The authors consider the movement of an inertial particle, i.e. a particle with mass, in a periodic velocity field \(v\) described by the equation \[ \tau \ddot x = v(x) -\dot x + \sigma \dot \beta \tag{1} \] where \(\beta(t)\) denotes the standard Brownian motion in \({\mathbb R}^d\), \(d\geq 1\). Within a formal framework it is shown that the behaviour at large scales and long times is governed by an effective diffusion equation. Furthermore, the effective diffusivity tensor is calculated.
In order to derive the effective differential equation time and space are rescaled according to \(t\to{t/ \varepsilon^2}\) and \(x\to x/\varepsilon\) modifying thereby equation (1). For the solution \(u^{\varepsilon}\) of the associated backward Kolmogorov equation a multiple scale expansion ansatz is made: \[ u^{\varepsilon}= u_{0} + \varepsilon u_{1} + \varepsilon^2 u_{2} + \ldots \] leading under appropriate conditions on the velocity field to the results. The expansion is justified by the hypoellipticity of the underlying differential equation which is proved in an appendix.
The results are compared with those for a passive tracer, that is a massless particle, as well as with numerical results obtained for the Taylor-Green field \(v_{TG}\) using among other things Monte Carlo simulations. Eventually, details for special fields (incompressible fluid, potential field) and the dependence on the Stokes number \(\tau\) are discussed.