Diffusion

Diffusion is the process behind Brownian motion. It was described by Albert Einstein in one of his annus mirabilis papers of 1905. What follows applies to homogeneous systems, see diffusion at interfaces for a non-homogeneous case.

The diffusion equation that describes this process is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is the (self-)diffusion coefficient. For initial conditions for a Dirac delta function at the origin, and boundary conditions that force the vanishing of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(r,t)} and its gradient at large distances, the solution factorizes as $P(r,t)=P(x,t)P(y,t)P(z,t)$, with a spreading Gaussian for each of the Cartesian components:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp \left[ - \frac{x^2}{4 D t} \right]. }

Einstein relation

It follows from the previous equation that, for each of the Cartesian components, e.g. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = \lim_{t \rightarrow \infty} \frac{1}{2t} \langle \vert x_i(t) - x_i(0) \vert^2\rangle } ,

for every particle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} . Therefore, an average over all particles can be employed in order to improve statistics. The same applies to time averaging: in equilibrium the average from $0$ to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} must equal the average from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t+\tau} , so several time segments from the same simulation may be averaged for a given interval [2]. Adding all components, the following also applies:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = \lim_{t \rightarrow \infty} \frac{1}{6t } \langle \vert \mathbf{r}_i(t) - \mathbf{r}_i(0) \vert^2\rangle }

Green-Kubo relation

Main article: Green-Kubo relations

$D={\frac {1}{3}}\int _{0}^{\infty }\langle v_{i}(t)\cdot v_{i}(0)\rangle ~dt$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_i(t)} is the center of mass velocity of molecule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} . Note that this connect the diffusion coefficient with the velocity autocorrelation.

See also

• Rotational diffusion

References

Related reading

• Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications". Academic Press 2002
• Karsten Meier, Arno Laesecke, and Stephan Kabelac "Transport coefficients of the Lennard-Jones model fluid. II Self-diffusion" J. Chem. Phys. 121 pp. 9526-9535 (2004)
• G. L. Aranovich and M. D. Donohue "Limitations and generalizations of the classical phenomenological model for diffusion in fluids", Molecular Physics 105 1085-1093 (2007)
• P.-A. Artola and B. Rousseau "Thermal diffusion in simple liquid mixtures: what have we learnt from molecular dynamics simulations?", Molecular Physics 111 pp. 3394-3403 (2013)
• Sunghan Roh, Juyeon Yi and Yong Woon Kim "Analysis of diffusion trajectories of anisotropic objects", Journal of Chemical Physics 142 214302 (2015)