Nonequilibrium states in soft condensed matter require a systematic approach to characterize and model materials, enhancing predictability and applications. Among the tools, X-ray photon correlation spectroscopy (XPCS) provides exceptional temporal and spatial resolution to extract dynamic insight into the properties of the material. However, existing models might overlook intricate details. We introduce an approach for extracting the transport coefficient, denoted as J ( t ) , from the XPCS studies. This coefficient is a fundamental parameter in nonequilibrium statistical mechanics and is crucial for characterizing transport processes within a system. Our method unifies the Green–Kubo formulas associated with various transport coefficients, including gradient flows, particle–particle interactions, friction matrices, and continuous noise. We achieve this by integrating the collective influence of random and systematic forces acting on the particles within the framework of a Markov chain. We initially validated this method using molecular dynamics simulations of a system subjected to changes in temperatures over time. Subsequently, we conducted further verification using experimental systems reported in the literature and known for their complex nonequilibrium characteristics. The results, including the derived J ( t ) and other relevant physical parameters, align with the previous observations and reveal detailed dynamical information in nonequilibrium states. This approach represents an advancement in XPCS analysis, addressing the growing demand to extract intricate nonequilibrium dynamics. Further, the methods presented are agnostic to the nature of the material system and can be potentially expanded to hard condensed matter systems. Soft materials, including colloidal suspensions ( 1, 2), polymers( 3, 4), gels ( 5, 6), and biological materials ( 7, 8), often exist in nonequilibrium states, a crucial aspect of their practical applications and scientific studies. These nonequilibrium states arise due to external factors, including mechanical forces, temperature fluctuations, or chemical reactions, necessitating an understanding of how these materials respond to such stimuli. Although classical theories such as mean-field theory ( 9, 10), renormalization group ( 11, 12), and universality ( 11, 13) have contributed significantly to our understanding of particle and atomic behavior within complex systems, they fall short when these materials deviate from equilibrium under external influences. Nonequilibrium dynamics introduce intriguing phenomena, including transitions between metastable states ( 14, 15), dynamical heterogeneity ( 16, 17), aging ( 18, 19), yielding ( 20, 21), avalanches ( 22, 23), and phase reentrance ( 24, 25). Real-time measurements serve as a valuable tool to unravel the kinetics and mechanisms governing these transitions, facilitating improved control and predictability ( 26, 27). Consequently, it becomes imperative to develop robust in situ tools and analysis models that enable us to investigate the dynamic behavior of complex materials ( 28, 29), validate theoretical constructs ( 30, 31), and engineer functional materials for a variety of applications within these soft matter systems.
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