# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## 4 Results  
In Figure 2, we present the distribution of heat-flow fluctuation for the three models with \(N=128\) and \(\tau=100\). Throughout our simulation in this paper, we fix the average temperature, \(T\), of two heat baths as 0.5 and the temperature difference, \(\Delta T\), as 0.1. We note that they exhibit Gaussian-like distribution for all three models, with no significant difference. Gaussian-like distributions of heat-flow fluctuation have been observed in [13], and their reason has been discussed theoretically [32; 33]. Figure 3 shows \(F(J)\) vs. \(J\) as a function of \(\tau\) for several lattice sizes for the three models. It can be observed that, at short time intervals, the prediction of the steady-state fluctuation relation is violated, although \(F(J)\) and \(J\) shows a linear dependence. When \(\tau\) is large enough, the prediction is always approached for all three models. However, there are significant disparities. In the case of the harmonic oscillator model and the FPUT-\(\beta\) model, we require increasingly larger values of \(\tau\) to align with the theoretical predictions as the chain's length grows. Notably, for a given length, the time window required is greater for the former than for the latter, as illustrated in the first two lines of Figure 3. This is in line with the integrability of the purely harmonic chains, which are at the opposite end of thermodynamics. It is important to highlight that testing the theorem becomes quite challenging for lengthy chains, given the infrequent occurrence of negative heat flow events. Consequently, one must allow the system to evolve for an extended period to obtain an effective distribution function. Indeed, when dealing with the harmonic model, this task is nearly beyond the computational capacity for \(N>1000\). As a result, direct observation of results converging to the theoretical prediction becomes infeasible. From a theoretical standpoint, even though these models can eventually fit the prediction, our analysis reveals that they defeat conventional wisdom. Figure 2: Heat flow distribution across three models. 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 trajectory segment of length \(\tau\) is 100. (**a**) The harmonic oscillator model. (**b**) The FPUT-\(\beta\) model. (**c**) The \(\phi^{4}\) model. According to this, which larger systems adhere better to the standard statistical descriptions characterizing thermodynamic systems? This, perhaps perplexing, point will be clarified by the analysis of the other models. What is more, we observe a counterintuitive trend in this context, which may be related to the growth fluctuation magnitude with \(N\). For the \(\phi^{4}\) model, once the size of the chain exceeds a threshold, \(\tau=200\) in this case, the dependence of \(F(J)\) to \(J\) for a given \(\tau\) becomes identical, as shown in the last line of Figure 3. As a result, for this model, the steady-state fluctuation relation can be generally satisfied for relatively long chains (with about \(N>200\)) and relatively large \(\tau\) (with about \(\tau>200\)). In practical applications involving real-world materials and measurable time scales, one can conclude that the \(\phi^{4}\) model consistently satisfies the theorem. To understand the mechanism, we calculated the correlation function \(C(t)=\left(\langle J(t)\rangle(0)\right)-\left\langle J(0)\right\rangle^{2})/ \left(\langle J(0)^{2}\rangle-\left\langle J(0)\right\rangle^{2}\right)\) of heat flow fluctuation of the three models under a nonequilibrium steady state. We observe that, although \(C(t)\) decays to zero with the increase of time, the behavior is different for each model. In the case of the harmonic model, \(C(t)\) decays linearly with time, and the rate of decay decreases as a function of the chain's length. In  
Figure 3: Global flow fluctuation analysis over increasing time intervals in diverse models.