Far tails of the biased CTRW model under the short time limit

TL;DR

This paper investigates the far tail behavior of biased continuous-time random walk models with Gaussian and discrete displacements, revealing exponential decay and analyzing convergence properties under short time limits.

Contribution

It provides a detailed analysis of the tail distributions in biased CTRW models, highlighting exponential decay and the effects of bias in different asymptotic regimes.

Findings

Exponential decay in the far tails of biased CTRW models.

Convergence of theoretical predictions for position distributions.

Relationship between biased and unbiased distributions in asymptotic limits.

Abstract

It has been observed in numerous experiments, simulations, and various theoretical treatments that the spreading of particles can be modeled by the continuous-time random walk. We consider two well-known cases, i.e., Gaussian displacements and discrete displacements, to compute the position distribution and demonstrate the emergence of exponential decay in the far tails when a bias is introduced. We further analyze the temporal rate function and the positional rate function to examine the convergence of the theoretical predictions. For Gaussian displacements, we further discuss the relationship between the position distributions with and without bias in different asymptotic limits.