# Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths
## 1 Introduction  
The 1990s witnessed the advent of fluctuation relations by authors like [PERSON], [PERSON], [PERSON] and [PERSON], revolutionizing statistical physics and nonequilibrium thermodynamics [1; 2; 3; 4; 5; 6]. These relations extended the second law of thermodynamics into a statistical-physical framework, embracing both increases and decreases in entropy. Their ratios of positive and negative entropy events are shown to rise exponentially with larger measurement times and system sizes, seamlessly bridging micro and macro thermodynamic theories and earmarking the fluctuation relation as a pivotal development in nonequilibrium statistical physics. Transient fluctuation relations for **time-reversible** dynamics are exact for systems of any size, near and far from equilibrium, and over any observation time [4; 5; 7; 8; 9]. On the other hand, **steady-state** fluctuation relations, apparently similar or directly derivable from transient relations, are only asymptotically valid hence not exact, and require proper conditions to be verified concerning the time correlation function of the dissipation [8; 10; 11]. We will see, however, that exact relations can be verified at any finite time, when the proper correlation functions can be evaluated, cf. [8; 12]. Nonequilibrium steady-state heat flow fluctuations are very convenient for testing the fluctuation relations. Under the local equilibrium hypothesis, local entropy generationlinks directly to heat flow, connecting to the system's total entropy change. [PERSON], [PERSON] and [PERSON]'s 1998 application of fluctuation relations for the dissipation function [13] to nonequilibrium steady-state heat flow fluctuations, validated by numerical simulations in the Fermi-Pasta-Ulam-Tsingou(FPUT)\(-\beta\) lattice model, laid the groundwork for further exploration [14]. Subsequent studies by [PERSON], [PERSON], [PERSON] and [PERSON] et al. expanded this theorem's applicability, to a variety of systems, including the harmonic oscillator chain [10; 15; 16; 17; 18]. These findings, however, raise a key question: does compliance reflect the system's dynamical characteristics or the influence of the heat bath? Furthermore, the need for larger time windows in larger systems seems to contradict the expectation that larger systems should more readily adhere to thermodynamic laws. Our research addresses these complexities by examining three distinct one-dimensional lattice models: the harmonic oscillator chain, FPUT-\(\beta\) and the \(\phi^{4}\) lattice model, each representing a unique thermal conduction class. The harmonic oscillator chain lacks chaotic motion, exhibiting a system-size proportional thermal conductivity, \(\kappa\)\(\sim\)\(N\). The FPUT-\(\beta\) and \(\phi^{4}\) lattice models, with chaotic motion, show divergent and size-independent thermal conductivity, respectively [19; 20; 21; 22; 23; 24]. This divergence in heat conduction behaviors presents a compelling study contrast. We demonstrate that, for a fixed system size, each model satisfies the steady-state fluctuation relations for the dissipation function within sufficiently large observational time windows [25], albeit through differing mechanisms. In lattices with anomalous heat conduction, compliance stems from the heat baths at the chain ends, where bath fluctuations permeate the entire chain given adequate time. Conversely, in lattices with normal heat conduction, bath fluctuations diminish rapidly, and theorem compliance is dictated by the system's statistical physical behavior. Notably, when the observation time is scaled with the characteristic correlation time, a universal, size-independent fluctuation relation emerges. This discovery underscores the correlation's disappearance as indicative of steady-state fluctuation relations satisfaction, and enabling predictions of fluctuation relations in longer chains based on shorter ones. The structure of this paper is organized as follows. The next section presents an introduction to the fluctuation relation for dissipation function.