Fluctuations of the driven tracer at arbitrary force

Characterize the fluctuations and full distribution of the tracer displacement X_t at arbitrary driving force, including determining the variance of a driven tracer in the symmetric exclusion process beyond the small-drive regime.

Background

The paper derives the exact nonlinear mean displacement amplitude and bath-density profile for general diffusive single-file systems using hydrodynamic transport coefficients and single-file duality. It notes that the corresponding fluctuation problem is substantially less developed: even for the symmetric exclusion process (SEP), the variance of a driven tracer is unavailable at arbitrary force, with only its small-drive behavior known.

The formalism developed in the paper may provide access to the full distribution of the tracer position X_t for general single-file systems, but the associated equations remain difficult to solve. The authors explicitly identify this difficulty as an open problem, including for the SEP.

References

Our work opens the question of the characterization of the fluctuations around the mean displacement~eq:mean_displacement. Even for the SEP, the variance of a driven tracer is not known at arbitrary force; only its small-drive behavior has recently been determined. The formalism presented in this work in principle gives a way to access the full distribution of $X_t$ for general systems, as shown in SM. However, solving these equations, even for a simple model like the SEP, remains a challenging open problem.

— Exact Nonlinear Active Microrheology in Diffusive Single-File Systems  (2609.18733 - Grabsch et al., 16 Sep 2026) in Conclusion