Ergodic time scale and transitive dynamics in single-particle tracking.

Area of Science:

• Physics
• Physical Chemistry
• Statistical Mechanics

Background:

• Single-particle tracking (SPT) is vital for observing molecular dynamics.
• Ergodicity is a fundamental concept linking time averages to ensemble averages.
• Understanding ergodic time scales is crucial for interpreting SPT data accurately.

Purpose of the Study:

• To develop a novel method for quantifying ergodic time scales in SPT experiments.
• To establish a connection between experimental observations and theoretical models of particle dynamics.
• To provide guidelines for optimizing measurement parameters in SPT.

Main Methods:

• Introduction of a covariance measure, Ω(Δ;t), for time-averaged relative square displacement.
• Modeling based on the generalized Langevin equation with a power-law memory function.
• Derivation of a finite-time-averaged Green-Kubo relation.

Main Results:

• A universal scaling law for Ω(Δ;Δ)/Ω(Δ;t) is identified for long, finite times.
• The effective ergodic time can be extracted using this scaling law.
• Optimal measurement conditions are determined by the ratio of lag-time (Δ) to elapsed time (t).

Conclusions:

• The study provides a robust method to determine ergodic time scales from SPT data.
• It bridges the gap between experimental self-averaging properties and theoretical velocity autocorrelation functions.
• The findings offer insights into the transition to ergodicity in complex systems.