2.1. Linearised Boltzmann equation

It is convenient to introduce the following dimensionless variables:

(2.6) \begin{align} \begin{aligned} \tilde {x}_2&=\frac {x_2}{\lambda }, \quad (\tilde {\boldsymbol{v}}, \tilde {\boldsymbol{u}}, \tilde {\boldsymbol{c}})=\frac {(\boldsymbol{v},\boldsymbol{u},\boldsymbol{c})}{v_m}, \quad (\tilde {t},\tilde {\tau })=\frac {v_m}{\lambda }(t,\tau ), \quad \tilde {a}_2=\frac {\lambda }{v_m^2}\frac {F_2}{m}, \\[3pt] \tilde {\omega }_{\text{min}}&={\omega }_{\text{min}}\frac {\lambda }{v_m}, \quad \tilde {\beta }=\beta \frac {\lambda ^2}{v^2_m}, \quad \tilde {n}=\frac {n}{n_0}, \quad \tilde {f}=\frac {v_m^3}{n_0}f, \end{aligned} \end{align}

where $n_0$ is the average number density of gas molecules, $\lambda =2\pi /k_L$ is the scattering wave length, $v_m=\sqrt {2k_BT_0/m}$ is the most probable speed at the reference temperature $T_0$ . For simplicity, the tilde will be removed.

Since the normalised acceleration in CRBS is very small, the distribution function can be expressed as $f(t,x_2,\boldsymbol{v})=f_{eq}(\boldsymbol{v})+ 2({\pi {}I_0\alpha }/{c\epsilon _0k_BT_0}) h(t,x_2,\boldsymbol{v})$ , where $f_{eq}(\boldsymbol{v})=\pi ^{-3/2}\exp (-v^{2})$ is the global equilibrium distribution function. Furthermore, for the optical lattice (2.1), a Fourier transform can be applied in the $x_2$ direction. As a result, the Boltzmann equation reduces to the following ordinary differential equation with respect to time (note that here we use the symbol $h(t,\boldsymbol{v})$ to represent the Fourier transform of $h(t,x_2,\boldsymbol{v})$ ):

(2.7) \begin{equation} \frac {\partial {h}}{\partial t} +2\pi {i}v_2{h} -\exp \left [i\left (\omega _{\text{min}}t+\frac {\beta }{2}t^2\right )\right ]v_2f_{eq} =\underbrace {\mathcal{L}^+(h) - \nu _{eq}(\boldsymbol{v}){h}}_{\mathcal{L}(h)}, \end{equation}

where $i$ is the imaginary unit, $ \mathcal{L}^+(h)=\iint B' [f_{eq}(\boldsymbol{v}')h({\boldsymbol{v}}'_{\ast })+f_{eq}(\boldsymbol{v}'_\ast )h({\boldsymbol{v}}')-f_{eq}(\boldsymbol{v})h({\boldsymbol{v}}_\ast )]\,\text{d}\Omega \,\text{d}{\boldsymbol{v}}_\ast$ is the gain term of the linearised Boltzmann collision operator, with $B'=n_0\lambda {v^{1-2\omega }_m} B_0(\omega )\times \sin ^{\frac {1-2\omega }{2}}(\theta ) \times {v}_r^{2(1-\omega )}$ , and

(2.8) \begin{equation} \nu _{eq}(\boldsymbol{v})=\iint {}B'f_{eq}(\boldsymbol{v}_{\ast }) \,\text{d}\Omega { \,\text{d}\boldsymbol{v}_\ast } \end{equation}

is the equilibrium collision frequency.

The constant $B_0(\omega )$ is determined by the shear viscosity via the equation $\mu (T_0)={mv_m} \int h_\mu (v)v_1v_2\,\text{d}v$ , where $h_\mu (v)$ satisfies the integral equation $\mathcal{L}(h_\mu )=-2f_{eq}v_1v_2$ . The fast spectral method and iterative scheme can be used to determine the constant $B_0(\omega )$ (Wu et al. Reference Wu, Liu, Zhang and Reese2015 b). This process is provided in the supplementary Matlab code.

2.2. Linearised Shakhov kinetic model

The Boltzmann collision operator is notoriously difficult to solve and is therefore often simplified through kinetic models, such as the Gross–Jackson model (Gross & Jackson Reference Gross and Jackson1959), the elliptic-statistic BGK model (Holway Reference Suzuki, Gerakis and Hara1966), the Shakhov model (Shakhov Reference Wu, White, Scanlon, Reese and Zhang1968), and the Tenti S6 and S7 models (Tenti et al. 1974; Pan et al. Reference Wu, White, Scanlon, Reese and Zhang2005). In the linearised case, the collision operators in these kinetic models can be expressed as linear combinations of the eigenfunctions of the Boltzmann equation for a Maxwellian gas (Wang-Chang & Uhlenbeck Reference Wang-Chang and Uhlenbeck1952), although the combination coefficients differ among models. In our monograph (Wu Reference Wu2022), we showed that, for a monatomic gas, Tenti’s S6 model is equivalent to the linearised Shakhov model, while Tenti’s S7 model corresponds to the Gross–Jackson model. Although the S7 model incorporates stress terms in the collision operator, both experimental and numerical studies have demonstrated that the S6 model provides higher accuracy (Pan et al. Reference Wu2004).

In the linearised Shakhov model, the linearised Boltzmann collision operator $\mathcal{L}$ in (2.7) is replaced by

(2.9) \begin{equation} \mathcal{L}_s=\frac {\sqrt {\pi }}{2 {\textrm{Kn}}}\left \{ \left [\rho +2u_2v_2+T_t\left (v^2-\frac {3}{2}\right )+\frac {4q_{t2}v_2}{15}\left (c^2-\frac {5}{2}\right ) \right ]f_{eq} -h\right \}, \end{equation}

where Kn is the Knudsen number, defined as the ratio of the mean free path of gas molecules to the scattering wavelength (it should be noted that, in the CRBS community, the $y$ -parameter is often used (Tenti et al. 1974), which is related to the Knudsen number as $1/4\sqrt {\pi }{\textrm{Kn}}$ ):

(2.10) \begin{equation} {\textrm{Kn}}=\sqrt {\frac {\pi }{4}} \frac {\mu (T_0){v_m}}{n_0k_BT_0\lambda }. \end{equation}

Additionally, the macroscopic quantities deviating from their corresponding values in equilibrium state, such as the number density $\rho$ , bulk velocity $u_2$ , temperature $T$ and translational heat flux $q_{t2}$ , can be calculated as

(2.11) \begin{equation} [\rho ,u_2,T_t,q_{t2}]= \int \left [1,v_2,\frac {2}{3}\left (v^2-\frac {3}{2}\right ), v_2\left ({v^2}-\frac {5}{2}\right )\right ]{h}\,\text{d}\boldsymbol{v}, \end{equation}

which, on top of the normalisation (2.6), are further normalised by $2{\pi {}I_0\alpha }/{c\epsilon _0k_BT_0}$ .

From the form of its collision operator, it is evident that the linearised Shakhov model does not capture the influence of the intermolecular potential. Running the supplementary Matlab code (by setting the collision type to 2 around the 14th line) confirms that the Shakhov model produces line shapes similar to those obtained from the Boltzmann equation for a Maxwellian gas.

3.1. Influence of intermolecular potential

We first consider the case of a chirp-free optical lattice. To obtain the CRBS line shape for a chirp-free pulse, pump beams of different frequencies are superimposed and applied continuously throughout the simulation. In this case, the linearised Boltzmann equation (2.7) is modified to include a new acceleration term (see the Appendix of Wu et al. (Reference Wu, White, Scanlon, Reese and Zhang2015)):

(3.1) \begin{equation} \frac {\partial {h}}{\partial t} +2\pi {i}v_2{h} -\frac {\sin (20\pi {t})}{t}v_2f_{eq} =\mathcal{L}(h). \end{equation}

The linearised Boltzmann collision operator $\mathcal{L}$ is efficiently solved using the fast spectral method (Wu et al. Reference Wu, White, Scanlon, Reese and Zhang2013, Reference Wu, White, Scanlon, Reese and Zhang2015), while the time-dependent ordinary differential equation is integrated by the second-order Heun’s scheme, with the initial condition $h=0$ , a normalised time duration of $\tau = 30$ and time step $0.002$ . During the simulation, the perturbed gas density $\rho (t)$ is recorded at each time step. After the simulation, a fast Fourier transform is applied to the time series of $\rho$ and the CRBS spectrum $S$ is obtained as the squared magnitude of the Fourier transform, i.e.

(3.2) \begin{equation} S(\omega _a)=\left |\int \rho (t)\exp (-i\omega _a t)\,\text{d}t\right |^2, \end{equation}

where $\omega _a$ is the angular frequency. For a given set of the Knudsen number and viscosity index, each line shape can be computed in approximately one minute using the in-house MATLAB programme $\text{CRBS}_{-}\text{monatomic}$ provided in the supplementary material.

Figure 1. Chirp-free CRBS spectra of a Maxwellian gas with $\omega = 1$ at different Knudsen numbers. In this and following figures, all spectra are normalised to their maximum magnitude.

Figure 1 shows the typical CRBS spectra of Maxwellian gas at different Knudsen numbers. When Kn = 0.01, the side Brillouin peaks dominate, while the central Rayleigh peak is almost negligible. The Brillouin peaks appear at the normalised angular frequency of $2\pi \sqrt {5/6}$ , corresponding to an optical lattice velocity equal to the sound speed of a monatomic gas. This indicates that the Brillouin peaks arise from the propagation of sound waves in the gas. As the Knudsen number increases, kinetic effects become significant. The Brillouin peaks broaden, while the Rayleigh peak grows in magnitude. The enhancement of the Rayleigh peak is attributed to the suppression of collective sound-wave propagation at higher Knudsen numbers, where random thermal motions dominate over coherent acoustic oscillations. When Kn increases beyond approximately 0.15, the Rayleigh and Brillouin peaks start to merge, and the spectrum takes on a bell-shaped profile, dominated by stronger Rayleigh scattering from purely diffusive density fluctuations rather than from well-defined sound waves. When ${\textrm{Kn}} \gtrapprox 0.5$ , the CRBS line shape is close to Gaussian:

(3.3) \begin{equation} S\approx \exp \left (-\frac {\omega _a^2}{4.84\pi ^2}\right )\!. \end{equation}

Now, we analyse the role of intermolecular potential. When the Knudsen number is small, the gas dynamics is in the hydrodynamic regime, determined entirely by the Navier–Stokes equation and the transport coefficients (viscosity and thermal conductivity). In this case, the intermolecular potential has no influence. Similarly, when the Knudsen number is large and the gas dynamics enters the free-molecular regime, the Boltzmann collision operator $\mathcal{L}$ vanishes, hence the influence of the intermolecular potential is absent, and the spectrum becomes Gaussian. Thus, the intermolecular potential only plays a role in the kinetic regime, see Figure 2. For example, when Kn = 0.09, as the viscosity index increases from 0.5 (hard-sphere gas) to 2.4 (representing a soft potential close to the Coulomb potentialFootnote ¹ ), the magnitude of the Rayleigh peak decreases. Moreover, the closer the viscosity index is to that of the Coulomb potential, the more rapidly the spectrum around zero frequency decreases.

Figure 2. Comparisons of chirp-free CRBS spectra for different power-law intermolecular potentials.

This phenomenon can be explained in terms of the collision frequency defined in (2.8). For Maxwellian molecules ( $\omega = 1$ ), the collision kernel (2.4) is independent of the relative collision speed and, consequently, the equilibrium collision frequency (2.8) does not depend on the molecular velocity. In contrast, for a hard-sphere gas ( $\omega = 0.5$ ), the equilibrium collision frequency increases with the molecular speed $v$ , see figure 13 of Wu et al. (Reference Wu, White, Scanlon, Reese and Zhang2013). This implies that for $v \approx 0$ , the collision frequency is lower than that of a Maxwellian gas, corresponding to an effectively larger Knudsen number (fewer collisions, larger viscosity and, hence, a larger Knudsen number). Since at ${\textrm{Kn}} \approx 0.09$ the magnitude of the Rayleigh peak increases with the Knudsen number (see Figure 1), the Rayleigh peak of the hard-sphere gas is higher than that of the Maxwellian gas, as shown in Figure 2a. By contrast, for a soft potential with $\omega \gt 1$ , the equilibrium collision frequency decreases with increasing molecular speed $v$ . Consequently, for $v \sim 0$ , the effective Knudsen number is smaller and the magnitude of Rayleigh peak is reduced.

3.2. Influence of chirp rate

Now, we solve the linearised Boltzmann equation (2.7) with the chirped optical lattice to investigate the effect of the chirp rate $\beta$ . We choose the normalised angular frequency range $\omega _{\text{min}}=-8\pi$ and $\omega _{\text{max}}=8\pi$ , corresponding to lattice speeds in the range $[-4v_m,4v_m]$ , which is sufficiently wide to cover the entire line shape. The chirp rate is controlled by the chirp duration $\tau$ in (2.2), which is normalised by $\lambda /v_m$ . The Maxwellian gas is considered.

Figure 3. Chirped CRBS spectra for Maxwell gas at different chirp duration $\tau$ .

Figure 3a shows the CRBS spectra at three different chirp durations for Kn = 0.01. At a fast chirp rate ( $\tau = 10$ ), the line shape becomes clearly asymmetric: the right Brillouin peak is significantly lower than the left one, consistent with the observations reported by Suzuki et al. (Reference Suzuki, Gerakis and Hara2024). Furthermore, we observe that the central Rayleigh peak develops fine ripples. When $\tau = 30$ , the CRBS spectrum regains symmetry and these ripples nearly vanish. At $\tau = 50$ , however, a slight asymmetry reemerges, but now the right Brillouin peak becoming slightly higher than the left.

Figure 3b shows the CRBS spectra at ${\textrm{Kn}} = 0.09$ . For the chirped laser, slight asymmetries are observed, with the right Brillouin peak appearing lower than the left. In addition, fine ripples emerge in the Rayleigh peak. As the Knudsen number increases further, the Rayleigh and Brillouin peaks begin to overlap, leading to a flattened line shape near the central frequency, as shown in Figures 3c and 3d. In this regime, the spectral asymmetry becomes less discernible, although the fine ripples persist. These results indicate that the pronounced spectral asymmetry originates from collisional effects of the gas molecules.

In all cases, the oscillation amplitude of the fine ripples gradually decreases as the chirp duration increases (i.e. as the chirp rate decreases). However, the oscillation frequency tends to increase with the chirp duration. In fact, Figure 3 indicates that the oscillation period (i.e. the number of ripples) is approximately equal to the normalised chirp duration $\tau$ , independent of the Knudsen number. Thus, this behaviour cannot be attributed to collisional effects of the gas molecules. At sufficiently large values of  $\tau$ , the spectrum eventually converges to the chirp-free case described in Figure 2a.