Research - Takuma Akimoto

• Infinite ergodic theory

Infinite ergodic theory has been progressed by mathematicians. In particular, Jon Aaroson has showed a distributional limit theorem for the time average of the L^1 function (Mittag-Leffler distribution), and Maximilian Thaler has showed another distributional limit theorem for the time average of a characteristic function (Generalized arcsine distribution).

In my paper, I show the generalized arcsine law and stable law for the time average of the non-$L^1$ function. From the viewpoint of physics, these distributional limit behaviour would be closely related to random behaviour of macroscopic observables in nature. Infinite ergodic theory could provide a new framework of ergodicity in statistical mechanics.

• Anomalous diffusion

Anomalous diffusion is characterized by nonlinear scaling of the mean square displacement (MSD), which is classified into two anomalous diffusions.

Subdiffusion: MSD increases slower than normal diffusion.
Superdiffusion: MSD increases faster than normal diffusion.

Using infinite ergodic theory, I show that a distributional limit behaviour exists in the time-averaged diffusion coefficient. Namely, the diffusion coefficients are random variables. In particular, the Mittag-Leffler distribution and the generalized arcsine distribution are observed in subdiffusion and superdiffusion, respectively.

• Earthquakes

We study interevent times between the occurrence of earthquakes. In our studies, we find that the distribution of interevent times obeys the superposition of the Weibull and log-Weibull distributions. Using a one-dimensional map, we characterize the intermittency of the occurrence of earthquakes. Generally, the occurrence of earthquakes can not be described as a Markov process. In the paper, we show that the occurrence of earthquakes can be described as a semi-Markov or Markov renewal process by the return map of interevent times.