Lévy flights of photons with infinite mean free path

Splitting Probabilities of Jump Processes

“…Calculating V (x 0 ) from ( 10), one verifies explicitly the agreement of this exact result with (11). Note that in this example f (k) = k→∞ o(k −1 ), so that one verifies in the x 0 a 2 regime the linear dependence on x 0 predicted by (13) (with the correct prefactor, see SM).…”

“…These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3d slab geometries. In this context, our explicit predictions of the transmission probability (23), which can be directly measured experimentally, provides in principle a quantitative determination of not only the Levy exponent µ, as already proposed in [12,13], but also of the microscopic length scale a µ . This significantly refines the characterization of the effective random process describing light scattering in the medium.…”

“…In the case of α-stable jump distributions, which have been shown recently to be relevant to photon scattering in hot atomic vapors [12,13], we obtain f (k) = Γ(µ −1 )−Γ(µ −1 ,(aµk) µ ) aµµk (see SM). As above, this provides the asymptotic expression (for x → ∞) of the splitting probability for any x 0 thanks to (11), and making use of (13) one obtains for x 0 a µ :…”

“…A celebrated example is given by the Gambler's ruin problem [1], schematically quantified by the splitting probability that a 1-dimensional random walker (figuring the gambler's fortune) reaches 0 (complete ruin) before a fixed given threshold; other examples are given by the fixation probability of a mutant in the context of population dynamics [7], or the melting probability of a heteropolymer [8], which can be re-expressed in terms of splitting probabilities. A key example, to which we will refer through this paper, is given by the transmission probability of particles (eg photons or neutrons) through a slab of a scattering medium, which has important applications in various fields [9][10][11][12][13]; in this case the transmission probability is nothing but the splitting probability for the particle to reach the exit side rather than being back-scattered.…”

Collisional redistribution of hydrogen line radiation in low- and moderate-density magnetized plasmas

“…One main advantage of these functions is that they involve analytical expressions or integrals that can be evaluated numerically within a reasonable CPU time; applications include inertial fusion [7] and related research works in astrophysics (e.g., see Refs. [8][9][10] for reviews), laboratory plasma experiments [11,12], and general transport theory [13,14]. In each of these domains, the radiation field is investigated based on a numerical solving of the radiative transfer equation and redistribution functions occur therein through an integral scattering term.…”