# A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables

## TL;DR

This paper investigates a memory-driven random walk model exhibiting a transition from diffusive to superdiffusive behavior, revealing non-ergodic properties and aging effects, with implications for understanding anomalous diffusion processes.

## Contribution

It introduces a novel memory-dependent model showing a diffusive-superdiffusive transition and analyzes its ensemble and time-averaged properties, highlighting non-ergodicity and aging phenomena.

## Key findings

- Ensemble behavior varies from normal to ballistic diffusion depending on memory parameters.
- Time-averaged mean squared displacement remains normal, indicating non-ergodicity.
- Response to bias diminishes asymptotically, contrasting with Levy walk behavior.

## Abstract

The ensemble properties and time-averaged observables of a memory-induced diffusive-superdiffusive transition are studied. The model consists in a random walker whose transitions in a given direction depend on a weighted linear combination of the number of both right and left previous transitions. The diffusion process is nonstationary and its probability develops the phenomenon of aging. Depending on the characteristic memory parameters, the ensemble behavior may be normal, superdiffusive, or ballistic. In contrast, the time-averaged mean squared displacement is equal to that of a normal undriven random walk, which renders the process non-ergodic. In addition, and similarly to Levy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)], for trajectories of finite duration the time-averaged displacement apparently become random with properties that depend on the measurement time and also on the memory properties. These features are related to the non-stationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Levy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)], the response always vanishes asymptotically.