## 3 Results and Discussion  
### Thermal Hysteresis and Dynamic Characteristics in PCM Thermal Conductivity of the RTM  
In this analysis, we investigated the hysteresis in the thermal conductivity of the PCM over a temperature range from 297 K to 320 K. Utilizing Equation (3) and incorporating the parameters outlined in Table 1 across various applied temperatures, we observed distinct thermal hysteresis characteristics within the PCM. These variations are visually represented in Figure 3a. The diagram reveals distinct thermal conductivities during the cooling and heating processes, with the PCM exhibiting low thermal conductivity during cooling and high thermal conductivity during heating. Essentially, the PCM acts as an insulator during the cooling phase and transforms into a metal-like conductor during the heating phase. The thermal conductivity of the PCM is confined within the range established by the values during the insulating and metallic phases, as previously mentioned. The vertical lines \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**S/N** & **Physical Parameters** & **Symbol** & **Value** & **Ref.** \\ \hline
1 & Thermal conductivity for metallic phase & \(k_{m}\) & 5.3 & \\  & & & Wm\({}^{-1}\)K\({}^{-1}\) & [25] \\
2 & Thermal conductivity for insulating phase & \(k_{i}\) & 3.4 & \\
3 & Cooling temperature & \(T_{c}\) & 301 K & [25] \\
4 & Heating temperature & \(T_{h}\) & 310 K & [25] \\
5 & SiC thermal conductivity & \(k_{2}\) & 3.88 & \\
6 & Fitting parameter & \(\beta\) & 1.57 K\({}^{-1}\) & [28] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters and their values. represent the experimentally determined cooling (\(T_{c}\)) and heating (\(T_{h}\)) temperatures [25]. Similarly, Figure 3b shows the corresponding effective thermal conductivity (\(k_{eff}\)) for the system in Figure 2. This is obtained by using Equation (4) at different input temperatures as previously described. In this case, \(k_{eff}\) is constrained by thermal conductivities with values lower than those observed for the PCM. It is evident that \(k_{eff}\) plays an important role in the enhancement of the radiative heat flux as established in Equation (1). Subsequently, we analyzed the time-varying characteristics of the effective thermal conductivity (\(k_{eff}\)), radiative heat flux (Q), and temperature difference driving thermal radiation in the thermal memristor, as depicted in Figure 4. The behavior of \(k_{eff}(t)\) reveals four distinct phases within one period, categorized into four cases. In case 1 (\(0.01\leq t/\tau<0.14\)), a rapid surge in \(k_{eff}\) occurs due to a phase transition (insulating to metallic) in the PCM as the temperature difference (\(\Delta\)T) increases. Subsequent variations in \(\Delta\)T, maintaining the PCM in its metallic phase, lead to a sustained high \(k_{eff}\) observed in case 2 (\(0.14\leq t/\tau<0.53\)). Case 3, covering the interval \(0.53\leq t/\tau<0.66\), witnesses a decline in \(\Delta\)T, triggering a metal to insulating phase transition in the PCM, resulting in a sudden drop in \(k_{eff}\). Finally, in case 4 (\(0.66\leq t/\tau<1\)), further changes in \(\Delta\)T maintain \(k_{eff}\) in the insulating phase (i.e., with a low \(k_{eff}\) value). It is crucial to elucidate how the variations of both \(\Delta\)T and \(k_{eff}\) induce a radiative heat flux with a strong nonlinear dependence (NLD) (cases 1 and 3) on \(\Delta\)T despite that \(\Delta\)T is significantly less than the reference temperature (\(T_{av}=305.5\) K). Conversely, for cases 2 and 4, where the value of \(k_{eff}\) remains constant, Q exhibits a linear dependence (LD) on \(\Delta\)T, as illustrated in Figure 4. This characteristic is associated with the simultaneous increases and decreases in Q and \(\Delta\)T (Figure 4b), as evidenced in Equation (1). Notably, these sharp variations in Q and \(\Delta\)T are generated by the PCM. Figure 4c displays the curves of Q versus \(\Delta\)T for the proposed thermal memristor at different values of \(\theta\).