# A comparison of dynamical fluctuations of biased diffusion and run-and-tumble dynamics in one dimension

## TL;DR

This paper compares fluctuations in biased diffusion and run-and-tumble dynamics in 1D, revealing conditions under which their velocity fluctuation statistics become identical despite different underlying processes.

## Contribution

It identifies a unique rate condition where velocity fluctuations of passive and active particles match, while their effective dynamics remain distinct.

## Key findings

- Velocity rate functions become identical under specific rate rescaling.
- Spectra of generator operators show remarkable parity at this condition.
- Both models exhibit a dynamical phase transition in occupation time when average velocity is non-zero.

## Abstract

We compare the fluctuations in the velocity and in the fraction of time spent at a given position for minimal models of a passive and an active particle: an asymmetric random walker and a run-and-tumble particle in continuous time and on a 1D lattice. We compute rate functions and effective dynamics conditioned on large deviations for these observables. While generally different, for a unique and non-trivial choice of rates (up to a rescaling of time) the velocity rate functions for the two models become identical, whereas the effective processes generating the fluctuations remain distinct. This equivalence coincides with a remarkable parity of the spectra of the processes' generators. For the occupation-time problem, we show that both the passive and active particles undergo a prototypical dynamical phase transition when the average velocity is non-vanishing in the long-time limit.