# Vibrational Model of Heat Conduction in a Fluid of Hard Spheres
## 5 Results  
Figure 1 shows the comparison between the MD data on the reduced thermal conductivity coefficient and the discussed theoretical approximations. Symbols correspond to the MD data [48]. The quantitative picture is conventional and resembles that in other simple fluids, see e.g., Refs. [30,73]. In the low-density domain \(\lambda_{\rm R}\) decreases with density. In the first approximation, the thermal conductivity coefficient of a dilute HS gas is \(\lambda_{0}=(75/64\sigma^{2})(T/\pi m)^{1/2}\)[74], which leads to \(\lambda_{\rm R}\simeq 0.661/(\rho^{*})^{2/3}\)[28]. The minimum of \(\lambda_{\rm R}\) is reached at \(\rho^{*}\simeq 0.25\). The amplitude of this minimum is known to be quasi-universal for many model and real fluids. Numerically, a rough estimte \(\lambda_{\rm R}^{\rm min}\sim 3\) holds for various liquids, except for plasma-related fluids with very soft Coulomb-like interactions [30,73]. This minimum corresponds to the crossover between two different mechanisms of energy transfer: pairwise collisions in the dilute gas regime and collective vibrations in the dense fluid regime. Note that a similar minimum appears in the reduced viscosity coefficient [30,73]. It is the location of the reduced viscosity minimum that previously served as one of the definitions of the gas-liquid dynamical crossover [75,76]. After the minimum, \(\lambda_{\rm R}\) increases monotonously with density. The typical value of the thermal conductivity coefficient at the freezing point of various simple liquids is \(\lambda_{\rm R}^{\rm fr}\sim 10\)[30,73]. We observe in Figure 1 that the thermal conductivity coefficient of the HS fluid at the freezing point is around \(\simeq 30\%\) higher. A similar observation was previously made in Ref. [28].  
The dashed curve in Figure 1 corresponds to the vibrational model of heat transfer as formulated above. The agreement is reasonably good at moderate densities (after minimum in \(\lambda_{\rm R}\) is reached), but becomes worse at \(\rho^{*}\gtrsim 0.7\). As the freezing density (\(\rho_{\rm fr}^{*}\simeq 0.939\) according to Ref. [77]) is approached the vibrational model overestimates \(\lambda_{\rm R}\) by \(\simeq 50\%\). This reflects the fact that the vibrational picture is not very appropriate for the dense HS fluid. It is to some extent surprising that the expression based on the averaging over collective modes remains a relatively good approximation in the moderately dense regime.  
The dash-dotted curve corresponds to the Bridgman's expression. Plotted is the value \(\lambda_{\rm R}=c_{s}/\tau_{\rm T}\), where \(c_{s}\) is evaluated using Equation (11). We see that the numerical factor equal to unity is appropriate for the dense HS fluid. Only near the freezing point it leads to some underestimation of \(\lambda_{\rm R}\).  
The solid curve corresponds to the Enskog theory, which is an extension of the gas kinetic theory for the transport coefficients to finite densities. It accounts for the excluded volume effects, but assumes that the successive collisions between the atoms are uncorrelated. The Enskog expression for the thermal conductivity coefficient of the HS fluid reads [48]:  
\[\lambda=1.02513\lambda_{0}\rho B_{2}\bigg{(}\frac{1}{Z_{\rm ex}}+1.2+0.7574Z_{ \rm ex}\bigg{)}, \tag{16}\]  
where \(\lambda_{0}\) is the first approximation for \(\lambda\) in the infinitely dilute gaseous limit (see above), the coefficient \(1.02513\) is the Sonine polynomial correction up to 4 th term, \(B_{2}=2\pi\sigma^{3}/3\) is the second virial coefficient, and \(Z_{\rm ex}=Z-1\) is the excess compressibility factor. It is observed that overall the thermal conductivity coefficient of the HS fluid is rather well described by the [PERSON] theory. Deviations from MD data of Ref. [48] do not exceed \(\sim\)5% in the entire fluid regime. This is in contrast to other transport properties: Agreement with [PERSON] theory for the self-diffusion and viscosity coefficients only takes place at sufficiently low densities [77].