Scale invariant Green-Kubo relation for time averaged diffusivity

Abstract

In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time and ensemble average mean squared displacement are remarkably different. The ensemble average diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale invariant non-stationary velocity correlation function with the transport coefficient. Here we obtain the relation between time averaged diffusivity, usually recorded in single particle tracking experiments, and the underlying scale invariant velocity correlation function. The time averaged mean squared displacement is given by $\overline{{\delta }^2}\sim 2D_{\nu }{t}^{\beta }{\Delta }^{\nu -\beta }$ where $t$ is the total measurement time and $\Delta$ the lag time. Here $\nu >1$ is the anomalous diffusion exponent obtained from ensemble averaged measurements $⟨x^2⟩\sim t^{\nu }$ while $\beta \ge -1$ marks the growth or decline of the kinetic energy $⟨v^2⟩\sim t^{\beta }$. Thus we establish a connection between exponents which can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant $D_{\nu }$. We demonstrate our results with non-stationary scale invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. This is the case, for example, if averaged kinetic energy is finite, i.e. $\beta =0$, where $⟨\overline{{\delta }^2}⟩\ne ⟨x^2⟩$.

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• Anomalous diffusion and the Moses effect in a model of aging
• Ergodic property of random diffusivity system with trapping events
• Aging two-state process with Lévy walk and Brownian motion
• Langevin picture of Lévy walk in a constant force field
• Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential
• Langevin picture of anomalous diffusion processes in expanding medium
• Lévy-walk-like Langevin dynamics affected by a time-dependent force
• Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models
• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Autocorrelation functions and ergodicity in diffusion with stochastic resetting
• Complete Decomposition of Anomalous Diffusion in Variable Speed Generalized Lévy Walks