**1. Introduction**

The classical material laws such as Fourier and Navier-Stokes are acceptable for tasks concerning homogeneous materials, dense gases, and far from low-temperatures (20 K). In the engineering practice, these constitutive equations are well-known and widely used. Nevertheless, there are situations where some generalizations must be applied. Such a case could occur on small (micro or nano) length scales, short time scales, near low-temperature or far from equilibrium.

It is easier to understand the origin of the present continuum model of non-equilibrium thermodynamics with internal variables (NET-IV) for rarefied gases together with its properties if a purely heat conduction problem is presented first, since the essential attributes are inherited. Moreover, the most visible differences are analogous in both cases, for instance, the available degrees of freedom, the structure of the equations and the interpretation of the parameters. All these differences originate at the roots of these approaches.

Considering only heat conduction, various forms of heat propagation are experimentally found [1–9]. These are called second sound and ballistic propagation [9–12]. Their modeling background is diverse, and one can find many extended heat conduction equations with many different interpretations in the literature [13–20]. One of them is related to the approach of Rational Extended Thermodynamics (RET) [9,21], it considers kinetic theory rigorously and uses phonon hydrodynamics to describe these deviations from Fourier's law [22,23]. Another approach uses non-equilibrium thermodynamics with internal variables and current multipliers (NET-IV) [12]. Both are tested on the same heat conduction experiment, and the latter one seems to be more effective [23].

The main difference between these two approaches is routed to the physics that lies behind the system of constitutive equations. Using kinetic theory, one always has to assume apriori a mechanism occurring between the particles and describe the interaction among them. On the other side, non-equilibrium thermodynamics is phenomenological, the derivation of constitutive equations does not require any assumption regarding the microstructure, which makes the model more general and, in parallel, offers more degrees of freedom by not restricting the coupling coefficients. In the kinetic theory, due to the prescribed interaction model, most of the coefficients can be calculated, and only a few of them have to be fitted to the experimental data. Although its mathematical structure is advantageous, it is symmetric hyperbolic [21,24], the fixed parameters lead to its weakness: e.g., in a previously mentioned heat conduction problem, one has to use at least 30 momentum equations with increasing tensorial order to obtain the ballistic propagation speed approximately. The approach of NET-IV can resolve this problem, also preserving the structure of momentum equation; however, in order to fit, it requires more parameters [12,22,23,25]. All these approaches have advantages and disadvantages, and their detailed comparison is presented in [26].

In the case of investigating room temperature non-Fourier phenomenon, the phonon picture is not applicable [25]. One advantage of NET-IV is that it is applicable and tested on room temperature experiments that show over-diffusive type non-Fourier heat propagation [27–29]. It makes the kinetic approach of heat conduction more challenging to apply for practical problems; however, there are situations where its predictive power is useful (e.g., estimating transport coefficients). Such a situation is related to the topic of rarefied gases [30]. In some senses, the behavior of a rarefied gas (i.e., a gas under low pressure) is analogous with a rarefied phonon gas that applied in case of heat conduction. The difference among them is the type of the particle and the interpretation of some physical quantities. In order to understand the analogy, the ballistic conduction must be defined.

Using phonon hydrodynamics, the ballistic heat conduction is interpreted as non-interacting particles that scatter on the boundary only, i.e., traveling through the material without any collision [9]. It is important to emphasize that the following Equations (1) and (2) describe not merely a ballistic phenomenon but together with the diffusion and second sound propagation modes. This assumption leads to the system of equations in one spatial dimension:

$$\begin{aligned} \partial\_t \boldsymbol{\varepsilon} + \boldsymbol{c}^2 \partial\_x \boldsymbol{p} &= 0, \\ \partial\_t \boldsymbol{p} + \frac{1}{3} \partial\_x \boldsymbol{\varepsilon} + \partial\_x \boldsymbol{N} &= -\frac{1}{\tau\_R} \boldsymbol{p}\_\prime \\ \partial\_t \boldsymbol{N} + \frac{4}{15} \boldsymbol{c}^2 \partial\_x \boldsymbol{p} &= -\left(\frac{1}{\tau\_R} + \frac{1}{\tau\_N}\right) \boldsymbol{N}\_\prime \end{aligned} \tag{1}$$

where *e* being the energy density, *p* is momentum density, *c* stands for the Debye speed, *τ<sup>R</sup>* and *τ<sup>N</sup>* are the relaxation times referring to the resistive and normal processes [9], furthermore, *∂<sup>t</sup>* denotes the partial time derivative, applied for a rigid heat conductor. Here, *N* is the deviatoric part of the pressure tensor. In phonon hydrodynamics, it can be identified as a current density of the heat flux. The key aspect to include ballistic contributions into the modeling is achieving coupling between the heat flux and the pressure. This is one merit of this approach: such coupling was not realized in any other theories before. That was the motivation for the approach of NET-IV, this coupling is obtained using current multipliers [31], and the same structure can be reproduced [12,25]:

$$\begin{aligned} \rho c \partial\_t T + \partial\_x q &= 0, \\ \pi\_q \partial\_t q + q + \lambda \partial\_x T + \kappa \partial\_x Q &= 0, \\ \pi\_Q \partial\_t Q + Q + \kappa \partial\_x q &= 0, \end{aligned} \tag{2}$$

where *Q* plays the role of *N*, *q* is the heat flux, *c* denotes the specific heat and the coefficient *κ* is not fixed on contrary to (1), this property allows to adjust the exact propagation speed using only 3 equations instead of 30. The properties of the models above are discussed deeply by Jou et al. [32], Alvarez et al. [33] and Guo et al. [34].

Despite the numerous differences between phonons and real molecules, the situation is similar for rarefied gases, at least at the level of entropy; Equation (3) does not contain restrictions about the type of the fluid, i.e., there is an "entropic equivalence" between them. Here, a gas under low pressure consists few enough particles to observe the ballistic contribution. In NET-IV, the starting point is the generalization of entropy density and its current:

$$\begin{aligned} s(\boldsymbol{\varepsilon}, \rho, q\_{\boldsymbol{i}}, \Pi\_{\boldsymbol{i}\boldsymbol{j}}) &= s\_{\boldsymbol{\varepsilon}}(\boldsymbol{\varepsilon}, \rho) - \frac{m\_1}{2} q\_{\boldsymbol{i}} q\_{\boldsymbol{i}} - \frac{m\_2}{2} \Pi\_{\langle \boldsymbol{i} \boldsymbol{j} \rangle} \Pi\_{\langle \boldsymbol{i} \boldsymbol{j} \rangle} - \frac{m\_3}{6} \Pi\_{\boldsymbol{i}\boldsymbol{i}} \Pi\_{\boldsymbol{j}\boldsymbol{i}}, \\\ J\_{\boldsymbol{i}} &= (b\_{\langle \boldsymbol{i} \boldsymbol{j} \rangle} + b\_{\boldsymbol{k}\boldsymbol{k}} \delta\_{\langle \boldsymbol{i} \rangle} / 3) q\_{\boldsymbol{j}}, \end{aligned} \tag{3}$$

then exploiting the entropy production inequality of second law [12], one obtains a continuum model compatible with the kinetic theory to model rarefied gases [25,26,35]. Einstein's summation convention is applied. Here Π*ij* is an internal variable [36–38], it is identified as the viscous pressure, Π*ij* = *Pij* − *pδij* with *p* being the hydrostatic pressure, in accordance with EIT [18,19]. This is the usual assumption in theories of Extended Thermodynamics, as a consequence of the compatibility with kinetic theory [21,32], it also includes Meixner's theory [39], the first extension of Navier-Stokes equation. Besides, *bij* is called Nyíri-multiplier (or current multiplier) [31] which permits obtaining coupling between the heat flux and the pressure. Furthermore, the form of entropy flux (3) is compatible with the one proposed by Sellitto et al. [40], where the gradient of heat flux acts as a multiplier. Since the proper description requires the separation of deviatoric and spherical parts, in Equation (3) denotes the traceless part of the pressure. Equation (3) presents the same generalization as used for modeling complex heat conduction processes that include diffusive and wave propagation modes together, thus, hereinafter it is called *non-local generalization* of entropy and its current density [25]. The linearized-generalized Navier-Stokes-Fourier system reads in one dimension [25]:

$$\begin{aligned} \pi\_q \partial\_l q + q + \lambda \partial\_x T - a\_{21} \partial\_x \Pi\_s - \beta\_{21} \partial\_x \Pi\_d &= 0, \\ \pi\_d \partial\_l \Pi\_d + \Pi\_d + \nu \partial\_x \upsilon + \beta\_{12} \partial\_x q &= 0, \\ \pi\_s \partial\_l \Pi\_s + \Pi\_s + \eta \partial\_x \upsilon + a\_{12} \partial\_x q &= 0, \end{aligned} \tag{4}$$

where the lower indices *d* and *s* denote the deviatoric and spherical parts, respectively. The *η* is the bulk viscosity, *ν* denotes the shear viscosity, *αab*, *βab* (*a*, *b* = 1, 2) are the coupling parameters between the heat flux and the pressure and *τ<sup>m</sup>* (*m* = *q*, *d*,*s*) are the relaxation times, here the coupling parameters and the relaxation times are to be fit. This structure is equivalent to the 1D linearized version model from RET [41–43]:

$$\begin{aligned} \pi\_{\theta}\partial\_{l}\eta + q + \lambda\partial\_{x}T - RT\_{0}\pi\_{\theta}\partial\_{x}\Pi\_{d} + RT\_{0}\pi\_{\theta}\partial\_{x}\Pi\_{s} &= 0, \\ \pi\_{d}\partial\_{t}\Pi\_{d} + \Pi\_{d} + 2\nu\partial\_{x}v - \frac{2\pi\_{d}}{1+c\_{v}^{\*}}\partial\_{x}q &= 0, \\ \pi\_{s}\partial\_{t}\Pi\_{s} + \Pi\_{s} + \eta\partial\_{x}v + \frac{\pi\_{s}(2c\_{v}^{\*}-3)}{3c\_{v}^{\*}(1+c\_{v}^{\*})}\partial\_{x}q &= 0, \end{aligned} \tag{5}$$

with *R* being the gas constant and *c*∗ *<sup>v</sup>* denotes the dimensionless specific heat: *c*<sup>∗</sup> *<sup>v</sup>* = *cv*/*R*. As it is apparent, only the relaxation times are free parameters, all the other coefficients are fixed. It is interesting to note that the system (5) is derived by considering a doubled hierarchy of balance equations [24,44]. The reason behind that fact is related to the more degrees of freedom within polyatomic gases [26]. It is also important to note that it is not the only way for the kinetic theory: Lebon and Cloot derived a possible generalization using gradient terms as new variables [45] to

model nonlocal phenomena. In order to obtain a complete (closed) system of equations, beside the constitutive equations above, one has to use the balance laws as well:

$$\begin{aligned} \partial\_t \rho + \rho\_0 \partial\_x v &= 0, \\ \rho\_0 \partial\_t v + \partial\_x \Pi\_d + \partial\_x \Pi\_s + RT\_0 \partial\_x \rho + R \rho\_0 \partial\_x T &= 0, \\ \rho\_0 c \partial\_t T + \partial\_x q + R \rho\_0 T\_0 \partial\_x v &= 0, \end{aligned} \tag{6}$$

i.e., the mass, momentum and energy balances, respectively.

It is worth mentioning the earlier works of Lebon and Cloot [45] and Carrassi and Morro [46,47] where a similar comparison is performed. In these papers, other experiments are analyzed that are conducted by Meyer and Sessler [48], which slightly differ from the following one.

Before discussing the experimental observations about rarefied gases, one must define what a rarefied gas is. According to Klimontovich, a density parameter *ε* should be small for rarefied gases: *ε* << 1. It expresses the ratio of the occupied volume by the molecules and the overall available volume. Using the air properties at atmospheric pressure and room temperature, it turns out that *<sup>ε</sup>* <sup>≈</sup> <sup>10</sup>−<sup>4</sup> [49]. In other words, air at 10 atm is also rarefied or at least close to a rarefied state. Eventually this definition is not appropriate as it takes the volume corrections only into account, leaving the Knudsen number out of sight. The validity limit of the classical transport equations can be given more appropriately using the Knudsen number since it includes the mean free path as a characteristic quantity of a process. Above *Kn* ≈ 0.05 − 0.1 the generalizations of the Navier-Stokes-Fourier equations must be applied [30,35].

In the following, the particular experimental observations of Rhodes are presented. That experiment highlights two essential aspects which are not independent of each other, namely, the density dependence of material parameters and the frequency/pressure (*ω*/*p*) scaling properties of the RET and NET-IV models. The discussion aims to present the necessary requirements of obtaining *ω*/*p*-dependence from continuum point of view. At the end, that measurement is evaluated using the framework of NET-IV and compared to the approach of RET.
