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Fig. 5 | SpringerPlus

Fig. 5

From: Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

Fig. 5

Diffusion propagator according to Eq. (119) around a cylinder with radius \(R=1\,\mu \text {m}\), diffusion coefficient \(D=1\, \mu \text {m}^2 / \text {ms}\) and \(\gamma =8\). The initial position is at \(r_0=4\,\mu \text {m}\) and \(\phi _0=\pi /4\). For small times (exemplary for \(t=1\,\text {ms}\)) the transition probability is concentrated around the initial position. For large times (exemplary for \(t=100\,\text {ms}\)) the diffusion propagator takes the constant value \(\lim _{t \rightarrow \infty } p(r,\phi ,r_0,\phi _0,t)=\frac{1}{\pi R^2 \left[ \gamma ^2-1 \right] } = 5.05254 \times 10^9 \,\text {m}^{-2}\).

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