Diffusion: Difference between revisions

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It follows from the previous equation that, for each of the Cartesian components, e.g. <math>x</math>:
It follows from the previous equation that, for each of the Cartesian components, e.g. <math>x</math>:
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{2} \langle \vert x_i(t) \cdot x_i(0) \vert^2\rangle  </math>,
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{2t} \langle \vert x_i(t) - x_i(0) \vert^2\rangle  </math>,
for every particle <math>i</math>. Therefore, an average over all particles can be employed in
for every particle <math>i</math>. Therefore, an average over all particles can be employed in
order to improve statistics. The same applies to time averaging: in equilibrium the average
order to improve statistics. The same applies to time averaging: in equilibrium the average
Line 23: Line 23:
so several time segments from the same simulation may be averaged for a given interval [2].
so several time segments from the same simulation may be averaged for a given interval [2].
Adding all components, the following also applies:
Adding all components, the following also applies:
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle  </math>
:<math>D = \lim_{t \rightarrow \infty} \frac{1}{6t } \langle \vert \mathbf{r}_i(t) - \mathbf{r}_i(0) \vert^2\rangle  </math>
 


==Green-Kubo relation==
==Green-Kubo relation==
:''Main article: [[Green-Kubo relations]]''
:<math>D = \frac{1}{3} \int_0^\infty \langle v_i(t) \cdot v_i(0)\rangle ~dt</math>
:<math>D = \frac{1}{3} \int_0^\infty \langle v_i(t) \cdot v_i(0)\rangle ~dt</math>


where <math>v_i(t)</math> is the center of mass velocity of molecule <math>i</math>. Note
where <math>v_i(t)</math> is the center of mass velocity of molecule <math>i</math>. Note
that this connect the diffusion coefficient with the velocity [[autocorrelation]].
that this connect the diffusion coefficient with the velocity [[autocorrelation]].
==See also==
*[[Rotational diffusion]]
==References==
<references/>
;Related reading
*[http://books.google.es/books?id=XmyO2oRUg0cC&dq=understanding+molecular+simulations&psp=1 Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications". Academic Press 2002]
*[http://dx.doi.org/10.1063/1.1786579 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)]
*[http://dx.doi.org/10.1080/00268970701348758 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)]
*[http://dx.doi.org/10.1080/00268976.2013.837534 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)]
*[http://dx.doi.org/10.1063/1.4921958  Sunghan Roh, Juyeon Yi and Yong Woon Kim "Analysis of diffusion trajectories of anisotropic objects", Journal of Chemical Physics '''142''' 214302 (2015)]


==References==
#[http://books.google.es/books?id=XmyO2oRUg0cC&dq=understanding+molecular+simulations&psp=1 Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications". Academic Press 2002]
#[http://dx.doi.org/10.1063/1.1786579 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)]
#[http://dx.doi.org/10.1080/00268970701348758 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)]
[[Category: Non-equilibrium thermodynamics]]
[[Category: Non-equilibrium thermodynamics]]

Latest revision as of 11:05, 23 June 2015

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

where is the (self-)diffusion coefficient. For initial conditions for a Dirac delta function at the origin, and boundary conditions that force the vanishing of and its gradient at large distances, the solution factorizes as , with a spreading Gaussian for each of the Cartesian components:

Einstein relation[edit]

It follows from the previous equation that, for each of the Cartesian components, e.g. :

,

for every particle . 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 to must equal the average from to , so several time segments from the same simulation may be averaged for a given interval [2]. Adding all components, the following also applies:

Green-Kubo relation[edit]

Main article: Green-Kubo relations

where is the center of mass velocity of molecule . Note that this connect the diffusion coefficient with the velocity autocorrelation.

See also[edit]

References[edit]

Related reading