# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## 4 Results
Due to the non-vanishing correlation within the system, the heat baths continue to play a role in eliminating the correlation, resulting in a timescale of \(t_{r}\sim N^{0.5}\) to ensure the system satisfies the fluctuation relation. In the \(\phi^{4}\) model, the system is also chaotic but the fluctuation of heat flow decays as \(C^{\prime}(t)\sim e^{-t}\). Even without attributing to the heat baths, \(C(t)\) decays to zero before \(t=N\). This explains the results of Figure 4c. Consequently, once the chain length exceeds a threshold, the correlation vanishes after a critical time that is independent of the system size, and, thus, the fluctuation relation applies hereafter. The reason why the fluctuation relation is satisfied after the vanishing of the correlation heat flow fluctuations is as follows. When there is no correlation between the fluctuations, the distribution of fluctuations appears as a Gaussian distribution \(P_{t}(I)=c\epsilon^{\prime}-\frac{(J-(j)t)^{2}}{2\sigma^{2}}\), according to the central limit theorem [32; 33; 39]. We then obtain \(\ln\frac{P_{t}(I)}{P_{t}(-I)}=\frac{(J+(j)t)^{2}-(J-(j)t)^{2}}{2\sigma^{2}}= \frac{2(j)t}{\sigma^{2}}\). With \(\sigma^{2}=\frac{2(j)}{(\frac{1}{R}-\frac{1}{L})}t\), we obtain \(\ln\frac{P_{t}(I)}{P_{t}(-I)}=J(\frac{1}{T_{R}}-\frac{1}{T_{L}})\), i.e., the steady-state fluctuation relation. Based on the foregoing analysis, we can re-scale \(F(J)\) to render it system-size independent. Denote the slope of the \(F(J)\) curve in Figure 5 as \(k(N,\tau)\). We re-scale \(\tau\) with respect to \(t_{r}\) and plot \(k(N,\tau/t_{r})\) as a function of \(\tau/t_{r}\) in Figure 5. It can be observed that \(k(N,\tau/t_{r})\) manifests as a system-size independent curve, \(k(t)=c_{1}t^{-a}+e^{b/t}\). As such, it can be regarded as an extension of the fluctuation relation for an arbitrary \(\tau\). The limit of \(k(N,\tau/t_{r})\) approaching unity as \(\tau\) tends to infinity indicates the satisfaction of the original steady-state fluctuation relation. The universality of the harmonic oscillator model is exemplary, and the other two are also fundamentally consistent for larger sizes. Technically, one can employ the universal curve to extrapolate the results obtained for a fixed chain length to other sizes without the need for further calculations when the scalings apply. Figure 5: Scaling of heat flow fluctuation behavior in three model types over varied time intervals. This figure presents the scaling of heat flow fluctuation behavior in three different types of models, utilizing various time windows normalized by correlation time. System parameters: consistent across all models. The average temperature at both ends of the heat reservoir \(T=0.5\), and the temperature difference \(\Delta T=0.1\). The number of particles are denoted in the plots. The results from molecular dynamics simulations are depicted as point plots. An orange solid line represents the fitting curve. (**a**) The harmonic oscillator model. (**b**) The FPUT-\(\beta\) model. (**c**) The \(\phi^{4}\) model. The relationship of \(k(N,\tau/t_{r})\) with \(\tau\) arises from persistent correlations. Notably, the global heat flow distribution retains its Gaussian character, even at smaller values of \(\tau\). This phenomenon is explained by the process of calculating global heat flow: the averaging of local heat flows from \(N\) particles, each exhibiting random heat flow characteristics. Consequently, the global heat flow represents an aggregation of numerous small, random variables, aligning with a Gaussian distribution, as dictated by the central limit theorem. The variance of the instantaneous heat flow distribution, measured over a unit time, is denoted as \(\sigma_{0}^{2}\). Taking the harmonic chain as an example, and assuming a steady heat flow correlation over an exceedingly brief \(\tau\) (\(C(t)\approx constant\), while neglecting decay due to heat baths), we find that the variance in heat flow fluctuations is represented by \(\sigma(\tau)^{2}=\tau^{2}\cdot\sigma_{0}^{2}\).