produces a pressure when $\rho_{\text{rad}} \ne 0$, and radiation reacts to matter in a manner so its usual vacuum relation is modified.

So I have a few questions:

1. Since non-linear EoS are possible (relation (2) is an example), is it possible to mix several EoS as (4) and (5), for mixtures of interacting fluids, by adding crossed terms to their EoS? Since I never saw this before, I would like to see some specific examples from thermodynamics.

2. Are EoS (6) and (7) actually make any physical sense, especially for the radiation pressure which isn't in the usual $\frac{1}{3} \, \rho_{\text{rad}}$ form? How can we interpret or justify them in a better way than what I wrote?

This question is related to two other questions about the same subject:

Is it valid to add energy densities of *interacting* perfect fluids?

Interacting multi-fluids in FLRW cosmology?

Answer 1

1. Basically just typical air in atmosphere is such a mixture of interacting fluids, and as these "several fluids" are practically just single gas molecules colliding to each others in a "energy density equilibrium", simply said in the same pressure and temperature. In practical means this causes some lighter molecules to be ionized to plasma while other heavier particles still remains neutral and thus gives the practical meaning for the "mixture of EoS".

2. Equations (6) and (7) can be written; \begin{align} p_{\text{mat}} &=\rho_{\text{mat}} \frac{ \, \rho_{\text{rad}}}{3(\rho_{\text{mat}} + \rho_{\text{rad}})} = w \rho_{\text{mat}}, \tag{6} \\[1ex] p_{\text{rad}} &= {\rho_{\text{rad}}}\frac{\rho_{\text{rad}}}{3(\rho_{\text{mat}} + \rho_{\text{rad}})}= w {\rho_{\text{rad}}}. \tag{7} \end{align} which pretty much shows, that $w$ is the correcting parameter for the radiation pressure "captured" in the matter. This capturing must be seen like an additional degree of freedom for (radiation-)"collisions", as the photon emission and absorption doesn't cause similar continuity of movement as the molecular collisions does. The molecules can have many different excited levels and after the absorption the emission can been caused by a collision and not by an absorption of a second photon. These interactions then cancel out in some amount, thus causing the internal "captured" pressure which can't be measured as external pressure, though it's included to energy density.

As I read your related questions, which are showing your open minded approach to the topic, I want to share you this rough draft of mine, where introduce how the states of matters can be defined from the speed of the light. Hope this helps you forward.