Ultraslow vacancy-mediated tracer diffusion in two dimensions: the Einstein relation verified

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics

Background:

• Studying particle dynamics on lattices is crucial for understanding transport phenomena.
• Hard-core exclusion and vacancy-mediated movement are key features in many physical systems.

Purpose of the Study:

• To investigate the anomalous diffusion of a charged tracer particle (TP) on a 2D lattice.
• To determine the impact of an external electric field on TP dynamics.
• To verify the Einstein relation under biased and unbiased conditions.

Main Methods:

• Exact calculation of the limiting probability distribution of the TP position.
• Analysis of the leading large-time behavior of the TP mean displacement.
• Comparison with existing theoretical predictions.

Main Results:

• The biased TP exhibits anomalous logarithmic diffusion: |X(n)|=α(0)(|E|)ln(n).
• The Einstein relation between diffusivity and mobility holds even when both vanish as time increases.
• A ballistic-type law |X(n)|=π(α)(0)(|E|)ρ(v)n is observed for small vacancy concentrations.

Conclusions:

• The Einstein relation is robust and applies to systems with vanishing transport coefficients.
• The study provides exact results for biased tracer particle dynamics on a lattice.
• The findings generalize understanding of transport phenomena in interacting particle systems.