Transport and entropy production due to chaos or turbulence

A projection-operator method is developed for the statistical-mechanical formulation of chaotic or turbulent transport, such as chaos-induced friction in a forced damped pendulum and turbulent viscosity in a turbulent fluid. Then the nonlinear deterministic equations of motion for these dynamical systems are transformed into linear stochastic equations with chaotic or turbulent fluctuating forces. This leads to a fluctuation-dissipation formula which relates the chaotic or turbulent transport coefficients to the time correlation of the fluctuating forces. Applying this theory to the forced damped pendulum, we explore the chaos-induced friction and the power spectra of chaotic orbits. Applying it to the fluid turbulence governed by the Navier-Stokes equation, we find that the turbulent viscosity in the inertial subrange depends on wave number k as k(-beta) with beta=4 / 3+1 / 2/micro(2/3)/, micro (q) being the intermittency exponent of order q.