2. Polytropic index of longitudinal waves in a one-component system

Our analysis is based on the (non-relativistic) Boltzmann equation for a one-component system, including macroscopic forces:

(2.1) \begin{align} \frac {\partial f}{\partial t} + \boldsymbol{v} \boldsymbol{\cdot } \frac {\partial f}{\partial \boldsymbol{x}} + \frac {\boldsymbol{F}}{m} \boldsymbol{\cdot } \frac {\partial f}{\partial \boldsymbol{v}} = \left ( \frac {\partial f}{\partial t} \right )_{\mathrm{col}}. \end{align}

In (2.1), $f$ is the particle distribution function, $(\partial f / \partial t)_{\mathrm{col}}$ is the collision operator, $\boldsymbol{x}$ is the position, $\boldsymbol{v}$ is the velocity and $\boldsymbol{F}$ is the macroscopic force (all vector quantities are $D$ -dimensional). For a one-component system, conservation of mass and momentum implies that $(\partial f / \partial t)_{\mathrm{col}}$ satisfies the following relations:

(2.2) \begin{align} \int m \left ( \frac {\partial f}{\partial t} \right )_{\mathrm{col}} \! \! \! \! \mathrm{d}\boldsymbol{v} = 0 , \quad \int m\boldsymbol{v} \left ( \frac {\partial f}{\partial t} \right )_{\mathrm{col}} \! \! \! \! \mathrm{d}\boldsymbol{v} = \boldsymbol{0}; \end{align}

all integrals are over the full ( $D$ -dimensional) velocity space. Taking the zeroth and first $\boldsymbol{v}$ moments of (2.1), while using (2.2), results in the continuity and momentum equations:

(2.3) \begin{align} \frac {\partial n}{\partial t} + \frac {\partial }{\partial \boldsymbol{x}} \boldsymbol{\cdot } (n \boldsymbol{u}) = 0, \end{align}

(2.4) \begin{align} n \left ( \frac {\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot } \frac {\partial \boldsymbol{u}}{\partial \boldsymbol{x}} \right ) + \frac {1}{m} \frac {\partial }{\partial \boldsymbol{x}} \boldsymbol{\cdot } \unicode{x1D64B} = \frac {n \langle \boldsymbol{F} \rangle }{m}. \end{align}

Here, $n = \int f \, \mathrm{d}\boldsymbol{v}$ , $\boldsymbol{u} = \int \boldsymbol{v} f \, \mathrm{d}\boldsymbol{v} / n$ is the fluid velocity, $ \unicode{x1D64B} = \int m (\boldsymbol{v} - \boldsymbol{u})(\boldsymbol{v} - \boldsymbol{u}) f \, \mathrm{d}\boldsymbol{v}$ is the pressure tensor and $\langle \boldsymbol{F} \rangle = \int \boldsymbol{F} f \, \mathrm{d}\boldsymbol{v} / n$ is the macroscopic fluid force. We note that in (2.4), we have assumed $(\partial / \partial \boldsymbol{v}) \boldsymbol{\cdot } \boldsymbol{F} = 0$ , which may be considered a way of separating the macroscopic force from frictional forces due to the collision operator in a general system. Additionally, this condition applies to the Lorentz and gravitational forces, which are the macroscopic forces of practical importance.

Our analysis utilises a BGK collision operator (Bhatnagar et al. Reference Bhatnagar, Gross and Krook1954):

(2.5) \begin{align} \left ( \frac {\partial f}{\partial t} \right )_{\mathrm{col}} \! \! \! \! = \nu (f_R - f), \end{align}

where the collision frequency, $\nu$ , is independent of $\boldsymbol{v}$ , but may depend on $n$ and $T$ . In (2.5),

(2.6) \begin{align} f_R = \frac {n \, \mathrm{e}^{-|\boldsymbol{v} - \boldsymbol{u}|^2 / v_T^2}}{\pi ^{D/2} v_T^D}, \end{align}

with $v_T = \sqrt {2 T / m}$ being the (non-perturbative) thermal particle speed, represents the Maxwellian distribution towards which the system relaxes due to collisions. This form of $f_R$ , using $n$ and $\boldsymbol{u}$ of the actual $f$ , ensures that $(\partial f / \partial t)_{\mathrm{col}}$ satisfies the mass and momentum conservation laws of (2.2). Further, assuming elastic collisions implies local conservation of kinetic energy in the fluid frame of the one-component system, such that

(2.7) \begin{align} \int \frac {m |\boldsymbol{v} - \boldsymbol{u}|^2}{2} \left ( \frac {\partial f}{\partial t} \right )_{\mathrm{col}} \! \! \! \! \mathrm{d}\boldsymbol{v} = 0. \end{align}

Inserting (2.5), (2.6) in (2.7) and carrying out the $D$ -dimensional integral yields

(2.8) \begin{align} \mathrm{Trace}( \unicode{x1D64B}) = D n T, \end{align}

which is a generalised ideal gas law when the system deviates from thermal equilibrium.

Next, we perform the standard perturbative expansion for wave analyses, $f = f_0 + f_1$ , $n = n_0 + n_1$ , $ \unicode{x1D64B} = \unicode{x1D64B}_0 + \unicode{x1D64B}_1$ , $\boldsymbol{F} = \boldsymbol{F}_0 + \boldsymbol{F}_1$ and $\boldsymbol{u} = \boldsymbol{u}_1$ . This allows (2.1), (2.3) and (2.4) to be expanded in order 0 and 1 terms as follows:

(2.9) \begin{align} \boldsymbol{F}_0 \boldsymbol{\cdot } \frac {\partial f_0}{\partial \boldsymbol{v}} = 0, \end{align}

(2.10) \begin{align} \frac {\partial f_1}{\partial t} + \boldsymbol{v} \boldsymbol{\cdot } \frac {\partial f_1}{\partial \boldsymbol{x}} + \frac {\boldsymbol{F}_1}{m} \boldsymbol{\cdot } \frac {\partial f_0}{\partial \boldsymbol{v}} + \frac {\boldsymbol{F}_0}{m} \boldsymbol{\cdot } \frac {\partial f_1}{\partial \boldsymbol{v}} = \left ( \frac {\partial f}{\partial t} \right )_{\mathrm{col},1}, \end{align}

(2.11) \begin{align} \frac {\partial n_1}{\partial t} + n_0 \frac {\partial }{\partial \boldsymbol{x}} \boldsymbol{\cdot } \boldsymbol{u}_1 = 0, \end{align}

(2.12) \begin{align} \langle \boldsymbol{F}_0 \rangle = \boldsymbol{0}, \end{align}

(2.13) \begin{align} n_0 \frac {\partial \boldsymbol{u}_1}{\partial t} + \frac {1}{m} \frac {\partial }{\partial \boldsymbol{x}} \boldsymbol{\cdot } \unicode{x1D64B}_1 = \frac {n_0 \langle \boldsymbol{F}_1 \rangle }{m}. \end{align}

We can further Fourier–Laplace transform the order-1 equations to obtain

(2.14) \begin{align} (-\mathrm{i} \omega + \mathrm{i}\boldsymbol{k}\boldsymbol{\cdot } \boldsymbol{v}) \hat {f}_1 + \frac {\,\hat{\!\boldsymbol{F}}_1}{m} \boldsymbol{\cdot } \frac {\partial f_0}{\partial \boldsymbol{v}} + \frac {\boldsymbol{F}_0}{m} \boldsymbol{\cdot } \frac {\partial \hat {f}_1}{\partial \boldsymbol{v}} = \hat {\left ( \frac {\partial f}{\partial t} \right )}_{\mathrm{col},1}, \end{align}

(2.15) \begin{align} -\mathrm{i}\omega \hat {n}_1 + \mathrm{i} n_0 \boldsymbol{k} \boldsymbol{\cdot } \hat {\boldsymbol{u}}_1 = 0, \end{align}

(2.16) \begin{align} -\mathrm{i} \omega n_0 \hat {\boldsymbol{u}}_1 + \frac {\mathrm{i}}{m} \boldsymbol{k} \boldsymbol{\cdot }\,\, \hat{\!\!\unicode{x1D64B}}_1 = \frac {n_0 \langle \,\hat{\!\boldsymbol{F}}_1 \rangle }{m}. \end{align}

Unlike regular Fourier transforms, the Fourier–Laplace transform allows complex $\omega$ and wave vectors, $\boldsymbol{k}$ , while neglecting the boundary terms of Laplace transforms, permitting us to study general dispersion relations without specific boundary or initial conditions.

Although the above set of equations is required to describe the general case with an order-0 macroscopic force, such as a plasma embedded in an external magnetic field, we are only interested in the simple case of longitudinal waves in an isotropic medium. Thus, we can assume $\boldsymbol{F}_0 = 0$ and let $\boldsymbol{F}_1$ be independent of $\boldsymbol{v}$ , such that $\langle \,\hat{\!\boldsymbol{F}}_1 \rangle = \,\hat{\!\boldsymbol{F}}_1$ . Additionally, $\hat {\boldsymbol{u}}_1 \propto \boldsymbol{k}$ , meaning that (2.15) implies $\hat {\boldsymbol{u}}_1 = (\omega / k) (\hat {n}_1 / n_0) \boldsymbol{e}_k$ , where $\boldsymbol{e}_k = \boldsymbol{k}/k$ . If we finally assume that $\,\,\hat{\!\!\unicode{x1D64B}}_1 = \hat {p}_{k1} \boldsymbol{e}_k \boldsymbol{e}_k + \hat {p}_{k\perp 1} (\boldsymbol{1} - \boldsymbol{e}_k \boldsymbol{e}_k)$ , (2.16) gives

(2.17) \begin{align} \,\hat{\!\boldsymbol{F}}_1 = - \mathrm{i} m \frac {\omega ^2}{k} \frac {\hat {n}_1}{n_0} \boldsymbol{e}_k + \frac {\mathrm{i} k \hat {p}_{k1}}{n_0} \boldsymbol{e}_k ; \end{align}

the above form of $\,\,\hat{\!\!\unicode{x1D64B}}_1$ can be justified by showing its consistency with the $\hat {f}_1$ resulting from the analysis. Taking $f_0$ to be Maxwellian, $f_0 = n_0 \, \mathrm{e}^{-v^2 / v_{0}^2} / (\pi ^{1/2} v_{T0})^D$ , recalling that $v_{T0} = \sqrt {2 T_0 / m}$ and substituting the above results into (2.14) then yields

(2.18) \begin{align} \mathrm{i} (k v_k - \omega ) \hat {f}_1 = \hat {\left ( \frac {\partial f}{\partial t} \right )}_{\mathrm{col},1} + \frac {\mathrm{i} 2 v_k}{v_{T0}^2} \left ( \frac {k \hat {p}_{k1}}{m n_0} - \frac {\omega ^2}{k} \frac {\hat {n}_1}{n_0} \right ) f_0 , \end{align}

where $v_k$ is the $\boldsymbol{e}_k$ component of $\boldsymbol{v}$ . To evaluate $\hat {(\partial f / \partial t)}_{\mathrm{col,1}}$ based on (2.5), we first evaluate the order-1 term of $f_R = f_0 + f_{R1}$ from (2.6):

(2.19) \begin{align} f_{R1} = \left [ \frac {n_1}{n_0} + \frac {2 v_k u_1}{v_{T0}^2} + \left (\frac {v^2}{v_{T0}^2} - \frac {D}{2} \right ) \frac {T_1}{T_0} \right ] f_0 . \end{align}

Using (2.8) and $\,\,\hat{\!\!\unicode{x1D64B}}_1 = \hat {p}_{k1} \boldsymbol{e}_k \boldsymbol{e}_k + \hat {p}_{k\perp 1} (\boldsymbol{1} - \boldsymbol{e}_k \boldsymbol{e}_k)$ yields $\hat {T}_1 = \hat {p}_{k1}/(Dn_0) + (D-1)$ $\hat {p}_{k\perp 1} / (D n_0) - (\hat {n}_1 / n_0) T_0$ . Combining this and $\hat {u}_1 = (\omega / k) (\hat {n}_1 / n_0)$ , $\hat {(\partial f / \partial t)}_{\mathrm{col,1}}$ becomes

(2.20) \begin{align} \hat {\left ( \frac {\partial f}{\partial t} \right )}_{\mathrm{col},1} = & \: \nu _0 f_0 \left \lbrace \left (\frac {D + 2}{2} - \frac {v^2}{v_{T0}^2} + \frac {2 v_k \omega }{v_{T0}^2 k} \right ) \frac {\hat {n}_1}{n_0} \right . \nonumber \\[5pt] & \left . + \left ( \frac {v^2}{D v_{T0}^2} - \frac {1}{2} \right ) \left [\frac {\hat {p}_{k1}}{n_0 T_0} + (D-1)\frac {\hat {p}_{k\perp 1}}{n_0 T_0} \right ] \right \rbrace - \nu _0 \hat {f}_1 , \end{align}

with $\nu _0$ being $\nu$ evaluated at $n_0$ and $T_0$ . Taking $\hat {(\partial f / \partial t)}_{\mathrm{col,1}}$ from (2.20), (2.18) may be recast as

(2.21) \begin{align} \hat {f}_1 & = \dfrac {f_0}{\mathrm{i}(k v_k - \omega _R)} \left \lbrace \left [ \nu _0 \left (\dfrac {D + 2}{2} - \dfrac {v^2}{v_{T0}^2} \right ) - \mathrm{i} \omega \dfrac {2 v_k \omega _R }{v_{T0}^2 k} \right ] \dfrac {\hat {n}_1}{n_0} \right .\nonumber \\[5pt] & \quad\left . + \left [\mathrm{i}k v_k + \nu _0 \left (\dfrac {v^2}{D v_{T0}^2} - \dfrac {1}{2} \right ) \right ] \dfrac {\hat {p}_{k1}}{n_0 T_0} + (D-1) \nu _0 \left (\dfrac {v^2}{D v_{T0}^2} - \dfrac {1}{2} \right ) \dfrac {\hat {p}_{k\perp 1}}{n_0 T_0} \right \rbrace , \end{align}

where $\omega _R = \omega + \mathrm{i}\nu _0$ . This form of $\hat {f}_1$ is consistent with the off-diagonal elements of $\,\,\hat{\!\!\unicode{x1D64B}}_1$ vanishing, since it is an even function of $\boldsymbol{v}_{k\perp }$ (the part of $\boldsymbol{v}$ perpendicular to $\boldsymbol{k}$ ).

To derive $\gamma$ from (2.21), we use the relation of $\hat {f}_1$ to the order-1 moments, $\hat {n}_1$ and $\hat {p}_{k\perp 1}$ , to obtain a single equation connecting $\hat {p}_{k 1}$ and $\hat {n}_1$ , as done by Stubbe (Reference Stubbe1994). Specifically, since polytropic behaviour implies $p_1 / p_0 = \gamma (n_1 / n_0)$ , we look for an expression like

(2.22) \begin{align} \frac {\hat {p}_{k 1}}{n_0 T_0} = \gamma \frac {\hat {n}_1}{n_0}, \end{align}

remembering that $p_0 = n_0 T_0$ based on (2.8). We note that (2.22) only describes local polytropic behaviour in limits where $\gamma$ can be considered independent of $\omega$ and $\boldsymbol{k}$ , as a dependence on these parameters in Fourier–Laplace space implies a non-local temporal and spatial response, respectively (Stubbe Reference Stubbe1994; Swanson Reference Swanson2003).

For $\hat {n}_1$ , it holds that

(2.23) \begin{align} \hat {n}_1 = \int \hat {f}_1 \, \mathrm{d}\boldsymbol{v}. \end{align}

We first evaluate the integral on the right-hand side over $\boldsymbol{v}_{k \perp }$ , using the $(D-1)$ -dimensional Gaussian integrals (valid for $D \geqslant 2$ )

(2.24) \begin{align} \int \mathrm{e}^{-v_{k\perp }^2 / v_{T0}^2} \, \mathrm{d}\boldsymbol{v}_{k\perp } = \pi ^{(D-1)/2} v_{T0}^{D-1}, \end{align}

(2.25) \begin{align} \int v_{k\perp }^2 \, \mathrm{e}^{-v_{k\perp }^2 / v_{T0}^2} \, \mathrm{d}\boldsymbol{v}_{k\perp } = \frac {D-1}{2} \pi ^{(D-1)/2} v_{T0}^{D+1}, \end{align}

while recalling that $v^2 = v_k^2 + v_{k\perp }^2$ . With this, (2.23) yields

(2.26) \begin{align} \hat {n}_1 & = \frac {n_0}{\sqrt {\pi }} \int \frac {\mathrm{e}^{-v_k^2 / v_{T0}^2}}{\mathrm{i}(k v_k - \omega _R)} \left \lbrace \left [ \nu _0 \left (\frac {3}{2} - \frac {v_k^2}{v_{T0}^2} \right ) - \mathrm{i}\omega \frac {2 v_k \omega _R}{v_{T0}^2 k} \right ] \frac {\hat {n}_1}{n_0}\right .\nonumber \\[5pt] & \quad\left . + \left [\mathrm{i}k v_k + \frac {\nu _0}{D} \left (\frac {v_k^2}{v_{T0}^2} - \frac {1}{2} \right ) \right ] \frac {\hat {p}_{k1}}{n_0 T_0} + \frac {(D-1)\nu _0}{D} \left (\frac {v_k^2}{v_{T0}^2} - \frac {1}{2} \right ) \frac {\hat {p}_{k\perp 1}}{n_0 T_0} \right \rbrace \frac {\mathrm{d}v_k}{v_{T0}}. \end{align}

Although (2.24) and (2.25) are only valid for $D \geqslant 2$ , (2.26) is valid for $D = 1$ as well. Additionally, we note that assuming a strictly scalar pressure, $\hat {p}_{k1} = \hat {p}_{k\perp 1}$ , results in the elimination of $D$ from (2.26), meaning that only the $D = 1$ case can be recovered by this assumption. The integral over $v_k$ in (2.26) may be evaluated using the equality

(2.27) \begin{align} \frac {1}{\sqrt {\pi }} \int \frac {v_k^N \, \mathrm{e}^{-v_k^2 / v_{T0}^2}}{k v_k - \omega _R} \frac {\mathrm{d}v_k}{v_{T0}} = \frac {v_{T0}^{N-1}}{k} Z_N\left ( \frac {\omega _R}{k v_{T0}} \right )\!, \end{align}

where $Z_N$ is the generalised plasma dispersion function (Swanson Reference Swanson2003):

(2.28) \begin{align} Z_N(\zeta ) = \frac {1}{\sqrt {\pi }} \int \frac {\xi ^N \mathrm{e}^{-\xi ^2}}{\xi - \zeta } \, \mathrm{d}\xi , \end{align}

with the integral performed over the Landau contour (Landau Reference Landau1946). Using (2.27) allows (2.26) to be rewritten as

(2.29) \begin{align} \frac {\hat {n}_1}{n_0} & = \frac {1}{\mathrm{i}k v_{T0}} \left \lbrace \left [ \nu _0 \left ( \frac {3}{2} Z_0 - Z_2 \right ) - \mathrm{i}\omega \frac {2 \omega _R}{k v_{T0}} Z_1 \right ] \frac {\hat {n}_1}{n_0}\right . \nonumber \\[8pt] & \quad \left . + \left [ \mathrm{i}k v_{T0} Z_1 + \frac {\nu _0}{D} \left (Z_2 - \frac {1}{2} Z_0 \right ) \right ] \frac {\hat {p}_{k1}}{n_0 T_0} + \frac {(D-1) \nu _0 }{D} \left (Z_2 - \frac {1}{2} Z_0 \right ) \frac {\hat {p}_{k\perp 1}}{n_0 T_0} \right \rbrace\! , \end{align}

which can be further recast with the identities following from (B.17) of Swanson (Reference Swanson2003):

(2.30) \begin{align} Z_2 = \frac {\omega _R}{k v_{T0}} Z_1 , \quad Z_0 = \frac {k v_{T0}}{\omega _R} (Z_1 - 1) , \quad Z_2 = \frac {k v_{T0}}{\omega _R} \left (Z_3 - \frac {1}{2}\right )\!, \end{align}

to eliminate multiplicative factors involving $k$ , yielding

(2.31) \begin{align} 0 & = \: \left [ 3 \frac {\mathrm{i} \nu _0}{\omega _R} \left ( Z_3 - \frac {1}{2} Z_1 \right ) - 2 Z_3 \right ] \frac {\hat {n}_1}{n_0} + \left [Z_1 + \frac {\mathrm{i} \nu _0}{D \omega _R} \left (\frac {1}{2}Z_1 - Z_3 \right ) \right ] \frac {\hat {p}_{k1}}{n_0 T_0} \nonumber \\[5pt] & \quad + \frac {\mathrm{i} (D-1) \nu _0}{D \omega _R} \left (\frac {1}{2}Z_1 - Z_3 \right ) \frac {\hat {p}_{k\perp 1}}{n_0 T_0}. \end{align}

Next, we evaluate $\hat {p}_{k\perp 1}$ based on $\hat {f}_1$ :

(2.32) \begin{align} \hat {p}_{k\perp 1} = \int m v_{k' \neq k}^2 \hat {f}_1 \, \mathrm{d}\boldsymbol{v}, \end{align}

where $v_{k' \neq k}$ is any component of $\boldsymbol{v}$ perpendicular to $\boldsymbol{k}$ . The integral over $\boldsymbol{v}_{k \perp }$ on the right-hand side may be computed using the $(D-1)$ -dimensional Gaussian integrals (again valid for $D \geqslant 2$ , which is the only case where $p_{k\perp 1}$ is defined):

(2.33) \begin{align} \int v_{k' \neq k}^2 \, \mathrm{e}^{-v_{k\perp }^2 / v_{T0}^2} \, \mathrm{d}\boldsymbol{v}_{k\perp } = \frac {1}{2} \pi ^{(D-1)/2} v_{T0}^{D+1}, \end{align}

(2.34) \begin{align} \int v_{k\perp }^2 v_{k' \neq k}^2 \, \mathrm{e}^{-v_{k\perp }^2 / v_{T0}^2} \, \mathrm{d}\boldsymbol{v}_{k\perp } = \frac {D + 1}{4} \pi ^{(D-1)/2} v_{T0}^{D + 3}, \end{align}

giving

(2.35) \begin{align} \hat {p}_{k\perp 1} & = \frac {n_0 T_0}{\sqrt {\pi }} \int \frac {\mathrm{e}^{-v_k^2 / v_{T0}^2}}{\mathrm{i}(k v_k - \omega _R)} \left \lbrace \left [ \nu _0 \left ( \frac {1}{2} - \frac {v_k^2}{v_{T0}^2} \right ) - \mathrm{i}\omega \frac {2 v_k \omega _R}{v_{T0}^2 k} \right ] \frac {\hat {n}_1}{n_0} \right .\nonumber \\[8pt] & \quad \left . + \left [\mathrm{i} k v_{T0} + \frac {\nu _0}{D} \left (\frac {v_k^2}{v_{T0}^2} + \frac {1}{2} \right ) \right ] \frac {\hat {p}_{k1}}{n_0 T_0} + \frac {(D-1)\nu _0}{D} \left (\frac {v_k^2}{v_{T0}^2} + \frac {1}{2} \right ) \frac {\hat {p}_{k\perp 1}}{n_0 T_0} \right \rbrace \frac {\mathrm{d}v_k}{v_{T0}}.\nonumber\\[5pt] \end{align}

Evaluating the $v_k$ integral using (2.27), we find

(2.36) \begin{align} \frac {\hat {p}_{k\perp 1}}{n_0 T_0} & = \frac {1}{\mathrm{i}k v_{T0}} \left \lbrace \left [ \nu _0 \left (\frac {1}{2} Z_0 - Z_2 \right ) - \mathrm{i}\omega \frac {2 \omega _R}{k v_{T0}} Z_1 \right ] \frac {\hat {n}_1}{n_0} \right . \nonumber\\[8pt] & \quad \left . + \left [\mathrm{i} k v_{T0} Z_1 + \frac {\nu _0}{D} \left (Z_2 + \frac {1}{2} Z_0 \right ) \right ] \frac {\hat {p}_{k1}}{n_0 T_0} + \frac {(D-1)\nu _0}{D} \left (Z_2 + \frac {1}{2} Z_0 \right ) \frac {\hat {p}_{k\perp 1}}{n_0 T_0} \right \rbrace ,\nonumber\\[5pt] \end{align}

and eliminating the multiplicative factors involving $k$ with (2.30) finally yields

(2.37) \begin{align} 0 & = \left [1 - 2 Z_3 + \frac {\mathrm{i}\nu _0}{\omega _R} \left (3 Z_3 - \frac {1}{2} Z_1 - 1 \right ) \right ] \frac {\hat {n}_1}{n_0} + \left [ Z_1 + \frac {\mathrm{i} \nu _0}{D \omega _R} \left (1 - Z_3 - \frac {1}{2} Z_1 \right ) \right ] \frac {\hat {p}_{k1}}{n_0 T_0} \nonumber \\[8pt] & \quad + \left [\frac {\mathrm{i}(D-1) \nu _0}{D \omega _R} \left (1 - Z_3 - \frac {1}{2}Z_1 \right ) - 1 \right ] \frac {\hat {p}_{k\perp 1}}{n_0 T_0}. \end{align}

Using (2.37) to express $\hat {p}_{k\perp 1}/(n_0 T_0)$ in terms of $\hat {p}_{k1}/(n_0 T_0)$ and $\hat {n}_1 / n_0$ , then substituting the resulting expression into (2.31), gives

(2.38) \begin{align} \frac {\hat {p}_{k1}}{n_0 T_0} = \frac {\left \lbrace\begin{array}{c} 2 Z_3 + 3 ({\mathrm{i}\nu _0}/{\omega _R}) \! \left[(Z_1/2) - Z_3 \right] + [{\mathrm{i}(D-1)\nu _0}/({D \omega _R})] \! \left ( 2 Z_1 Z_3 - Z_3 - Z_1/2 \right )\\ + \big[{(D-1)\nu _0^2}/({D \omega _R^2})\big] \! \left (Z_1 - 2 Z_3 \right )(1 - Z_1) \end{array}\right \rbrace}{Z_1 + [{\mathrm{i}\nu _0/(D \omega _R})] \! \left[ (Z_1/2) - Z_3 \right] + [{\mathrm{i}(D-1) \nu _0}/({D \omega _R})] Z_1 (Z_1 - 1)} \frac{\hat {n}_1}{n_0}, \end{align}

from which (2.22) implies that

(2.39) \begin{align} \gamma = \frac {\left \lbrace\begin{array}{c} 2 Z_3 + 3 ({\mathrm{i}\nu _0}/{\omega _R})\! \left[(Z_1/2) - Z_3 \right] + [{\mathrm{i}(D-1)\nu _0}/({D \omega _R})] \! \left ( 2 Z_1 Z_3 - Z_3 - Z_1/2 \right )\\ + \big[{(D-1)\nu _0^2}/({D \omega _R^2})\big] \! \left (Z_1 - 2 Z_3 \right )(1 - Z_1) \end{array}\right \rbrace}{Z_1 + [{\mathrm{i}\nu _0/(D \omega _R})] \! \left[ (Z_1/2) - Z_3 \right] + [{\mathrm{i}(D-1) \nu _0}/({D \omega _R})] Z_1 (Z_1 - 1)}. \end{align}

A more illuminating form of $\gamma$ is obtained by substituting $\omega _R = \omega + \mathrm{i}\nu _0$ into (2.39):

(2.40) \begin{align} \gamma = \frac {\omega ^2 2 Z_3 + ({\mathrm{i}\nu _0 \omega }/{D}) \! \left \lbrace [ ({2D + 1})/{2}]Z_1 + Z_3 + 2(D-1) Z_1 Z_3 \right \rbrace + \left({\nu _0^2}/{D}\right) \! \left [Z_3 - ({3}/{2})Z_1 - (D-1) Z_1^2 \right ]}{\omega ^2 Z_1 + ({\mathrm{i}\nu _0 \omega }/{D}) \! \left\lbrace [({2D + 3})/{2}] Z_1 - Z_3 + (D-1) Z_1^2 \right\rbrace + \left({\nu _0^2}/{D}\right) \! \left[Z_3 - ({3}/{2})Z_1 - (D-1) Z_1^2 \right]}.\nonumber\\[5pt] \end{align}

The above $\gamma$ is fully kinetic and valid for $D \geqslant 1$ . Therefore, this $\gamma$ can be used to investigate the polytropic behaviour of longitudinal waves in one-component systems with arbitrary $\omega$ , $\nu _0$ , $k$ , $v_{T0}$ and $D$ . We first note that for slow phenomena ( $\omega \to 0$ ), we always obtain $\gamma \to 1$ . This corresponds to the isothermal limit and agrees with the result using a more advanced relaxation collision model for $D = 3$ by Stubbe (Reference Stubbe1994), as well as the intuitive expectation based on the ideal gas law for constant $T$ . When $\nu _0 \to 0$ , we find

(2.41) \begin{align} \gamma \to \frac {2 Z_3}{Z_1}, \end{align}

which agrees with the collisionless analysis of Swanson (Reference Swanson2003). In the isothermal limit of $\omega / (k v_{T0}) \to 0$ , (2.41) yields $\gamma \to 1$ , reproducing the isothermal collisional result. The collisionless adiabatic limit of $\omega /(k v_{T0}) \to \infty$ results in $\gamma \to 3$ based on (2.41), corresponding to the usual collisionless adiabatic coefficient from plasma physics. In the more general adiabatic limit at finite $\nu _0$ , we may substitute the limiting values of $2 Z_3 / Z_1 \to 3$ , $Z_3, Z_1 \to 0$ into (2.40) to obtain

(2.42) \begin{align} \gamma \to \frac {3 \omega + \mathrm{i}\nu _0 [{(D + 2)}/{D}]}{\omega + \mathrm{i}\nu _0}. \end{align}

This adiabatic coefficient reduces to the collisionless value of $\gamma \to 3$ for $\nu _0 / \omega \to 0$ , but also recovers the usual fluid value of $\gamma \to (D+2)/D$ for $\nu _0 / \omega \to \infty$ . The above result clearly supports the argument that the adiabatic coefficient involving all $D$ is appropriate at high $\nu _0 / \omega$ , while that corresponding to $D = 1$ is appropriate for low $\nu _0 / \omega$ (Spitzer Reference Spitzer1956; Nicholson Reference Nicholson1983; Goldston & Rutherford Reference Goldston and Rutherford1995; Stacey Reference Stacey2012; Salewski Reference Salewski2027).

Repeating the same analysis, but eliminating $\hat {p}_{k 1} / (n_0 T_0)$ instead of $\hat {p}_{k \perp 1}/ (n_0 T_0)$ , we can derive an alternative polytropic index, $\gamma _\perp$ , appearing in the equation

(2.43) \begin{align} \frac {\hat {p}_{k\perp 1}}{n_0 T_0} = \gamma _\perp \frac {\hat {n}_1}{n_0}. \end{align}

The value of $\gamma _\perp$ is found to be

(2.44) \begin{align} \gamma _\perp = \frac {\omega ^2 Z_1 + ({\mathrm{i}\nu _0 \omega }/{D}) \! \left \lbrace DZ_1^2 +[({2D + 1})/{2}]Z_1 + Z_3(1 - 2 Z_1) \right \rbrace + \left({\nu _0^2}/{D}\right) \! \left [Z_3 - ({3}/{2})Z_1 - (D-1)Z_1^2 \right ]}{\omega ^2 Z_1 + ({\mathrm{i}\nu _0 \omega }/{D}) \! \left \lbrace [({2D + 3})/{2}] Z_1 - Z_3 + (D-1) Z_1^2 \right \rbrace + \left({\nu _0^2}/{D}\right) \! \left [Z_3 - ({3}/{2})Z_1 - (D-1) Z_1^2 \right ]}.\nonumber\\[5pt] \end{align}

In the isothermal ( $\omega \to 0$ ) and collisionless ( $\nu _0 \to 0$ ) limits,

(2.45) \begin{align} \gamma _\perp \to 1 . \end{align}

The isothermal behaviour of $\hat {p}_{k\perp 1}$ in the absence of collisions makes sense, since this expresses the fact that changes in $p_{k\perp }$ are entirely due to changes in $n$ , while no energy is transferred to the degrees of freedom associated with $p_{k\perp }$ . In the adiabatic limit,

(2.46) \begin{align} \gamma _\perp \to \frac {\omega + \mathrm{i}\nu _0 [{(D + 2)}/{D}]}{\omega + \mathrm{i}\nu _0}, \end{align}

which again yields $\gamma _\perp \to 1$ for $\nu _0/\omega \to 0$ and $\gamma \to (D + 2) / D$ for $\nu _0 / \omega \to \infty$ . The correspondence of $\gamma$ and $\gamma _\perp$ in the isothermal and high- $\nu _0 / \omega$ adiabatic limits indicates that the usual assumption of a scalar pressure is justified in these cases. By contrast, the deviation of $\gamma$ from $\gamma _\perp$ in the low- $\nu _0 / \omega$ adiabatic limit indicates that the tensor nature of pressure cannot be ignored in this case. Additionally, $\gamma \to 3$ , $\gamma _\perp \to 1$ mimics the double adiabatic law in non-relativistic, collisionless, magnetised plasmas (Chew et al. Reference Chew, Goldberger and Low1956). We finally note that the adiabatic coefficient for the scalar pressure, $\mathrm{Trace}(\,\,\hat{\!\!\unicode{x1D64B}}_1)/D = \hat {p}_{k1} / (D n_0 T_0) + (D-1) \hat {p}_{k\perp 1} / (D n_0 T_0) = \{[\gamma + (D-1) \gamma _\perp ] / D \} \hat {n}_1 / n_0$ , is $(D+2)/D$ for all $\nu _0 / \omega$ . This also mimics the general result for a non-relativistic, collisionless, magnetised plasma (Belmont & Mazelle Reference Belmont and Mazelle1992), indicating that it may be valid for any adiabatic process involving $\leqslant D$ degrees of freedom in $D$ -dimensional (velocity) space.

Figure 1.

Behaviour of $\mathrm{Re}(2 Z_3 / Z_1)$ (upper left), $\mathrm{Im}(2 Z_3 / Z_1)$ (upper right), $|Z_3|$ (lower left) and $|Z_1|$ (lower right) versus $\mathrm{Re}[\omega _R / (k v_{T0})]$ and $\mathrm{Im}[\omega _R / (k v_{T0})]$ . The adiabatic limit of $2 Z_3 / Z_1 \to 3$ and $Z_3, Z_1 \to 0$ is found for both $\mathrm{Re}[\omega _R / (k v_{T0})] \gg 1$ and $\mathrm{Im}[\omega _R / (k v_{T0})] \gg 1$ .

Figure 1. Long description

Panel A: A heat map shows the real part of the ratio of two variables, Re(2Z3/Z1), plotted against the real and imaginary parts of another variable, ωR, normalized by kvT0. The color scale ranges from 1 to 3.5. Panel B: A heat map displays the imaginary part of the ratio of two variables, Im(2Z3/Z1), plotted against the real and imaginary parts of ωR, normalized by kvT0. The color scale ranges from -1.5 to 0. Panel C: A heat map illustrates the magnitude of a variable, |Z3|, plotted against the real and imaginary parts of ωR, normalized by kvT0. The color scale ranges from 0 to 0.8. Panel D: A heat map shows the magnitude of another variable, |Z1|, plotted against the real and imaginary parts of ωR, normalized by kvT0. The color scale ranges from 0 to 1.

To gain further insight into when collisionless and collisional adiabatic responses may be expected, we compute $Z_1$ and $Z_3$ using the recurrence relations of Swanson (Reference Swanson2003):

(2.47) \begin{align} & Z_1\left ( \frac {\omega _R}{k v_{T0}} \right ) = 1 + \mathrm{i}\sqrt {\pi } \, \frac {\omega _R}{k v_{T0}} \, w\! \left ( \frac {\omega _R}{k v_{T0}} \right ), \nonumber \\[5pt] & Z_3\left ( \frac {\omega _R}{k v_{T0}} \right ) = \frac {1}{2} + \left ( \frac {\omega _R}{k v_{T0}} \right )^2 + \mathrm{i}\sqrt {\pi } \left ( \frac {\omega _R}{k v_{T0}} \right )^3 w\! \left ( \frac {\omega _R}{k v_{T0}} \right ), \end{align}

where $w = Z_0 / (\mathrm{i}\sqrt {\pi })$ is the Faddeeva function (Gautschi Reference Gautschi1970; Abrarov & Quine Reference Abrarov and Quine2011). We specifically employ the Matlab implementation of $w$ from Abrarov (Reference Abrarov2016), which utilises the approximations of Gautschi (Reference Gautschi1970) and Abrarov & Quine (Reference Abrarov and Quine2011) to obtain accurate results throughout the complex plane. Based on the behaviour of $2 Z_3 / Z_1$ , $|Z_3|$ and $|Z_1|$ with complex arguments in figure 1, we conclude that the adiabatic limit is obtained when $|\omega _R /(k v_{T0})| \to \infty$ , independently of whether $\mathrm{Re}[\omega _R / (k v_{T0})]$ or $\mathrm{Im}[\omega _R / (k v_{T0})]$ dominates. Thus, in addition to the collisionless adiabatic limit from plasma physics, $\omega /(k v_{T0}) \to \infty$ , which corresponds to phase velocities, $\omega / k$ , far larger than $v_{T0}$ , a collisional adiabatic limit exists when $\nu _0 / (k v_{T0}) \to \infty$ . This adiabatic limit corresponds to a mean free path ( ${\sim}v_{T0} / \nu _0$ ) far shorter than the wavelength ( ${\sim}1 / k$ ) and is essential for propagation of sound waves in neutral gases, which have $\omega /(k v_{T0}) = \sqrt {\gamma / 2} \sim 1$ based on (1.4). The $\gamma$ of (2.40) is identical for charged and uncharged particles, meaning that sound waves in neutral gases experience Landau damping, similar to that of longitudinal plasma waves, when $\omega / (k v_{T0}) \sim 1$ at low $\nu _0 / \omega$ , as noted by Stubbe (Reference Stubbe1994) and Sukhorukov & Stubbe (Reference Sukhorukov and Stubbe1995). However, an adiabatic response with negligible Landau damping is recovered even for $\omega / (k v_{T0}) \sim 1$ if $\nu _0 / (k v_{T0}) \to \infty$ . This may be understood heuristically by noting that $\nu _0 / (k v_{T0}) \to \infty$ results in resonant particles for Landau damping, satisfying $\omega = \boldsymbol{k}\boldsymbol{\cdot } \boldsymbol{v}$ , being scattered within a distance far less than a wavelength and on a time scale far shorter than the wave period. The rapid scattering prevents particles from exchanging energy with the wave through the Landau mechanism, since the resonance condition cannot be maintained on the length and time scales of the wave.

We finally note that $\nu _0 / (k v_{T0}) \to \infty$ may also hold in the isothermal limit. Thus, an additional requirement for the adiabatic limit to apply at high $\nu _0 / \omega$ is that the $\nu _0 \omega$ terms dominate over the $\nu _0^2$ terms in (2.40). Since $2 Z_3 / Z_1 \to 3 + \mathcal{O}(|k v_{T0} / \nu _0|^2)$ and $Z_3 , Z_1 \to \mathcal{O}(|k v_{T0} / \nu _0|^2)$ when $\nu _0 / (k v_{T0}) \to \infty$ at high $\nu _0 / \omega$ , based on the asymptotic approximations in Swanson (Reference Swanson2003), the ratio of the $\nu _0 \omega$ terms to the $\nu _0^2$ terms in (2.40) is $\sim (\omega / \nu _0) |\nu _0 / (k v_{T0})|^2 = |\omega / (k v_{T0})| |\nu _0 / (k v_{T0}))|$ . An adiabatic response at high $\nu _0 / (k v_{T0})$ , therefore, requires $|\omega / (k v_{T0})| \gg |k v_{T0} / \nu _0| \ll 1$ . This requirement is always satisfied by sound waves, for which $\omega / (k v_{T0}) = \sqrt {\gamma / 2} \sim 1$ , when $\nu _0 / (k v_{T0}) \to \infty$ .

Figure 2.

Plots of $\sqrt {5/3} \, \mathrm{Re}(\kappa )$ (upper) and $\sqrt {5/3} \, \mathrm{Im}(\kappa )$ (lower) versus $r$ for monatomic gases. Results obtained by solving $\kappa ^2 \gamma = 1$ , with $\gamma$ from (2.40) and $D = 3$ , are shown using solid and dashed lines; the dashed lines only include $\mathrm{Re}(\kappa )$ when computing $\gamma$ , as done by Stubbe (Reference Stubbe1994). The scattered points show the experimental results of Greenspan (Reference Greenspan1956), Meyer & Sessler (Reference Meyer and Sessler1957) and Schotter (Reference Schotter1974) for comparison, while the dotted line indicates $r = 0.04$ below which the experiment cannot be described using a single Fourier–Laplace mode.

3. Sound waves in neutral gases

The transition from $\gamma \to 3$ at low $\nu _0/\omega$ to $\gamma \to (D + 2)/D$ at high $\nu _0/\omega$ is not directly observable for longitudinal waves in typical one-component systems. However, the developed theory does permit the study of sound wave propagation in rarefied monatomic, diatomic and polyatomic gases, of which the former two are discussed in this section. In a neutral gas macroscopic forces are negligible, so $\,\hat{\!\boldsymbol{F}}_1 = 0$ , which recasts (2.17) as $\hat {p}_{k1} = m \hat {n}_1 (\omega / k)^2$ . When combined with (2.22), $\hat {p}_{k1} = \gamma T_0 \hat {n}_1$ , this yields a dispersion relation for sound waves identical to the one based on (1.4):

(3.1) \begin{align} \frac {2 \omega ^2}{k^2 v_{T0}^2} = \gamma , \end{align}

which can be rewritten using a normalised wavenumber, $\kappa = k v_{T0} / (\sqrt {2} \omega )$ , and $r = \nu _0 /\omega$ :

(3.2) \begin{align} \kappa ^2 \gamma (\kappa ,r) = 1, \end{align}

where the argument of $Z_1$ and $Z_3$ in $\gamma$ is $(1 + \mathrm{i}r)/(\sqrt {2} \kappa )$ . Experimental investigations of sound waves in rarefied gases typically employ a transducer operating at a fixed $\omega$ and observe the dispersion and damping away from the transducer (Greenspan Reference Greenspan1956, Reference Greenspan1959; Meyer & Sessler Reference Meyer and Sessler1957; Schotter Reference Schotter1974). We, therefore, assume $\omega$ (and $r$ ) to be a real number, while $k$ (and $\kappa$ ) is a complex number, with $\mathrm{Im}(k)$ describing spatial wave damping. The behaviour of $\mathrm{Re}(\kappa )$ and $\mathrm{Im}(\kappa )$ versus $r$ based on (3.2) and (2.40) for a monatomic gas ( $D = 3$ ) is shown in figure 2, which appears virtually identical to the BGK results of Marques (Reference Marques1999). Similarly, the behaviour for a diatomic gas with two active rotational degrees of freedom ( $D = 5$ ) is shown in figure 3. In addition to the solution of (3.2), figures 2 and 3 also contain solutions of $\kappa ^2 \gamma [\mathrm{Re}(\kappa ),r] = 1$ to allow comparisons with Stubbe (Reference Stubbe1994). The experimental results of Greenspan (Reference Greenspan1956), Meyer & Sessler (Reference Meyer and Sessler1957) and Schotter (Reference Schotter1974) for monatomic gases are shown in figure 2, while figure 3 displays the results of Meyer & Sessler (Reference Meyer and Sessler1957) and Greenspan (Reference Greenspan1959) for diatomic gases at room temperature. To compare the experimental rarefication parameters with the $r$ defined above, we note that based on (2.5), (2.6), (2.8) and $ \unicode{x1D64B} = \int (\boldsymbol{v} - \boldsymbol{u}) (\boldsymbol{v} - \boldsymbol{u}) f \, \mathrm{d}\boldsymbol{v}$ ,

(3.3) \begin{align} \left ( \frac {\partial \unicode{x1D64B}}{\partial t} \right )_{\mathrm{col}} = \nu \left [\frac {\mathrm{Trace}(\unicode{x1D64B}\kern0.5pt)}{D} \boldsymbol{1} - \unicode{x1D64B} \right ]. \end{align}

Comparing this with equation (7) of Stubbe (Reference Stubbe1994), we conclude that the experimental rarefication parameter equals $r$ for the BGK collision model, which also agrees with Marques (Reference Marques1999, Reference Marques2004). The main features of sound wave propagation and damping in monatomic and diatomic gases are reproduced by the BGK collision model in figures 2 and 3, respectively.

Figure 3.

Plots of $\sqrt {7/5} \, \mathrm{Re}(\kappa )$ (upper) and $\sqrt {7/5} \, \mathrm{Im}(\kappa )$ (lower) versus $r$ for diatomic gases. Results obtained by solving $\kappa ^2 \gamma = 1$ , with $\gamma$ from (2.40) and $D = 5$ , are shown using solid and dashed lines; the dashed lines only include $\mathrm{Re}(\kappa )$ when computing $\gamma$ , following Stubbe (Reference Stubbe1994). The scattered points show the experimental results of Meyer & Sessler (Reference Meyer and Sessler1957) and Greenspan (Reference Greenspan1959).

Specifically, $\mathrm{Re}(\kappa ) \to \sqrt {D/(D+2)}$ for $r \to \infty$ , indicating that the usual collisional adiabatic sound wave dispersion relation is recovered in this limit, i.e. $\omega /\mathrm{Re}(k) = \sqrt {[(D+2)/D](T_0 /m)}$ . When $0.1 \lesssim r \lesssim 10$ , $\mathrm{Re}(\kappa )$ decreases relative to the $r \to \infty$ limit for both monatomic and diatomic gases, in rough agreement with the experimental observations. Disagreement between the experimental and theoretical $\mathrm{Re}(\kappa )$ is mainly observed for $r \lt 0.1$ in figure 2, where the experimental $\mathrm{Re}(\kappa )$ approaches a constant value, while the theoretical $\mathrm{Re}(\kappa )$ continues to decrease. The reason for this disagreement is that the heavily damped oscillations at $r \lt 0.1$ cannot be described using a single Fourier–Laplace mode, as assumed in (3.2). Instead, the $\kappa$ value in this limit is determined by the specific experimental boundary conditions (Meyer & Sessler Reference Meyer and Sessler1957; Buckner & Ferziger Reference Buckner and Ferziger1966; Sukhorukov & Stubbe Reference Sukhorukov and Stubbe1995). For a monatomic gas, a lower limit of $r = 0.04$ is identified for the description using a single Fourier–Laplace mode (Meyer & Sessler Reference Meyer and Sessler1957). Figure 2 shows $r = 0.04$ , which coincides closely with the point where the theoretical $\mathrm{Re}(\kappa )$ starts deviating from the experimental value. The diatomic measurements in figure 3 only cover $r \gt 0.1$ and, therefore, no low- $r$ deviation is visible.

We similarly find that $\mathrm{Im}(\kappa ) \propto 1/r$ when $r \to \infty$ for both monatomic and diatomic gases. This corresponds to $\mathrm{Im}(k) \propto \omega ^2$ , in agreement with the fluid mechanical result (Landau & Lifshitz Reference Landau and Lifshitz1959). We note that the theoretical values of $\mathrm{Im}(\kappa )$ for $r \gg 1$ in figures 2 and 3 are slightly lower than the experimental ones. This is explained by the fact that the simple BGK collision model does not produce an accurate Prandtl number (Marques Reference Marques1999), as the discrepancy vanishes for more advanced collision models. When $r \lesssim 1$ , $\mathrm{Im}(\kappa )$ becomes insensitive to $r$ , approaching a relatively high approximately constant value as $r \to 0$ . The BGK collision model recovers this behaviour, which cannot be reproduced by fluid models, as the dominant damping mechanism is Landau damping (Stubbe Reference Stubbe1994; Sukhorukov & Stubbe Reference Sukhorukov and Stubbe1995). We note that the model considering only $\mathrm{Re}(\kappa )$ when computing $\gamma$ produces values of $\mathrm{Im}(\kappa )$ that are somewhat closer to the experimental ones for $r \ll 1$ . Based on this, the ability of the simple BGK collision model to compute $\mathrm{Im}(\kappa )$ appears to be on a par with that of the relaxation collision model used by Stubbe (Reference Stubbe1994). However, we note that the maximum of $\mathrm{Im}(\kappa )$ at $r \sim 1$ , as well as more accurate values of $\mathrm{Re}(\kappa )$ for $r \gt 0.04$ , may be obtained from the advanced collision models of Marques (Reference Marques1999, Reference Marques2004) with a $\gamma$ using the full $\kappa$ .

In summary, the simple collision model used in this work is capable of reproducing the qualitative features of sound wave dispersion and damping in rarefied monatomic and diatomic gases, while more advanced models allow quantitative agreement.