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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}{ | :<math>D = \lim_{t \rightarrow \infty} \frac{1}{2} \langle \vert x_i(t) \cdot 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 | ||
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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}{ | :<math>D = \lim_{t \rightarrow \infty} \frac{1}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle </math> | ||
==Green-Kubo relation== | ==Green-Kubo relation== | ||
:<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 | where <math>v_i(t)</math> is the center of mass velovity of molecule <math>i</math>. | ||
==References== | ==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]] |