# The Impacts of Random Distributed Vacancy Defects in Steady-State Thermal Conduction of Graphene
## 3 Results and Discussion
### Random Distributed Vacancy Defects
With the variance of thermal energy, the probability density is distributed in a narrow range when the number of random vacancy defects is small in both boundary conditions. When the amount of vacancy defects is amplified, the probability density covers a wider range with larger variances. When _Per_ is 0.2% for graphene with its heat source  
Figure 13: The statistical results of graphene with random distributed vacancy defects (the unit for thermal energy is \(10^{5}\) J; (**a–d**) are the minimum, maximum, mean, and variance values of the thermal energy, respectively. in armchair edges, the range of probability density distribution ranges from 114.9-115.7 kJ; the range becomes 95.71-100.6 kJ when _Per_ equals 5%. The same situation happens in graphene with its heat source in zigzag edges. The range of probability density distribution is at 92.3-93.0 kJ when _Per_ is 0.2% and 77.18-80.50 kJ when _Per_ equals to 5%. When the percentage of random vacancy defects is petite, the probability density is more condensed as a peak, while the probability density distribution is gentle with a longer drag when _Per_ is large. Importantly, even though vacancy defects are stochastically dispersed in the entire graphene, the range of thermal energy cannot cover the whole result domain and may be limited in certain intervals when _Per_ is determined. The range of thermal energy in graphene with two boundary conditions is not continuous but discrete in the specific intervals according to _Per_. The discontinuity (between _Per_ is 1%, 3%, and 5%) in thermal energy is useful to distinguish the amount of vacancy defects and also can be designed as control switch in precise devices and instruments. In addition, the ratio of thermal energy reduction in graphene with two boundary conditions is compared in Figure 16. In the graphene with its heat source in armchair edges, the minimum value of thermal energy has 1%, 2.37%, 3.16%, 3.85%, 10.86%, and 17.56% of reduction compared with the thermal energy in original pristine graphene, when _Per_ equals to 0.2%, 0.5%, 0.8%, 1%, 3%, and 5%. The decrease in the maximum value of thermal energy is smaller when _Per_ equals 0.28%, 0.98%, 1.65%,  
Figure 14: The histogram of probability density distribution of thermal energy in graphene with its heat source in armchair edges (P1–P6 represents the percentage of vacancy as 0.2%, 0.5%, 0.8%, 1%, 3%, and 5%, respectively, and the unit for thermal energy is J). Figure 15: The histogram of probability density distribution of thermal energy in graphene with its heat source in zigzag edges (P1–P6 represents the percentage of vacancy as 0.2%, 0.5%, 0.8%, 1%, 3%, and 5%, respectively, and the unit for thermal energy is J). 2.25%, 7.39%, and 13.32%, respectively. In graphene with its heat source in zigzag edges, the degrees of reduction in minimum and maximum values of thermal energy are similar to that in armchair edges. Precisely, the minimum values of thermal energy have 1.05%, 2.09%, 3.19%, 4.09%, 11.04%, and 17.28% of declines, while the maximum values of thermal energy have 0.31%, 1.03%, 1.81%, 2.14%, 7.94%, and 13.72% of declines, respectively, when _Per_ equals to 0.2%, 0.5%, 0.8%, 1%, 3% and 5%. Therefore, the difference caused by boundary conditions is not as obvious as that attributed to the random distribution of vacancy defects. The stochastic dispersion of vacancy defects plays a more important role in the deviation of thermal energy. In order to analyze the robustness of graphene with random distributed vacancy defects, the index \(\beta\) is introduced as:  
\[\beta=\frac{M}{\sqrt{V}} \tag{20}\]  
where \(M\) is the mean value of thermal energy and \(V\) is the variance of thermal energy. The divergence due to the randomness of vacancy defects in graphene is measured using the variance of thermal energy; the quadratic square root of variance is written as a denominator compared with the mean value of thermal energy. In Figure 17a, the equivalent coefficient of thermal conductivity is nearly identical in the two boundary conditions.