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Mobility and diffusivity

A sphere suspended in a fluid follows a random thermally-driven trajectory $ \vec r(t)$ satisfying

$\displaystyle \langle \vert r_\alpha(t) - r_\alpha(0)\vert^2 \rangle = 2 D_\alpha t,$ (29)

where the angle brackets indicate an ensemble average, and $ D_\alpha$ is the sphere's diffusion coefficient in the $ \alpha$ direction. In 1905, Einstein demonstrated that a sphere's diffusivity at temperature $ T$ is simply related to its mobility through the now-familiar Stokes-Einstein relation

$\displaystyle D_0 = k_B T \, b_0.$ (30)

This was the first statement of the more general fluctuation-dissipation theorem which plays an important role in statistical mechanics. More generally, components of the $ N$-particle diffusivity tensor

$\displaystyle \ensuremath{\mathsf{D}}\xspace = k_B T \, \ensuremath{\mathsf{b}}\xspace .$ (31)

parameterize generalized diffusion relations [59]

$\displaystyle \langle \Delta r_{i\alpha}(\tau) \Delta r_{j\beta}(\tau)\rangle = 2 \, D_\ensuremath{{i \alpha, j \beta}}\, \tau.$ (32)

describing how particle $ i$'s motion in the $ \alpha$ direction influences particle $ j$'s in the $ \beta$ direction. Off-diagonal terms in $ \mathsf{D}$ thus encode spheres' hydrodynamic interactions.

In general, $ \mathsf{D}$ can be diagonalized by projection onto a system of coordinates $ \psi$ consisting of linear combinations of the $ \vec r_i$. These normal modes evolve independently over time; they offer natural experimental probes of the particles' dynamics.


next up previous
Next: Normal mode diffusivity Up: Pair Diffusion Previous: Pair Diffusion
David G. Grier 2001-01-16