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A Hitch-hiker’s Guide to Stochastic Differential Equations

Solution Methods for Energetic Particle Transport in Space Physics and Astrophysics

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Abstract

In this review, an overview of the recent history of stochastic differential equations (SDEs) in application to particle transport problems in space physics and astrophysics is given. The aim is to present a helpful working guide to the literature and at the same time introduce key principles of the SDE approach via “toy models”. Using these examples, we hope to provide an easy way for newcomers to the field to use such methods in their own research. Aspects covered are the solar modulation of cosmic rays, diffusive shock acceleration, galactic cosmic ray propagation and solar energetic particle transport. We believe that the SDE method, due to its simplicity and computational efficiency on modern computer architectures, will be of significant relevance in energetic particle studies in the years to come.

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Notes

  1. As with any review paper, some references will inadvertently fall through the cracks and we would like to apologize for this in advance.

  2. The terms convection and advection are usually used interchangeably without any consensus on their potentially different meaning (e.g. an active vs. passive process). We do not attempt to use one over the other rigorously in this text.

  3. Various different types of SDEs exist, such as the widely used Stratonovich stochastic formulation. Here we choose to be as brief as possible and refer the interested reader to e.g. Gardiner (1983) for a comprehensive review of these different formulations. It must necessary be kept in mind that, due to the different temporal discretizations used in these SDE formulations, the numerical methods described here may not be applicable to solve SDE that are not of the Itō-type; this is certainly true for the Stratonovich formulation.

  4. As \(b_{ij}\) is basically the square root of \(C_{ij}\), calculating \(b_{ij}\) for some scenarios can be very tricky in higher dimensions, but is always possible as \(C_{ij}\) is a positive definite tensor (Gardiner 1983), and usually also symmetric (Kopp et al. 2012). It is also interesting to note that \(b_{ij}\) is not unique but different choices of \(b_{ij}\) lead to the same solution as they are all mathematically equivalent.

  5. Of course, the relative efficiency of both approaches depends on the relative sizes of the source and boundary surfaces compared to the size of the effective observer. As suggested by Milstein et al. (2004), some solutions are also hard to evaluate in the time backward scenario.

  6. Although for some of the simple problems that are considered here, some (semi) analytic solutions exist, we decided not to include them here, to keep the focus on the numerical approach.

  7. Earlier, Jokipii and Levy (1977) used the first- and second-order moments of the TPE to construct a random walk model for CRs. However, as this model did not implement a Wiener process, but rather a uniform distribution of random numbers, it is not an SDE model in a strict sense.

  8. A shock is, however, not a prerequisite for Fermi I acceleration to occur. See, e.g., the SDE modeling by Armstrong et al. (2012) and the references therein, where a beam of CRs can be accelerated via Fermi I acceleration when waves, traveling in a preferred direction, are present.

  9. Zhang (2000) explores a technique for simulating DSA, at a discontinuous TS by means of SDEs using what is termed to be “skew Brownian motion”, where the spatial coordinates are rescaled in order to remove the discontinuity.

  10. In an earlier paper by Jokipii (2001), the use of a Dirichlet boundary condition was motivated by analytical considerations.

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Acknowledgements

We thank Horst Fichtner for his careful reading of the draft manuscript and Marius Potgieter, Ingo Büsching, Andreas Kopp, Yuri Litvinenko and Phillip Dunzlaff for collaborating on different aspects of SDE modeling. RDS is funded through the National Research Foundation (NRF) of South Africa. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the NRF. Partial financial support from the Alexander von Humboldt Foundation and the Fulbright Visiting Scholar Program are acknowledged. FE is supported by NASA grant NNX14AG03G and part of the work was completed during a fellowship at the University of Waikato.

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  1. Center for Space Research, North-West University, Potchefstroom, 2531, South Africa

    R. Du Toit Strauss

  2. National Institute for Theoretical Physics (NITheP), Gauteng, Johannesburg, South Africa

    R. Du Toit Strauss

  3. Department of Physics and KIPAC, Stanford University, Stanford, CA, 94305, USA

    Frederic Effenberger

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Strauss, R.D.T., Effenberger, F. A Hitch-hiker’s Guide to Stochastic Differential Equations. Space Sci Rev 212, 151–192 (2017). https://doi.org/10.1007/s11214-017-0351-y

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