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The WKB Method: A User-Guide

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The Statistical Physics of Fixation and Equilibration in Individual-Based Models

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Abstract

The Wentzel–Kramers–Brillouin (WKB) method has been used to address a variety of problems in physics and at the interface of biosciences, from problems in optics, quantum mechanics and General Relativity to estimating the lifetime of a disease outbreak. In this chapter we explore the mathematical basis of the method in its application to stochastic processes.

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Notes

  1. 1.

    This was the approximation scheme used by physicists to investigate wave-scattering phenomena in problems such as optics and quantum mechanics.

  2. 2.

    The quantity \(M_k\) is found by enumerating the paths with k backwards steps. We have not found a general expression for this number.

  3. 3.

    We assume that this boundary is ‘regular’, i.e. the probability of reaching the boundary is non-zero and the expected time to reach the boundary is finite [29].

  4. 4.

    We could expand the reactions rates in further powers of \(\Omega \), such that \(T^r_{\Omega x} = \Omega f_r(x) + g_r(x) + h_r(x)/\Omega +\dots \), as described in Ref. [19]. However in this section we only consider the leading-order contributions.

  5. 5.

    The absorbing state can be removed by setting \(T^-_1=0\).

  6. 6.

    Pierre François Verhulst (1804–1849).

  7. 7.

    Analytical solutions are available for the two-type branching process [39], but, in general, such a description is usually lacking.

  8. 8.

    Leonhard Euler (1707–1783) and Gisiro Maruyama (1916–1986).

  9. 9.

    Lars Onsager (1903–1976) and Stefan Machlup (1927–2008).

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  1. Institute for Integrative Biology, ETH Zürich, Zürich, Switzerland

    Peter Ashcroft

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Ashcroft, P. (2016). The WKB Method: A User-Guide. In: The Statistical Physics of Fixation and Equilibration in Individual-Based Models. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41213-9_6

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