Thermostatted delta f

The delta f simulation method is revisited. Statistical coarse-graining is used to rigorously derive the equation for the fluctuation delta f in the particle distribution. It is argued that completely collisionless simulation is incompatible with the achievement of true statistically steady states with nonzero turbulent fluxes because the variance of the particle weights w grows with time. To ensure such steady states, it is shown that for dynamically collisionless situations a generalized thermostat or W-stat may be used in lieu of a full collision operator to absorb the flow of entropy to unresolved fine scales in velocity space. The simplest W-stat can be implemented as a self-consistently determined, time-dependent damping applied to w. A precise kinematic analogy to thermostatted nonequilibrium molecular dynamics (NEMD) is pointed out, and the justification of W-stats for simulations of turbulence is discussed. An extrapolation procedure is proposed such that the long-time, steady-state, collisionless flux can be deduced from several short W-statted runs with large effective collisionality, and a numerical demonstration is given