A non-equilibrium Alfvénic state of the Langevin system for single particles reproduces the linear relation between the cross helicity and the residual energy in the solar wind

As a minimal phenomenological model of unidirectional Alfvénic fluctuations in the solar wind, a non-equilibrium Alfvénic state (NAS) in the Langevin system for single particles is studied. The NAS is a nonequilibrium steady state of the Langevin system including both friction force and random force. It is shown that the overdamped Langevin system gives the NAS without the assumption of the monochromatic wave. The resultant NAS gives the relationship between the energy dissipation rate, the cross-helicity, and the residual energy with a single phenomenological parameter. It is shown that the energy dissipation rate can be evaluated using the non-equilibrium work, which satisfies the fluctuation dissipation-like relation.