2017/08/31 by Philipp Meyer, Philipp G. Meyer, Eli Barkai +2 · 1 citation
Materials Science · Mathematics · Physics and Astronomy · #Anomalous diffusion #Classical mechanics #Correlation function (quantum field theory) #Exponent #Fractional Differential Equations Solutions #Geometry #Invariant (physics) #Kinetic energy #Material Dynamics and Properties #Mathematical analysis #Mathematical physics #Mathematics #Mean squared displacement #Physics #Quantum mechanics #Scale invariance #Scaling #Statistical physics #Statistics #Theoretical and Computational Physics #Thermal diffusivity #Thermodynamics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.96.062122
published as Phys. Rev. E 96, 062122 (2017)
arxiv created 2017/08/31 · openalex publication_date 2017/12/15 · arxiv updated 2017/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time- and ensemble-averaged mean-squared displacement are remarkably different. The ensemble-averaged diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale-invariant nonstationary 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 \ensuremath⟨\ensuremathδ2\ensuremath⟩\ensuremath∼2D_\ensuremathνt^\ensuremathβ\mathrm\ensuremathΔ^\ensuremathν\ensuremath-\ensuremathβ, where t is the total measurement time and \mathrm\ensuremathΔ is the lag time. Here \ensuremathν is the anomalous diffusion exponent obtained from ensemble-averaged measurements \ensuremath⟨x2\ensuremath⟩\ensuremath∼t^\ensuremathν, while \ensuremathβ\ensuremath≥\ensuremath-1 marks the growth or decline of the kinetic energy \ensuremath⟨v2\ensuremath⟩\ensuremath∼t^\ensuremathβ. Thus, we establish a connection between exponents that can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant D_\ensuremathν. We demonstrate our results with nonstationary scale-invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. If the averaged kinetic energy is finite, \ensuremathβ=0, the time scaling of \ensuremath⟨\ensuremathδ2\ensuremath⟩ and \ensuremath⟨x2\ensuremath⟩ are identical; however, the time-averaged transport coefficient D_\ensuremathν is not identical to the corresponding ensemble-averaged diffusion constant.