Scaling Green-Kubo Relation and Application to Three Aging Systems

The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with long-range or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function $⟨v(t+\tau )v(t)⟩$ asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, $⟨x^2(t)⟩\simeq 2D_{\nu }{t}^{\nu }$. We show how both the anomalous diffusion coefficient $D_{\nu }$ and the exponent $\nu$ can be extracted from this scaling form. Our scaling Green-Kubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity $D_{\nu }$ is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

Popular Summary

In 1827, Robert Brown observed that pollen on the surface of water will move about and spread, even if the water seems perfectly still. The essence of that spreading, namely, that pollen particles move about randomly as a result of random collisions with water molecules that actually are not still, is captured by the so-called Green-Kubo formula. The formula relates the rate of the spatial spreading of the pollen particles, called the “diffusion constant,” to the statistical variations in their velocities and is one of the most fundamentally insightful and powerful results of statistical physics. Its validity rests on the assumption that the water is in thermal equilibrium. More recently, however, it has seen a fundamental limit in its applications to physical systems that exhibit “aging”—systems that take infinitely long to evolve toward their equilibrium state but never reach it. In this paper, we remove this limit by presenting a generalization of the Green-Kubo formula to aging systems.

The traditional paradigm of random diffusion underlying the classical Green-Kubo formula is that the radius squared of a spreading particle ensemble grows, on average, linearly with time. In contrast, aging systems generally exhibit anomalous diffusion, where the radius of a diffusing particle cloud grows faster with time. Our generalized Green-Kubo formula provides a concrete relation between the anomalous diffusion constant and the underlying particle velocity correlation function—a measure of the randomness in the particle velocities, in particular, the scaling of the correlation function with time. This new relation reveals one fundamental property of transport processes in an aging system: that the initial properties of the system can greatly influence its transport coefficients.

Our generalized Green-Kubo relation is able to treat a variety of models with numerous applications in physical systems. We have already applied it to a number of stochastic models describing cold atoms in optical lattices, active transport in living cells, and blinking quantum dots, and we expect even broader applications.

Figure 1

Computation of the diffusion exponent and coefficient for cold atoms in optical lattices from the correlation function using the scaling Green-Kubo formula Eq. (9). (a) Correlation function as a function of the total time $t+\tau$ for different values of $t$. (b) Correlation function plotted as a function of $\tau /t$. (c) After rescaling by $t^{\nu -2}$ with $\nu =2.37$. (d) Mean-square displacement obtained by integrating over the rescaled correlation function $\varphi (s)$ (empty circles) compared to the analytical result Eqs. (31) and (32) (red line) and the mean-square displacement obtained directly from the Langevin simulation (blue squares). From the rescaling and integration procedure, we obtain $\nu \simeq 2.37$ and $D_{\nu }\simeq 0.321$, which agrees well with the asymptotic analytical results $\nu =2.37$ and $D_{\nu }=0.323$. The parameters for the simulation are $D=3.25$, $\gamma =5.10$, and $v_c=1.20$ [see Eq. (22)], with $10^6$ trajectories and $2.5\times 10^5$ integration steps per trajectory.

Figure 2

Diffusion coefficient $D_{\nu }/({\xi }^2{\theta }^{-\nu })$ (32) for cold atoms in optical lattices as a function of the parameter $\alpha$ (red line); the result for the normal diffusive case $\alpha >3$ was taken from Ref. [53]. The result $D_{\nu ,\mathrm{s}}$ from using the stationary correlation function Eq. (34) is shown in blue. The coefficient $D_{\nu }$ diverges at the transition from normal to anomalous superdiffusion ($\alpha =3$) and vanishes at the transition from superdiffusion to $t^3$ diffusion ($\alpha =1$). While the values obtained from the aging and stationary correlation functions agree if the system is in the normal phase $\alpha >3$, they differ significantly as the system becomes more superdiffusive, clearly signifying the dependence on the initial condition. As $\alpha$ approaches the transition to aging behavior at $\alpha =2$, the stationary correlation function ceases to provide a good description of the finite time system and the corresponding diffusion coefficient diverges.

Figure 3

The anomalous diffusion coefficient $D_{\nu }/({\mathcal{F}}_{\lambda }{\gamma }_{\rho }^{-2})$ for the fractional Langevin equation as a function of the diffusion exponent $\nu$. The red line is the result using the aging correlation function Eq. (42), and the blue line is the one from the stationary correlation function Eq. (44). Close to the threshold to normal diffusion $\nu =1$, both values agree; however, as the diffusion exponent increases, the two results deviate from each other. At the transition to the nonstationary phase ($\nu =2$), the stationary result diverges.

Figure 4

Aging correlation function for the blinking quantum dots. The plots were obtained from numerical Lévy walk simulations with $t_{\mathrm{c}}=1$ and $\mu =0.6$. Since the correlation function is of the usual aging type ($\nu =2$), no rescaling is necessary for the curves to collapse onto a single one. Integrating this curve, we obtain the coefficient $D_2\simeq 0.043$ in good agreement with the result $D_2=0.05$ from Eq. (48).

Figure 5

The anomalous diffusion coefficient $D_{\nu }/({\mathcal{I}}_0^2{t}_{\mathrm{c}}^{\mu -1})$ for the variance of the photon number. Here, we assumed a waiting-time distribution of the form $\psi (t)=0$ for $t<t_{\mathrm{c}}$ and $\psi (t)=\mu /t_{\mathrm{c}}(t/t_{\mathrm{c}}{)}^{-\mu -1}$ for $t>t_{\mathrm{c}}$. The stationary result $D_{\nu ,\mathrm{s}}$ [Eq. (54), blue line] differs from the nonstationary one $D_{\nu }$ [Eq. (53), red line] but is finite at the transition to the aging phase $\mu <1$. The blue dashed line is the result obtained for long aging times $t_0$ in the aging regime.

Figure 6

Sample trajectories for the three systems discussed in Sec. 3. Top: Diffusion with a nonlinear $1/v$ friction force. Middle: Fractional Langevin equation with external noise. Bottom: Lévy walk alternating between $\pm v_0$. The left figure depicts the velocity of a single trajectory as a function of time; the right figure shows the corresponding position. The parameters were chosen such that the diffusion exponent for all three systems is $\nu \simeq 1.5$ and the (dimensionless) anomalous diffusion coefficient is $D_{\nu }\simeq 0.25$. Despite similar diffusive properties, the actual trajectories are apparently very different.