MULTIDIMENSIONAL GREENS-FUNCTIONS AND THE STATISTICS OF DIFFUSIVE SHOCK ACCELERATION

A three-dimensional, time-dependent Green's function for the diffusive shock acceleration of energetic charged particles at general oblique MHD shocks, without losses, is derived. By interpreting the Green's fu nction as a probability distribution, various statistical means and va riances describing the shock acceleration process are obtained. These include the mean time [tau] for particles to be accelerated from momen tum pi up to momentum p(i) the mean distance [zeta] traveled by partic les parallel to the electric field at the shock, and the mean distance [chi] traveled by particles along the shock surface in the (V, B) pla ne (a new result). We also obtain new second-order moments of the shoc k acceleration process, including the variances sigma(zeta zeta)(2),si gma(chi chi)(2), sigma(zeta tau)(2),sigma(zeta chi)(2), and sigma(chi tau)(2), as well as the only variance sigma(tau tau)(2) obtained in pr evious analyses. The distance, [zeta], traveled by the particle parall el to the electric field at the shock is related to the particle energ y changes at the shock owing to curvature and grad B drift in the elec tric held at the shock. The Green's function for the limiting case of no cross-held diffusion (K-perpendicular to = 0) consists of the produ ct of two Dirac delta distributions and the one-dimensional Green's fu nction for diffusive shock acceleration obtained by previous authors. One of the Dirac delta distributions expresses the fact that for K-per pendicular to = 0 particles are trapped on the same field line project ion on the plane spanned by the fluid velocity V and magnetic held B ( the x-y plane). The other Dirac delta distribution contains informatio n on the energy changes of particles owing to drift at the shock. A st udy of the means and variances of the probability distribution for a m odel diffusion tenser is used to determine the dependence of the shock acceleration process on the angle theta(1) between the upstream magne tic field B-1 and shock normal n, as well as on other physical paramet ers in the model. A discussion is given of the role of perpendicular d iffusion (K(perpendicular to)not equal 0) at general oblique shocks. E xamples of shock acceleration statistics are obtained for traveling in terplanetary shocks. Other possible astrophysical applications are als o discussed.