Tempered Aging Effects And The Diffusion Equation

In this paper, we mainly discuss the aging effects of random walk process, its waiting time obeys exponentially tempered distribution and jump length is Gaussian distribution. Firstly, we are based on the aging renewal theory to obtain the pth moment of the renewal process nα(tα, t) in the time interval [tα, tα+t]. In our paper, we give two different systems, namely, the strongly aged system and the slightly aged system. Especially, we focus on analysing the behaviors of the mean number of steps and the corresponding numerical simulation in this two limits. Then, we introduce one-dimensional lattice continuous time random walks(CTRW) with tem-pered aging effects. we can be seen an interesting population splitting phenomenon: a fraction of the particles are keeping immobile, while the complimentary fraction are mobile. And we get the mean square displacement also for two limit situations. Furthermore, we give a quite different method which can also analyze the aging effects, namely, applying a force to system at the aging time, it becomes a biased asymmetric lattice continuous time random walks. we discuss the strong correla-tions between fluctuation and response in this case of aging process. Then we can derive an tempered aging Einstein relation. In the end of our paper, we obtain the tempered fractional aging diffusion equation, it has tempered fractional operator and the meaning of its solution is the probability density of corresponding particle.