On stochastic evolution equations for chaos and turbulence

The chaotic orbits of dynamical systems have positive Liapunov exponents, and become stochastic and random on long timescales due to the orbital instability, leading to various remarkable phenomena, such as (1) ss of memory with respect to the initial states, ssipation of the kinetic energy into random fluctuations, ent transport phenomena (e.g., turbulent viscosity, turbulent thermal diffusivity). er to obtain a statistical-mechanical approach to these phenomena, we have formulated the randomization of chaotic orbits by deriving a non-Markovian stochastic evolution equation in terms of a nonlinear fluctuating force and a memory function. following, we outline its derivation and its application to the Boussinesq equations of turbulent Bénard convection. Then we find that turbulence produces an interference between the velocity flux and the heat flux which is similar to the interference between the electric current and the heat flux in the thermoelectric phenomena of metals.