## 3 Results and Discussions
### Diffusion Coefficient
#### 3.5.2 Diffusion Coefficient Dependence on Temperature  
Figure 9 shows the dependence of the diffusion coefficient \(D\) on temperature \(T\), where \(D\) values are calculated using the MDS in Equation (1) and the VACF in Equation (4). We compare the results from the QMD simulations with the data computed using the Yukawa-MD calculations and with the results obtained using the Coulomb logarithm (as described in Section 2.4). It can be seen that the diffusion coefficients calculated using the QMD method and the Yukawa model show good agreement with each other. This confirms that if the Yukawa model provides accurate structural properties, it can also adequately describe ion transport properties.  
In Figure 9, the relationship between \(D(T)\) and temperature exhibits a nearly linear behavior, which aligns with the expected physical principles. As temperature rises, the kinetic energy of the particles increases, resulting in a decrease in interatomic correlations and an increase in the mean free path length of particles. Consequently, particles can more easily overcome the barriers in the potential energy landscape created by interatomic forces.  
Finally, in Figure 9, a comparison of the results obtained using the Chapman-Enskog approach (which incorporates the Coulomb logarithm) and data from molecular dynamics simulations reveals that the Chapman-Enskog method significantly overestimates the diffusion coefficient at temperatures below \(16,000\) K. However, it provides accurate results at higher temperatures. To understand this, we note that the Chapman-Enskog method relies on the idea of pair collisions, as evidenced by the use of pair-collision cross sections. This simplification is only valid when the correlation between particles is sufficiently weak. We can illustrate this by comparing the RDF curves at different temperatures, as shown in Figure 5. At temperatures \(T\geq 16\),000 K, the RDF displays only one dominant peak.  
Figure 8: The diffusion coefficient \(D\) as a function of particle number \(N\) for dense beryllium computed using the QMD simulations at \(T=8000\) K. The results were computed using the MSD and VACF in the NVT and NVE ensembles.  
Thus, on the example of dense Be, we can confirm that the Chapman-Enskog approach is highly effective in regimes where ion-ion coupling is relatively weak, resulting in an RDF with fewer than two main peaks. Furthermore, in Figure 9 we see that the Thomas-Fermi potential-based results overestimate the diffusion coefficient and become more accurate with increasing temperature. As expected, we see that the ion-ion correlations (nonideality) lead to a reduction in the diffusion coefficient.  
Now, based on the QMD data, using the observation that the Yukawa model provides an adequate description of ion dynamics, we can define the general condition for the parameters at which the Chapman-Enskog method can be used. For that, we recall that the RDF of the Yukawa system can be uniquely defined by an effective coupling parameter in the entire liquid regime. We use the expression for the effective coupling parameter proposed in Ref. [55]:  
\[\Gamma_{\text{eff}}(\kappa,\Gamma)=f(\kappa)\Gamma, \tag{13}\]  
where  
\[f(\kappa)=1-0.309\kappa^{2}+0.0800\kappa^{3}. \tag{14}\]  
By utilizing the effective coupling parameter and observing the RDF features at temperatures where the results from the Chapman-Enskog method aligns with the data from the QMD simulations for ion diffusion in dense beryllium (Be), we conclude that the Chapman-Enskog method can accurately describe the system when \(\Gamma_{\text{eff}}<50\). We expect that this conclusion will generally hold true for dense disordered metallic systems at high temperatures.