Generalized Einstein relation for aging processes

The study addresses how the Einstein relation between mobility and diffusion, a cornerstone of equilibrium statistical mechanics, extends to non-stationary, aging systems. Traditional approaches built for stationary states are insufficient for processes exhibiting physical aging—characterized by slow non-exponential relaxation, broken time-translation invariance, and dynamical scaling—common in disordered materials and other complex systems. Prior experiments studied frequency-dependent effective temperatures, but explicit time-dependent temperatures in the fluctuation-dissipation context have been less explored. Motivated by dissipative granular gases where inelastic collisions cool the environment, the work asks whether a precise definition and minimal model of aging can be realized without spatial disorder, and whether a generalized Einstein relation can diagnose anomalous diffusion and weak ergodicity breaking in such non-stationary settings.

The Einstein relation and fluctuation-dissipation theorem have been validated and generalized in various equilibrium and near-equilibrium contexts (e.g., experiments on aging colloidal glasses and theoretical generalizations for anomalous transport). Prior works addressed frequency-dependent effective temperatures and non-stationary fluctuation-dissipation forms, but explicit time-dependent environmental temperatures remained underexplored. Aging has been widely discussed in disordered systems and via scaling relations (e.g., aging Wiener-Khinchin theorems and scaling Green-Kubo relations), yet simple stochastic models without spatial disorder treating aging comprehensively are scarce. Granular gases provide a pertinent platform, with Haff’s law describing cooling and prior theory/experiments often assuming homogeneity and isotropy. This paper positions its irreversible Langevin approach within these developments to extend Einstein’s relation to aging with time-dependent damping and temperature.

The authors construct an irreversible Langevin model with time-dependent friction and noise to capture aging dynamics: dot{x}=v, m dot{v} = -m g^2(t) v - α U(x) + g(t) ξ(t), where the Gaussian noise satisfies a time-dependent fluctuation-dissipation relation ⟨g(t)ξ(t) g(t′)ξ(t′)⟩ = 2 m g^2(t) k_B T(t) δ(t−t′). They choose g(t) = [γ_0/(1 + t/τ_0)]^{1/2} and T(t) = T_0/(1 + t/τ_0), yielding γ(t) = ∫_0^t g^2(t′) dt′ = γ_0 τ_0 ln(1 + t/τ_0) and λ = γ_0 τ_0. The velocity is solved for U=0: v(t) = v(0) e^{−γ(t)} + ∫0^t e^{−γ(t)+γ(t′)} g(t′) ξ(t′) dt′. Assuming ⟨v(0)⟩=0 and t_2 ≥ t_1, the velocity correlation function (VCF) exhibits aging (two-time dependence via ratios) and remains positive for λ > b/2 (with b the cooling exponent, 0≤b≤2). Under a constant force U(x)=F x, replacing g(t)ξ(t) by F+g(t)ξ(t) gives the mean displacement: ⟨x(t)⟩ = [F τ_0/(m(1+γ_0 τ_0))][(1+t/τ_0)^{γ_0 τ_0} − 1] + ⟨x(0)⟩ + ⟨v(0)⟩ τ_0[(1+t/τ_0)^{γ_0 τ_0} − 1]/(1−γ_0 τ_0), revealing dissipative acceleration and an effective force F_eff = F/(1+γ_0 τ_0). The mean-squared displacement (MSD) without potential is obtained from integrating the VCF: for 0≤b<2, ⟨x^2(t)⟩ = [4 k_B T_0 γ_0 τ_0/(m(2−b)(1+λ))] ((1+t/τ_0)^{λ−1}[(1+t/τ_0)^{2−b}−1]/(λ−1)) + A(t;x(0),v(0),b), with A collecting initial-condition terms; for b=2, ⟨x^2(t)⟩ = [2 k_B T_0 γ_0 τ_0/(m(λ−1)^2)] ln(1+t/τ_0) + [2 k_B T_0 γ_0 τ_0/(m(λ−1)^3)][(1+t/τ_0)^{λ−1}−1] + A(t; x(0), v(0),2). A generalized Einstein relation is probed via the dimensionless ratio R(t)= F ⟨x^2(t)⟩{F=0}/[2 k_B T_0(t) ⟨x(t)⟩F], where T_0(t)=T(t). Using the analytical forms yields, for 0<b<2, a finite asymptote R(∞)= 4λ(λ+1)/[(2λ−b)(2−b)(λ+1−b)] > 1; for b=2, R(t) ≈ 2λ(λ+1)/(λ−1)^2 + ln(1+t/τ_0), diverging ultra-slowly. They also define the system’s effective temperature via kinetic energy, k_B T_sys(t)= m ⟨v^2(t)⟩, and analyze its ratio to T_env(t)=T(t). Ergodicity is examined using irreversibility (lim{t→∞}⟨v(t)v(t_2)⟩=0) and a fluctuation metric Ω_A(t) for the time-averaged velocity A = (1/t)∫_0^t v(t′) dt′, testing convergence and highlighting weak ergodicity breaking.

• A generalized Einstein relation holds in aging media with time-dependent damping and temperature: the ratio R(t)= F ⟨x^2(t)⟩_{F=0}/[2 k_B T(t) ⟨x(t)⟩_F] approaches a finite value R(∞)= 4λ(λ+1)/[(2λ−b)(2−b)(λ+1−b)] > 1 for 0<b<2, indicating stronger diffusion relative to mobility than in equilibrium. For b=2 (Haff cooling), R(t) diverges ultra-slowly as ~ ln(1+t/τ_0), signaling a breakdown of the Einstein relation in the logarithmic phase.
• Mean displacement under constant force exhibits dissipative acceleration with an effective force F_eff= F/(1+γ_0 τ_0), positioning the dynamics between Newtonian (F_eff=F) and Langevin (F_eff→0) limits.
• MSD displays full-scale anomalous diffusion controlled by b: ballistic (∝ t^2) at b=0; normal diffusion (∝ t) at b=1; subdiffusion for 0<b<2, b≠1; and ultra-slow, Sinai-type diffusion (∝ ln t) at b=2.
• The velocity correlation function is aging (two-time scaling via ratios) and decays more slowly than exponentially due to irreversibility and environmental cooling; positivity requires λ>b/2 (practically λ>1).
• The system’s effective temperature exceeds the environment’s at long times, with lim_{t→∞} T_sys(t)/T_env(t) = 2/(2λ − b) > 1, consistent with experiments in cooling media.
• Irreversibility (vanishing long-time correlations) and convergence of time-averaged observables, while necessary, are not sufficient for ergodicity in aging systems; weak ergodicity breaking persists despite these conditions.
• The model maps naturally onto granular gases: b=2 (Haff’s law, T∼1/t^2) yields ultra-slow diffusion; impact-velocity-dependent restitution can give b=5/3 and subdiffusion, in agreement with kinetic theory expectations.

The results demonstrate that non-stationary environments with vanishing friction and cooling temperatures fundamentally alter the fluctuation-mobility balance. The generalized Einstein relation quantifies deviations from equilibrium through R(t), revealing regimes where diffusion per unit temperature exceeds mobility per unit force (R>1) and identifying a logarithmic regime (b=2) where the relation fails. This provides a practical diagnostic for aging dynamics and nonergodicity: measuring mean displacement under force versus force-free MSD and environmental temperature can reveal weak ergodicity breaking. The dissipative acceleration underscores how vanishing damping reshapes transport, producing dynamics intermediate between Newtonian and Langevin. The aging VCF’s scaling form links to broader theoretical frameworks (aging Wiener-Khinchin and Green-Kubo scaling). In granular gases, the framework captures known cooling laws and predicts corresponding diffusion behaviors, offering a bridge between single-particle stochastic models and dissipative many-body kinetics. The finding that T_sys>T_env at long times highlights non-equilibrium energy partitioning and connections to systems with negative specific heat, challenging equilibrium thermodynamic intuitions.

The study introduces an irreversible Langevin framework with time-dependent friction and temperature that captures the hallmarks of aging dynamics and yields a generalized Einstein relation. For 0<b<2, the relation holds with a finite correction factor R(∞)>1; for b=2, it breaks down via ultra-slow divergence. Transport exhibits full-scale anomalous diffusion, from ballistic to logarithmic Sinai-type, and drift shows dissipative acceleration with an effective force reduced by (1+γ_0 τ_0). Using the generalized Einstein relation as a probe, the work clarifies that irreversibility and convergence of time averages do not guarantee ergodicity in aging systems, evidencing weak ergodicity breaking. The model naturally applies to granular gases, reproducing Haff’s law and predicting diffusion behaviors under different restitution scenarios. Future work could test these predictions experimentally in cooling granular systems and extend the framework to more complex potentials, spatial disorder, or active matter with inertia.

• The analysis is based on a phenomenological single-particle Langevin model with specific power-law time dependences: g(t) ∝ (1+t/τ_0)^{-1/2} and T(t) ∝ (1+t/τ_0)^{-1}. Results rely on parameter constraints such as 0≤b≤2 and λ=γ_0 τ_0 > b/2 (practically λ>1) to ensure positivity and well-defined dynamics.
• Many results are derived for either no potential or a linear potential U(x)=F x; behaviors in general nonlinear or disordered potentials are not addressed.
• The noise is assumed Gaussian and delta-correlated (white) at each time with a time-dependent amplitude; colored noise or more complex bath memory effects are not treated.
• Experimental validation is discussed qualitatively via granular gas analogies, but direct empirical tests of R(t) and dissipative acceleration in real systems are not provided within this work.