Generalized Einstein relation for aging processes

Physical aging, characterized by slow, non-exponential relaxation, broken time-translation invariance, and dynamical scaling, is observed across diverse systems including glassy materials, quantum dots, and laser-cooled atoms. Existing understandings of aging primarily focus on disordered systems, leaving a gap in understanding its manifestation in simpler models. This study addresses this gap by exploring whether a precise definition of aging can be formulated and observed within a simple single-particle model, aiming to deepen our understanding of aging kinetics. The Einstein relation, a cornerstone of equilibrium statistical mechanics connecting diffusion and mobility, has been widely studied, but its generalization to non-equilibrium and aging scenarios remains a significant challenge. Previous research has explored frequency-dependent effective temperatures in the Einstein relation; however, the impact of time-dependent temperature, relevant to processes like cooling in granular materials, has been largely unaddressed. This paper proposes a novel approach using a time-dependent fluctuation-dissipation relation within a non-stationary framework to derive a generalized Einstein relation and subsequently construct a dynamical aging correlation function.

The Einstein relation, a specific form of the fluctuation-dissipation theorem, has been extensively validated experimentally and theoretically in equilibrium scenarios. However, its extension to out-of-equilibrium situations, particularly those exhibiting aging, is an active area of research. While aging is ubiquitous across various physical systems, a precise definition and a comprehensive theoretical framework remain elusive. Prior studies have explored aging in disordered systems, but a simple stochastic model without spatial disorder capable of exhibiting aging was lacking. Existing models often fail to capture all three key fingerprints of aging (slow non-exponential relaxation, time translation invariance breaking, and dynamical scaling). This paper addresses these limitations by developing a minimal model that successfully captures these characteristics.

The authors construct an irreversible Langevin equation to model aging dynamics. The equation incorporates time-dependent noise and friction to account for the changing environment. The time-dependent parameters are chosen based on the behavior observed in granular gases, where inelastic collisions lead to a decrease in average kinetic energy and a cooling effect. Specifically, the friction coefficient g(t) and temperature T(t) are modeled as power-law functions of time. The model considers a particle subject to a time-dependent friction coefficient and time-dependent Gaussian white noise representing the varying environmental temperature. The velocity correlation function (VCF) is analytically calculated, exhibiting characteristics of aging. The average displacement and mean squared displacement (MSD) are also calculated analytically. A non-equilibrium index, 'b', is introduced to quantify the deviation from equilibrium behavior. The generalized Einstein relation is then derived and used to analyze the anomalous diffusion properties of the model. The authors also analyze the connection between the generalized Einstein relation and weak ergodicity breaking, introducing a fluctuation metric to investigate the ergodic properties of the system. The model uses the following irreversible Langevin equation: ẋ = v, mẇ = -mg²(t)v - αU(x) + g(t)ξ(t) (1) where the zero-mean Gaussian noise ξ(t) satisfies a time-dependent fluctuation-dissipation relation: ⟨g(t)ξ(t)g(t')ξ(t')⟩ = 2mg²(t)kT(t) The analytical solutions for velocity, average displacement and MSD are derived using this model under specific assumptions of g(t) and T(t). This allows for a detailed analysis of the anomalous diffusion behavior and the relationship between the non-equilibrium index, the MSD, and the average displacement. A dimensionless ratio R(t) is defined to quantify the deviation from the standard Einstein relation in the aging system.

The study reveals several key findings. Firstly, a generalized Einstein relation is derived for aging processes, extending its applicability to non-equilibrium situations with time-dependent damping and temperature. This relation effectively probes the relationship between mobility and fluctuation in aging systems. Secondly, the model exhibits a full range of anomalous diffusion behaviors, transitioning from ballistic motion (MSD ~ t²) for b=0 to logarithmic Sinai-type ultra-slow diffusion (MSD ~ ln(t)) for b=2. Between these limits (0 < b < 2), the system displays full-scaling anomalous diffusion. The parameter 'b' serves as a non-equilibrium index, characterizing the extent of temperature decrease in the environment. Thirdly, the generalized Einstein relation reveals a correction factor R(t) which deviates from unity in non-equilibrium scenarios, highlighting a significant difference from the equilibrium Einstein relation. For b=2 (ultra-slow diffusion), R(t) diverges logarithmically with time. Fourthly, the study clarifies the conditions for ergodicity in aging systems, demonstrating that the commonly used criteria (irreversibility and convergence of time-averaged observables) are insufficient in this context. The model accurately describes the aging dynamics observed in granular gases, with the non-equilibrium index 'b' directly related to the inelastic collision parameter. The system exhibits dissipative acceleration, where the average velocity increases linearly with time, which contrasts sharply with standard Brownian motion. The effective temperature of the system is consistently higher than the environment's temperature due to the self-cooling nature of the environment.

The findings significantly advance our understanding of aging dynamics and the applicability of the Einstein relation to non-equilibrium processes. The generalized Einstein relation provides a powerful tool for characterizing anomalous diffusion in aging systems, bridging the gap between theoretical models and experimental observations. The identification of the non-equilibrium index 'b' provides a quantitative measure of the deviation from equilibrium, allowing for a more nuanced understanding of aging behavior. The study’s exploration of ergodicity in aging systems reveals the inadequacy of conventional criteria, emphasizing the need for a more refined theoretical framework. The model’s applicability to granular gases, where the parameters can be directly linked to experimental observables, provides a promising avenue for further investigation and validation. The observed dissipative acceleration further highlights the distinct non-equilibrium nature of aging processes, differing significantly from typical Brownian motion.

This research presents a generalized Einstein relation applicable to aging processes characterized by time-dependent damping and temperature. The minimal Langevin model used successfully captures the key features of aging dynamics, revealing a full range of anomalous diffusion from ballistic to ultra-slow diffusion. The study clarifies the conditions for ergodicity in aging systems and highlights the limitations of traditional criteria. Future research could extend this model to more complex systems and explore the effects of different forms of time-dependent noise and friction. Experimental validation of the generalized Einstein relation and the non-equilibrium index in various aging systems would further solidify the findings.

The model assumes a specific form of time-dependent damping and temperature, which might not capture the full complexity of all aging systems. The analysis relies on analytical calculations with certain simplifying assumptions, potentially limiting the applicability to systems with more complex interactions. The study focuses primarily on single-particle dynamics; extending the model to multiple interacting particles could provide further insights into collective behavior during aging.