[Paper Review] Ageing effects in single particle trajectory averages

Johannes H. P. Schulz, Eli Barkai+1 more|arXiv (Cornell University)|Apr 4, 2012

Stochastic processes and statistical mechanics2 references3 citations

TL;DR

This paper develops ageing renewal theory for continuous time random walks (CTRWs) with heavy-tailed waiting times, showing that time-averaged observables in anomalous diffusion are universally depressed by a single ageing function, Λₐ(tₐ/T), independent of external forces or boundary conditions. The key result is that ergodicity breaking increases with measurement age tₐ, quantified by a diverging ergodicity breaking parameter in the aged regime.

ABSTRACT

We study time averages of single particle trajectories in scale free anomalous diffusion processes, in which the measurement starts at some time t_a>0 after initiation of the process at the time origin, t=0. Using ageing renewal theory we show that for such non-stationary processes a large class of observables are affected by a unique ageing function, which is independent of boundary conditions or the external forces. We quantify the weakly non-ergodic nature of this process in terms of the distribution of time averages and the ergodicity breaking parameter which both explicitly depend on the ageing time t_a. Consequences for the interpretation of single particle tracking data are discussed.

Motivation & Objective

• To address the lack of theoretical treatment for time-averaged observables when measurements begin after system initiation (tₐ > 0), a common scenario in single-particle tracking experiments.
• To quantify how ageing—defined as dependence on the measurement start time tₐ—alters the statistical properties of time-averaged observables in non-stationary, scale-free anomalous diffusion processes.
• To establish a universal ageing depression function, Λₐ(tₐ/T), that governs the scaling of time-averaged mean squared displacement (TAMSD) across diverse physical conditions.
• To analyze the distribution of time averages and the ergodicity breaking parameter in the aged regime, showing their explicit dependence on tₐ/T.
• To provide a framework applicable beyond CTRW models, with implications for systems like blinking quantum dots and infinite ergodic theory.

Proposed method

• Derives an ageing renewal theory for continuous time random walks (CTRWs) with power-law distributed waiting times ψ(t) ∝ t⁻⁽¹⁺ᵃ⁾ (0 < α < 1), leading to anomalous diffusion.
• Introduces the ageing depression function Λₐ(z) = (1+z)ᵃ − zᵃ, where z = tₐ/T, as the universal prefactor governing time-averaged observables in the limit Δ ≪ T.
• Uses renewal theory to show that time-averaged observables, such as TAMSD, scale as ⟨TAMSD⟩ = Λₐ(tₐ/T) × g(Δ)/T¹⁻ᵃ, with g(Δ) encoding Δ-dependence.
• Demonstrates that the distribution of time-averaged observables, ξ = TAMSD / ⟨TAMSD⟩, is equivalent in distribution to the normalized number of jumps nₐ(tₐ,T) in the observation window.
• Applies extreme statistics and stable laws to derive the full distribution φ(ξ) of time-averaged observables, showing a bimodal structure with a growing immobile fraction (ξ=0) as tₐ/T increases.
• Computes the ergodicity breaking parameter EB as a function of tₐ/T and α, showing EB diverges in the limit tₐ/T → ∞, indicating enhanced non-ergodic fluctuations in aged systems.

Experimental results

Research questions

• RQ1 How does the statistical behavior of time-averaged observables in anomalous diffusion change when measurements begin at a finite time tₐ > 0 after system initiation?
• RQ2 What universal scaling function governs the ageing dependence of time-averaged mean squared displacement (TAMSD) in continuous time random walks with heavy-tailed waiting times?
• RQ3 How does the distribution of time-averaged observables evolve with ageing, and what fraction of trajectories remain immobile during the measurement window?
• RQ4 In what way does the ergodicity breaking parameter EB depend on the ratio tₐ/T, and can it diverge in the aged regime?
• RQ5 To what extent is the ageing effect independent of external forces, boundary conditions, or specific physical observables?

Key findings

• The time-averaged mean squared displacement (TAMSD) for a CTRW with heavy-tailed waiting times scales universally as ⟨TAMSD⟩ = Λₐ(tₐ/T) × g(Δ)/T¹⁻ᵃ, where Λₐ(z) = (1+z)ᵃ − zᵃ is the ageing depression function independent of Δ, g(Δ), or system details.
• The distribution of time-averaged observables φ(ξ) splits into two components: a delta peak at ξ=0 (immobile trajectories) with weight ≈ (T/tₐ)¹⁻ᵃ for tₐ ≫ T, and a continuous part described by a special function involving the H-function.
• The ergodicity breaking parameter EB diverges as tₐ/T → ∞, scaling as EB ∼ 2(tₐ/T)¹⁻ᵃ / [α(1+α)], indicating that non-ergodic fluctuations grow significantly in aged systems.
• For Δ ≪ T, the distribution of the time-averaged observable is statistically equivalent to the normalized number of jumps in the observation window, i.e., ξ ≡ TAMSD / ⟨TAMSD⟩ ≡ nₐ / ⟨nₐ⟩ in distribution.
• The ageing depression function Λₐ(tₐ/T) is universal: it applies to all physical observables in the CTRW model, regardless of external forces, boundary conditions, or the specific form of g(Δ).
• Numerical simulations confirm the analytical predictions for φ(ξ) and EB across different α values and system configurations (free particle, confined particle), validating the theoretical framework.