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# 4th IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis

November 29, 2021 to December 6, 2021
Virtual event
Europe/Vienna timezone
30 Nov - 3 Dec, 2021 Abstract submission open NOW

## Mori-Zwanzig projection operator method as a statistical correlation analysis of time-series data

Nov 29, 2021, 3:30 PM
15m
Virtual event

#### Virtual event

Regular Oral Data analysis preparation for ITER and Software Tools for ITER diagnostics

### Speaker

Dr Shinya Maeyama (Nagoya University)

### Description

Mori-Zwanzig projection operator method as a statistical correlation analysis of time-series data

S. Maeyama, M. Misawa, T.-H. Watanabe
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
e-mail: smaeyama@p.phys.nagoya-u.ac.jp

Mori-Zwanzig projection operator method is a mathematical method developed in non-equilibrium statistical physics [1]. A key is a decomposition of a time propagator by the Dyson decomposition, where the time propagator is split into the projected and complementary parts based on a definition of a projection operator [2]. Using the Mori’s linear projection on initial values [1] as the projection operator, we have developed a statistical correlation analysis method for two time-series ensemble data in a statistically steady state [3]. Based on the method, the analyzed time series data $f(t)$ is split into a correlated (Markov and memory) part and an uncorrelated part $r(t)$ with regard to the variable of interest $u(t)$. The uncorrelated part $r(t)$ is characterized by the no statistical correlation with the initial value $u(0)$, namely, $\langle r(t)u(0) \rangle = 0$, where the angle brackets denote the ensemble average. There is a relation between the memory function and the uncorrelated term, known as a generalized fluctuation-dissipation theorem of the second kind.

In this presentation, we will explain the formulation of the projection operator method, the validity range of the method, and the procedure to apply the method to analyze two time-series ensemble data as a practical data analysis tool. As an example, we have analyzed the nonlinear excitation and damping of zonal flows in Hasegawa-Wakatani resistive drift wave turbulence [4]. We found that Reynolds stress of turbulence on zonal flows acts not only a stochastic forcing but also damping via the correlation with the memory term.

[1] H. Mori, Prog. Theor. Phys. 33, 423 (1965).
[2] D. J. Evans, and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Cambridge Univ. Press, Cambridge, U.K., 2008).
[3] S. Maeyama, and T.-H. Watanabe, J. Phys. Soc. Jpn. 89, 024401 (2020); Source code is available from https://github.com/smaeyama/mzprojection.
[4] M. Wakatani, and A. Hasegawa, Phys. Fluids 27, 611 (1984).

Affiliation Nagoya University Japan

### Primary authors

Dr Shinya Maeyama (Nagoya University) Ms Megumi Misawa (Nagoya University)