Protheroe et al. (2003) investigated ways of accurately simulating the propagation of energetic charged particles over small times where the standard Monte Carlo approximation to diffusive transport breaks down. Protheroe et al. (2003) find that a small-angle scattering procedure with appropriately chosen step-lengths and scattering angles gives accurate results, and they apply this to the simulation of propagation upstream in relativistic shock acceleration. The matter is that in diffusive shock acceleration at relativistic shocks problems arise when simulating particle motion upstream of the shock because the particle speeds, v, and the shock speed vsh = c(1 -Y-h f (4.29.14)
are both close to c, and so very small deflections are sufficient to cause a particle to re-cross the shock. Clearly, Monte Carlo simulation by a random walk with mean free path X and large-angle scattering is inappropriate here, and in Monte Carlo simulations of relativistic shock acceleration at parallel shocks Achterberg et al. (2001) consider instead the diffusion of a particle's direction for a given angular diffusion coefficient Dq (rad2 s-1). Similarly, for a given spatial diffusion coefficient k, Protheroe (2001) and Meli and Quenby (2001) adopted a random walk with a smaller mean free path, l << A, followed by scattering at each step by a small angle with mean deflection, Q <1/yshock. (see Bednarz and Ostrowski, 2001 for a review of relativistic shock acceleration). Protheroe et al. (2003) consider propagation by small steps sampled from an exponential distribution with mean l << A , followed at each step by scattering through a small angle sampled from an exponential distribution with mean Q << n. The change in direction (61, 02) may then be described as two-dimensional diffusion with angular diffusion coefficient Dq = Qq/2 (rad2 s-1) where vQ = Qt, and t = l/v such that DQ = Q2v/(2/). The time tiso , which gives rise to a deflection equivalent to a large angle (isotropic) scattering, is determined by the equation
and a spatial diffusion coefficient
By using a Monte Carlo method it is straightforward to test this, determine the constant of proportionality, and thereby make the connection between diffusion and small angle scattering. The solution of the diffusion equation for a delta-function source in position and time q(r, t ) = )8(t) and an infinite diffusive medium is a three-dimensional Gaussian with standard deviation c = ^2Kt (Chandrasekhar,
1943). The results from several Monte Carlo random walk simulations are shown in Fig. 4.29.5, from which it can be find that the expected dependence occurs for e < 5o at times t > 105 l/v .
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