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The Fokker-Planck (FP) equation ∂tf+μ∂xf=∂μ(1-μ2)∂μf is solved analytically. Foremost among its applications, this equation describes the propagation of energetic particles through a scattering medium (in x-direction, with μ being the x-projection of particle velocity). The solution is found in terms of an infinite series of mixed moments of particle distribution, ⟨μjxk⟩. The second moment ⟨x2⟩ (j=0, k=2) was obtained by G. I. Taylor (1920) in his classical study of random walk: ⟨x2⟩=⟨x2⟩0+t/3+[exp(-2t)-1]/6 (where t is given in units of an average time between collisions). It characterizes a spatial dispersion of a particle cloud released at t=0, with √⟨x2⟩0 being its initial width. This formula distills a transition from ballistic (rectilinear) propagation phase, ⟨x2⟩-⟨x2⟩0≈t2/3 to a time-asymptotic, diffusive phase, ⟨x2⟩-⟨x2⟩0≈t/3. The present paper provides all the higher moments by a recurrence formula. The full set of moments is equivalent to the full solution of the FP equation, expressed in form of an infinite series in moments ⟨μjxk⟩. An explicit, easy-to-use approximation for a point source spreading of a pitch-angle averaged distribution f0(x,t) (starting from f0(x,0)=δ(x), i.e., Green's function), is also presented and verified by a numerical integration of the FP equation. [ABSTRACT FROM AUTHOR]

The unexpected energy spectrum of the positron/electron ratio is interpreted astrophysically, with a possible exception of the 100-300 GeV range. The data indicate that this ratio, after a decline between 0.5 and 8 GeV, rises steadily with a trend towards saturation at 200-400 GeV. These observations (except for the trend) appear to be in conflict with the diffusive shock acceleration (DSA) mechanism, operating in a single supernova remnant (SNR) shock. We argue that e+/e- ratio can still be explained by the diffusive shock acceleration if positrons are accelerated in a subset of SNR shocks which (i) propagate in clumpy gas media and (ii) are modified by accelerated cosmic ray protons. The protons penetrate into the dense gas clumps upstream to produce positrons and charge the clumps positively. The induced electric field expels positrons into the upstream plasma where they are shock accelerated. Since the shock is modified, these positrons develop a harder spectrum than that of the cosmic ray electrons accelerated in other SNRs. Mixing these populations explains the increase in the e+/e- ratio at E>8 GeV. It decreases at E<8 GeV because of a subshock weakening which also results from the shock modification. Contrary to the expelled positrons, most of the antiprotons, electrons, and heavier nuclei, are left unaccelerated inside the clumps. Scenarios for the 100-300 GeV AMS-02 fraction exceeding the model prediction, including, but not limited to, possible dark matter contribution, are also discussed. [ABSTRACT FROM AUTHOR]