Statistical Properties of the T-Exponential of Isotropically Distributed Random Matrices

Abstract

A functional method for calculating averages of the time-ordered exponential of a continuous isotropic random N × N matrix process is presented. The process is not assumed to be Gaussian. In particular, the Lyapunov exponents and higher correlation functions of the T-exponent are derived from the statistical properties of the process.The approach may be of use in a wide range of physical problems. For example, in theory of turbulence the account of non-gaussian statistics is very important since the non-Gaussian behavior is responsible for the time asymmetry of the energy flow.

Magnetic energy spectrum produced by turbulent dynamo: effect of time irreversibility

“…We are interested in the influence of the third order correlator. So, in the frame of the V 3 model (Il'yn et al 2016(Il'yn et al , 2019Kopyev et al 2021) we consider the non-zero second and third order correlators,…”

“…In this paper we develop the generalization of the Kazantsev-Kraichnan model for time asymmetric velocity statistics. The V 3 model was first proposed in Il'yn et al (2019Il'yn et al ( , 2016; it describes small time anisotropy by taking account of the non-zero third order correlator. Here we combine it with the Kazantsev approach and calculate the magnetic energy spectrum produced by a slightly non-Gaussian flow (which corresponds to real hydrodynamic turbulence).…”

Infinite Products of Random Isotropically Distributed Matrices

“…Since correlation functions is the only quantity that has physical meaning, it is more convenient to describe δ-processes by a cumulant function, thereby to avoid divergency problems. One more way to work with delta-correlated non-gaussian processes without divergences is introducing of smoothed variables, as it was done by [21].…”

Mixed Triple Two-Point Correlations of Velocity and Velocity Gradients in Homogeneous Isotropic Turbulence

“…Новая скалярная функция может быть с большой точностью аппроксимирована в инерционном диапазоне масштабов постоянным значением, зависящим только от скорости диссипации. Обсуждается значение полученных корреляторов в теории турбулентного транспорта (Il'yn, Sirota, Zybin;2016, Казанцев, 1967.…”

“…The function can be well approximated in the inertial range by a constant value that depends on the dissipation only. The application of the obtained correlators in the turbulent transport theory is discussed (Il'yn, Sirota, Zybin, 2016;Kazantsev, 1968).…”