Unfolding Tagged Particle Histories in Single-File Diffusion: Exact Single- and Two-Tag Local Times Beyond Large Deviation Theory

TL;DR

This paper provides exact analytical results for the fluctuations and correlations of tagged particle local times in single-file diffusion, revealing how particle histories become correlated due to collective rearrangements, beyond large deviation theory.

Contribution

It introduces rigorous results for the statistics of functionals of ergodic Markov processes and applies them to derive exact local time distributions in single-file diffusion.

Findings

Exact formulas for one- and two-tag local times.

Particle histories become correlated at higher densities.

Memory effects are linked to collective particle rearrangements.

Abstract

Strong positional correlations between particles render the diffusion of a tracer particle in a single file anomalous and non-Markovian. While ensemble average observables of tracer particles are nowadays well understood, little is known about the statistics of the corresponding functionals, i.e. the time-average observables. It even remains unclear how the non-Markovian nature emerges from correlations between particle trajectories at different times. Here, we first present rigorous results for fluctuations and two-tag correlations of general bounded functionals of ergodic Markov processes with a diagonalizable propagator. They relate the statistics of functionals on arbitrary time-scales to the relaxation eigenspectrum. Then we study tagged particle local times -- the time a tracer particle spends at some predefined location along a single trajectory up to a time t. Exact results are derived for one- and two-tag local times, which reveal how the individual particles' histories become correlated at higher densities because each consecutive displacement along a trajectory requires collective rearrangements. Our results unveil the intricate meaning of projection-induced memory on a trajectory level, invisible to ensemble-average observables, and allow for a detailed analysis of single-file experiments probing tagged particle exploration statistics.