Anomalous diffusions induced by enhancement of memory

Abstract

We introduced simple microscopic non-Markovian walk models which describe underlying mechanism of anomalous diffusions. In the models, we considered the competitions between randomness and memory effects of previous history by introducing the probability parameters. The memory effects were considered in two aspects, one is the perfect memory of whole history and the other is the latest memory improved with time. In the perfect memory model superdiffusion was induced with the relation the Hurst exponent $H$ to the controlling parameter $p$ as $H=p$ for $p>1/2$. While in the latest memory enhancement models, anomalous diffusions involving both superdiffusion and subdiffusion were induced with the relations $H=(1+α)/2$ and $H=(1-α)/2$ for $0 \le α\le 1$ where $α$ is the parameter controlling the degree of the latest memory enhancement. Also we found that although the latest memory was only considered, the memory improved with time results in the long-range correlations between steps and the correlations increases as time goes. Thus we suggest the memory enhancement as a novel key origin describing anomalous diffusions.