Diffusion - Revision history

Carl McBride: Added a recent publication

2015-06-23T10:05:14Z

Added a recent publication

← Older revision		Revision as of 12:05, 23 June 2015
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	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==		==See also==

	*[[Rotational diffusion]]		*[[Rotational diffusion]]


	==References==		==References==
	<references/>		<references/>
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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)]		*[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/10.1080/00268976.2013.837534 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)]

	[[Category: Non-equilibrium thermodynamics]]		[[Category: Non-equilibrium thermodynamics]]

Carl McBride: Added a recent publication

2014-02-21T13:37:34Z

Added a recent publication

← Older revision		Revision as of 15:37, 21 February 2014
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	==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]		<references/>
	#[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)]		;Related reading
	#[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://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)]

	[[Category: Non-equilibrium thermodynamics]]		[[Category: Non-equilibrium thermodynamics]]

Noe: /* Einstein relation */ auwful mistake (I guess) corrected (I hope)

2009-04-01T15:36:06Z

Einstein relation: auwful mistake (I guess) corrected (I hope)

← Older revision		Revision as of 17:36, 1 April 2009
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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
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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}{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==

Carl McBride: /* Green-Kubo relation */ Added internal link to main aricle for the Green-Kubo relations

2008-06-20T08:51:48Z

Green-Kubo relation: Added internal link to main aricle for the Green-Kubo relations

← Older revision		Revision as of 10:51, 20 June 2008
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	==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==		==See also==

Dduque: new section, link to rotational diffusion

2008-06-20T08:42:39Z

new section, link to rotational diffusion

← Older revision		Revision as of 10:42, 20 June 2008
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	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==

Dduque: /* Green-Kubo relation */

2008-06-20T08:31:47Z

Green-Kubo relation

← Older revision		Revision as of 10:31, 20 June 2008
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	:<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 ~~velovity~~ of molecule <math>i</math>.		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]].

	==References==		==References==

Dduque: added references

2007-12-04T11:53:07Z

added references

← Older revision		Revision as of 13:53, 4 December 2007
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	==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)]		#[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]]
	# [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]

Dduque: slightly rewritten, link to future diffusion at interfaces

2007-12-04T09:34:19Z

slightly rewritten, link to future diffusion at interfaces

← Older revision		Revision as of 11:34, 4 December 2007
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	'''Diffusion''' is the process behind [[Brownian motion]]. It was described		'''Diffusion''' is the process behind [[Brownian motion]]. It was described
	by [[Albert Einstein]] in one of his ''annus mirabilis'' papers of 1905.		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		The diffusion equation that describes this process is
	:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math>		:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math>
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	==Einstein relation==		==Einstein relation==

	~~For~~ a ~~homogeneous system~~,		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>,
			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
			from <math>0</math> to <math>t</math> must equal the average from <math>\tau</math> to <math>t+\tau</math>,
			so several time segments from the same simulation may be averaged for a given interval [2].
			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}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle </math>

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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)]		#[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]]
			# [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]

Carl McBride: Slight tidy up.

2007-11-13T12:48:32Z

Slight tidy up.

← Older revision		Revision as of 14:48, 13 November 2007
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	Diffusion is the process behind Brownian motion. It was described		'''Diffusion''' is the process behind [[Brownian motion]]. It was described
	by [[Albert Einstein]] in one of his annus mirabilis (1905~~) papers~~.		by [[Albert Einstein]] in one of his ''annus mirabilis'' papers of 1905.
	The diffusion equation is that describes ~~the~~ process is		The diffusion equation that describes this process is
	:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</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>D</math> is the (self-)'''diffusion coefficient'''.
	For initial conditions for a Dirac delta function at the origin, and		For initial conditions for a [[Dirac delta distribution \|Dirac delta function]] at the origin, and
	boundary conditions that force the vanishing of <math>P(r,t)</math>		boundary conditions that force the vanishing of <math>P(r,t)</math>
	and its gradient at large distances, the solution factorizes as <math>P(r,t)=P(x,t)P(y,t)P(z,t)</math>,		and its gradient at large distances, the solution factorizes as <math>P(r,t)=P(x,t)P(y,t)P(z,t)</math>,
	with a spreading Gaussian for each of the Cartesian components:		with a spreading [[Gaussian distribution \|Gaussian]] for each of the Cartesian components:
	:<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp		:<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp
	\left[ - \frac{x^2}{4 D t} \right]. </math>		\left[ - \frac{x^2}{4 D t} \right]. </math>

Dduque at 12:41, 13 November 2007

2007-11-13T12:41:25Z

← Older revision		Revision as of 14:41, 13 November 2007
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	The '''diffusion coefficient''', ~~is given by~~		Diffusion is the process behind Brownian motion. It was described
			by [[Albert Einstein]] in one of his annus mirabilis (1905) papers.
			The diffusion equation is that describes the process is
			:<math>\frac{\partial P(r,t)}{\partial t}= D \nabla^2 P(z,t),</math>
			where <math>D</math> is the (self-)'''diffusion coefficient'''.
			For initial conditions for a Dirac delta function at the origin, and
			boundary conditions that force the vanishing of <math>P(r,t)</math>
			and its gradient at large distances, the solution factorizes as <math>P(r,t)=P(x,t)P(y,t)P(z,t)</math>,
			with a spreading Gaussian for each of the Cartesian components:
			:<math> P(x,t)=\frac{1}{\sqrt{4\pi D t}} \exp
			\left[ - \frac{x^2}{4 D t} \right]. </math>

			==Einstein relation==

			For a homogeneous system,
			:<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==
	:<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 velovity of molecule <math>i</math>.		where <math>v_i(t)</math> is the center of mass velovity of molecule <math>i</math>.

	~~==Einstein relation==~~

	~~:<math>D = \lim_{t \rightarrow \infty} \frac{1}{6} \langle \vert \mathbf{r}_i(t) \cdot \mathbf{r}_i(0) \vert^2\rangle </math>~~

	==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)]		#[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]]