## 3 Results and Discussion
### Thermal Hysteresis and Dynamic Characteristics in PCM Thermal Conductivity of the RTM
The cycle direction is indicated by arrows corresponding to the four behavioral characteristics illustrated in Figure 4a,b. In the linear dependence (LD) regions (Cases 2 & 4), the straight lines affirm that Q is directly proportional to \(\Delta\)T (\(Q\propto\Delta\)T), i.e., \(Q=G\Delta\)T, where G is the proportionality constant representing thermal conductance. The linear characteristics of the system are linked to the absence of PCM phase transitions. Additionally, the amplitude and spread of the Lissajous curves depend on the variation of \(\theta\), which diminishes with further decreases in \(\theta\) and disappear when \(\theta=0\). Figure 3: Thermal conductivity model during cooling and heating process. (**a**) PCM thermal conductivity (\(k_{1}\)). (**b**) Effective thermal conductivity (\(K_{eff}\)). Analysis of Memristance and Thermal Conductance in PCM Systems: Insights into Nonlinear Behavior and Phase Transitions  
To facilitate the understanding of the dynamics within phase-change material (PCM) systems, we delve into the reciprocal relationship between memristance (M) and thermal conductance (G) within PCM systems, shedding light on their distinct characteristics across various operational regimes. Additionally, the influence of the PCM phase transitions on the nonlinear characteristics is examined. Figure 5a shows the plot of memristance (M) associated with the Lissajous curve in Figure 4c, while its associated thermal conductance (G) is shown in Figure S1 of the Supplementary Note. It can be observed that M is equivalent to the reciprocal of G (\(M=G^{-1}\)). In this context, a generalized expression for G in the linear region (cases 2 and 4) of the system is derived as:  
\[G=\frac{\alpha k_{m(j)}T_{\alpha\
u}}{d} \tag{7}\]  
Here, \(\alpha\) is a constant with a value of 0.0013587. The value of \(\alpha\) is determined through a systematic process involving the analysis of the linear region above the zero crossing. Initially, we calculated the value of G, which represents the slope within this linear segment. Subsequently, we substituted the values of other parameters listed in Table 1 into Equation (7) to obtain \(\alpha\). A detailed explanation of this calculation procedure is provided in the Supplementary Note. Equation (7) implies that the value of G for case 2 is greater than G for case 4, because \(k_{m}>k_{i}\). The nonlinear characteristics observed in cases 1 and 3 can be attributed to the phase transitions of the PCM, driven by variations in thermal conductivity, as previously discussed. The absence of these transitions close to the extreme values of \(\Delta\)T results in the pinched shape of the Lissajous curve. In Figure 5a, the linear segments denoted as 2 (top) and 4 (bottom) represent the regions without a phase transition in the PCM. The corresponding vertical lines (1 on the left and 3 on the right) depict the influence of the phase-change material (PCM), aligning with the observations in Figure 4. The M-\(\Delta\)T curve's height in Figure 5a is governed by the thermal conductivity contrast of the PCM, denoted as r (\(r=k_{m}/k_{i}\)), illustrated in Figure 5b. As r increases, so does the height of Figure 5b. This thermal contrast (r) significantly affects the shape of the Lissajous curve and can be utilized to regulate the duration of a binary memory being in its ON and OFF states, with the choice of M determining the encoding of binary states 0 and 1. Figure 4: Performance analysis of the proposed thermal memristor. **(a)** Time\(-\)dependent effective thermal conductivity showing different phase transitions. **(b)** Time\(-\)dependent temperature difference and net radiative heat flux. **(c)** Lissajous curve (Q vs \(\Delta\)T curve) at different \(\theta\). Impact of Thermal Contrast, Amplitude, and Temperature Variation on Characteristics of the Lissajous Curve  
In this section, we examine the impact of thermal contrast (r), \(\theta\), and temperature variation on the characteristics of Lissajous curves.