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| '''Diffusion''' is the process behind [[Brownian motion]]. It was described | | The '''diffusion coefficient''', is given by |
| by [[Albert Einstein]] in one of his ''annus mirabilis'' papers of 1905. | |
| What follows applies to homogeneous systems, see [[diffusion at interfaces]]
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| for a non-homogeneous case.
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| The diffusion equation that describes this process is
| | :<math>D = \frac{1}{3} \int_0^\infty \langle v_i(t) \cdot v_i(0)\rangle ~dt</math> |
| :<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math> | | |
| where <math>D</math> is the (self-)'''diffusion coefficient'''. | | where <math>v_i(t)</math> is the center of mass velovity of molecule <math>i</math>. |
| For initial conditions for a [[Dirac delta distribution |Dirac delta function]] at the origin, and
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| boundary conditions that force the vanishing of <math>P(r,t)</math>
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| and its gradient at large distances, the solution factorizes as <math>P(r,t)=P(x,t)P(y,t)P(z,t)</math>,
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| with a spreading [[Gaussian distribution |Gaussian]] for each of the Cartesian components:
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| :<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp
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| \left[ - \frac{x^2}{4 D t} \right]. </math>
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| ==Einstein relation== | | ==Einstein relation== |
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| It follows from the previous equation that, for each of the Cartesian components, e.g. <math>x</math>:
| | :<math>2tD = \frac{1}{3} \langle \vert r_i(t) \cdot r_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
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| order to improve statistics. The same applies to time averaging: in equilibrium the average
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| from <math>0</math> to <math>t</math> must equal the average from <math>\tau</math> to <math>t+\tau</math>,
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| so several time segments from the same simulation may be averaged for a given interval [2].
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| Adding all components, the following also applies:
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| :<math>D = \lim_{t \rightarrow \infty} \frac{1}{6t } \langle \vert \mathbf{r}_i(t) - \mathbf{r}_i(0) \vert^2\rangle </math>
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| ==Green-Kubo relation==
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| :''Main article: [[Green-Kubo relations]]''
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| :<math>D = \frac{1}{3} \int_0^\infty \langle v_i(t) \cdot v_i(0)\rangle ~dt</math>
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| where <math>v_i(t)</math> is the center of mass velocity of molecule <math>i</math>. Note
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| that this connect the diffusion coefficient with the velocity [[autocorrelation]].
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| ==See also==
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| *[[Rotational diffusion]]
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| ==References== | | ==References== |
| <references/>
| | #[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)] |
| ;Related reading
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| *[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]
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| *[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)]
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| *[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)]
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| *[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)]
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| *[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)]
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| [[Category: Non-equilibrium thermodynamics]] | | [[Category: Non-equilibrium thermodynamics]] |