# Vibrational Model of Heat Conduction in a Fluid of Hard Spheres
## 3 Vibrational Model
The HS fluid is extremely anharmonic and dynamics is dominated by short hard-core-like collision events [55]. Nevertheless, similar to other simple fluids, the dense HS fluid does support the longitudinal and transverse collective excitations [56] (with a forbidden long-wavelength region for the transverse mode, the so-called \"\(k\)-gap\" [56; 57; 58; 59; 60]). Therefore, averaging over collective modes can be performed and in this sense \(\left\langle\omega\right\rangle\) remains a meaningful quantity. The Stokes-Einstein relation holds in the HS fluid, just as it does in soft sphere fluids, where it emerges naturally as a consequence of the vibrational picture of atomic dynamics [7; 16]. Vibrational model allows to estimate the excess entropy of various fluids with pairwise repulsive interactions near the fluid-solid phase transition, including the HS limit [61]. Under these circumstances it is not very unreasonable to compare the predictions of Equation (6) with the available results from MD simulations in the HS limit. Since the actual frequency distribution is often not known in liquids, and can vary from one type of liquid to another, some simplifying assumptions are necessary to evaluate \(\left\langle\omega\right\rangle\) in Equation (6). In the simplest Einstein approximation all atoms vibrate with the same (Einstein) frequency \(\Omega_{\mathrm{E}}\). We arrive at the expression derived by [PERSON] and [PERSON] [62]. But this approximation is obviously not applicable to the HS fluid. The conventional expression for the Einstein frequency,  
\[\Omega_{\rm E}=\frac{\rho}{3m}\int d\pi\
abla^{2}\phi(r)g(r), \tag{7}\]  
where \(g(r)\) is the radial distribution function (RDF), is simply undefined in the HS limit. One can use an acoustic spectrum, \(\omega=kc_{s}\) with an appropriate maximum cutoff wavenumber \(k_{\rm max}\). Here \(c_{s}\) is the conventional adiabatic sound velocity. Then, the standard averaging procedure would result in:  
\[\lambda\sim\frac{c_{s}}{\Delta^{2}}. \tag{8}\]  
This can be considered as an analog of [PERSON]'s equation [63]. The numerical coefficient remains undefined, because of the qualitative character of the arguments involved (the values between 2 and 3 can be found in the literature, see e.g., Ref. [35]). We shall see that for the HS fluid the numerical coefficient is in fact close to unity. As a more physically sound approximation, assume that a dense liquid supports one longitudinal and two transverse collective modes. Debye-like averaging in \(k\)-space can be performed using acoustic asymptotes \(\omega_{l}(k)=c_{l}k\) and \(\omega_{l}(k)=c_{l}k\) in the long-wavelength domain, where \(c_{l}\) and \(c_{t}\) are the longitudinal and transverse sound velocities, respectively. The thermal conductivity coefficient becomes [37]:  
\[\lambda\simeq\frac{1}{4}\bigg{(}\frac{3}{4\pi}\bigg{)}^{1/3}c_{\rm p}\frac{c_ {l}+2c_{t}}{\Delta^{2}}. \tag{9}\]  
If we substitute \(c_{\rm p}\simeq c_{\rm v}\simeq 3\) near the freezing point (according to Dulong-Petit law), we get a formula, which is similar to that of the minimal thermal conductivity model proposed by [PERSON] and [PERSON] for amorphous solids [64; 65; 66]. It turns out that a similar formula obtained using the vibrational model works rather well in the liquid regime. It is the expression (9) that will be compared with the recent MD data on the thermal conductivity coefficient of the HS fluid. Note that there are no free parameters in the discussed approach.