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Unformatted text preview: 14. Diffusion models 14.1 The diffusion equation In the preceding sections we have discussed only the total volume-integrated particle spectra. Now we want to study the spatial distribution as well. We already now that energetic charged particles are coupled to the flow of the bulk plasma by interaction with possibly self-produced plasma waves, which results in diffusive propagation relative to that flow and convection with the flow. If the transport is purely convective, then there usually is a unique relationship between the spatial coordinates and time (or possibly energy), which implies that particles move on charac- teristics in vectorx- t-space (or vectorx- E-space). If the transport is diffusive, we have to solve a second-order equation again, which here we want to do for the electron component of galactic cosmic rays. We consider a continuity equation for the differential number density N ( p ) = dN/dp/dV with a view to determine a Green function for the problem, that is the solution to the equation −∇ parenleftBig D ( p, x ) ∇ G − vector V G parenrightBig + ∂ ∂p (˙ p G ) + G τ c = δ ( vectorx − vectorx ′ ) δ ( p − p ′ ) (14 . 1 . 1) that generally contains term for diffusion and convection, but also continuous and catastrophic losses. For simplicity we choose a one-dimensional problem and further D = D ( p ) ˙ p = − B ( p ) vector V = 0 (14 . 1 . 2) and for electrons τ c ≃ ∞ . As in sec.12 it is beneficial to rewrite the equation for g = B ( p ) G , so (14.1.1) simplifies to − ∂ 2 g ∂z 2 − ∂g ∂λ = δ ( z − z ′ ) δ ( λ − λ ′ ) mit λ = integraldisplay p dq D ( q ) B ( q ) (14 . 1 . 3) As boundary condition we require that g = 0 for z → ±∞ . Equation (14.1.3) is a heat conduction equation, whose solution is known and can be derived by Fourier transformation. Recall that δ ( z − z ′ ) = 1 2 π integraldisplay ∞ −∞ dk exp [ ık ( z − z ′ )] (14 . 1 . 4) Our boundary conditions make sure that g is integrable and square-integrable, and thus we can write g = 1 √ 2 π integraldisplay ∞ −∞ dk H ( k, λ ) exp[ ık ( z − z ′ )] (14 . 1 . 5) Then (14.1.3) can be rewritten as parenleftBigg k 2 − ∂ ∂λ parenrightBigg H = 1 √ 2 π δ ( λ − λ ′ ) (14 .....
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