# Time-averaged MSD for switching diffusion

## TL;DR

This paper analyzes a two-state switching diffusion model, deriving exact expressions for the mean and variance of the TAMSD, revealing ergodicity breaking effects in finite measurement times that align with experimental observations.

## Contribution

It provides the first exact calculations of TAMSD statistics for switching diffusions, highlighting conditions for ergodicity breaking and offering an ergodic alternative to non-ergodic models.

## Key findings

- Exact mean and variance of TAMSD derived.
- Ergodicity breaking parameter can be arbitrarily large.
- Switching diffusions can mimic non-ergodic behavior.

## Abstract

We consider a classic two-state switching diffusion model from a single-particle tracking perspective. The mean and the variance of the time-averaged mean square displacement (TAMSD) are computed exactly. When the measurement time (i.e., the trajectory duration) is comparable to or smaller than the mean residence times in each state, the ergodicity breaking parameter is shown to take arbitrarily large values, suggesting an apparent weak ergodicity breaking for this ergodic model. In this regime, individual random trajectories are not representative while the related TAMSD curves exhibit a broad spread, in agreement with experimental observations in living cells and complex fluids. Switching diffusions can thus present, in some cases, an ergodic alternative to commonly used and inherently non-ergodic continuous-time random walks that capture similar features.