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Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities
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10.1063/1.4828860
/content/aip/journal/jcp/139/18/10.1063/1.4828860
http://aip.metastore.ingenta.com/content/aip/journal/jcp/139/18/10.1063/1.4828860
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

The distribution of diffusivities (histogram) from a simulated trajectory of a homogeneous anisotropic diffusion process in two dimensions with diffusion tensor given by Eq. (40) agrees well with the analytic distribution of diffusivities (solid line) from Eq. (32) with = 5 and = 1 denoting the eigenvalues of . Additionally, the asymptotic function Eq. (36) (dotted line, = 5) agrees reasonably for large . Furthermore, a distribution of diffusivities (dashed line) of two-dimensional isotropic diffusion with the same mean diffusion coefficient = ⟨⟩ = ( + )/2 = 3 is shown for comparison. The different asymptotic decays are clearly visible and allow the distinction from homogeneous isotropic processes.

Image of FIG. 2.
FIG. 2.

The distribution of diffusivities (histogram) from one simulated trajectory of a homogeneous anisotropic diffusion process in three dimensions with diffusion tensor given by Eq. (54) agrees well with the distribution of diffusivities (solid line) obtained from numerical integration of Eq. (47) , using the eigenvalues = 5, = 3, and = 1 of tensor . For comparison, the distribution of diffusivities (dashed line) of an isotropic diffusion process in three dimensions, given by Eq. (55) , is shown, where the same mean diffusion coefficient = ⟨⟩ = ( + + )/3 = 3 as in the anisotropic process was used. The different asymptotic decays are clearly visible and allow the distinction from homogeneous isotropic processes. In the inset, the asymptotic function Eq. (51) (dotted line) agrees reasonably for large .

Image of FIG. 3.
FIG. 3.

Distribution of diffusivities (lines with open symbols) of different homogeneous anisotropic diffusion processes in three dimensions. A qualitative distinction between the oblate case (◊; (1) = 1, (2) = 5), the prolate case (⬠; (1) = 5, (2) = 1), and a general anisotropic case (◯; = 5, = 3, = 1) is possible since the decay after the maximum peak shows a concave curvature in the first case and a convex curvature in the latter cases. The inset shows that each anisotropic case obeys the same asymptotic decay given by the largest diffusion coefficient (dotted line as a guide to the eye). For comparison, the isotropic case with the same asymptotic decay (▲; = 5) is given, which always has a concave shape. Thus, it is qualitatively indistinguishable from the oblate case. However, a comparison of the first moment with the asymptotic decay offers a simple distinction between both cases.

Image of FIG. 4.
FIG. 4.

Distribution of diffusivities (lines with open symbols) for different ratios given by Eq. (62) and fixed (2) = 1. The crossover from oblate cases ( < 1, solid lines) to prolate cases ( > 1, dashed lines) shows a broadening of the peak for increasing ratios. Again, the behavior after the peak changes from concave to convex, respectively. For comparison, the distribution of diffusivities of the limiting isotropic cases are depicted for two-dimensional ( ; = 2/3) and three-dimensional processes (; = 1). The distinction of prolate cases from the isotropic limits is simpler than for the oblate cases.

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/content/aip/journal/jcp/139/18/10.1063/1.4828860
2013-11-11
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities
http://aip.metastore.ingenta.com/content/aip/journal/jcp/139/18/10.1063/1.4828860
10.1063/1.4828860
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