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Topic: Diffusion equation

Related Topics

Boundary condition

Diffusion

Geometric Brownian motion

Stochastic process

Ficks law of diffusion

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Chemical reaction

Allan Wilson

Net flux

Passive transport

Motoo Kimura

Active transport

Neutral theory of molecular evolution

Electrochemical potential

	Diffusion Coefficient - The relationship between the formation factor and the diffusion coefficient of porous ...
		This approach, however, yields a diffusion coefficient tensor that characterizes a particular experiment and cannot, in general, be applied to different scenarios.
		Although this equation neglects adsorption effects, it is otherwise a complete equilibrium thermodynamic description of nonreactive diffusive transport of charged species in concentrated electrolytes.
		In those cases where the self-diffusion coefficients of all the diffusing species are nearly identical, the diffusion potential will be nearly zero, and the apparent bulk diffusion coefficient will be nearly equal to the agglomerated diffusion coefficient.
	ciks.cbt.nist.gov /~garbocz/FFvDiff/FFvDiff/node3.html (417 words)

	Diffusion
		Diffusion fluxes can cause changes in the concentration distribution of the diffusing molecules, and to deal with these we need the full differential equation for diffusion, which can be written
		Let's assume that the atmosphere is well-mixed by wind and turbulance at a height of 100 meters, and that diffusion is called upon to move CO from 100 meters to the ground.
		In the steady state, we assume that all of the molecules reaching the surface of the cell at radius a are adsorbed, resulting in C=0 at r=a, and a constant rate, Q, of uptake of nutrient molecules.
	www.cco.caltech.edu /~brokawc/Bi145/Diffusion.html (1796 words)

	Molecular Diffusion: Model description (Site not responding. Last check: 2007-10-20)
		Diffusion in a medium with a constant diffusion coefficient is often described using the equation (Carslaw and Jaeger, 1967; Crank, 1956;Kirkham and Powers, 1976)
		Diffusion coefficient may vary with the concentration of a diffusing substance and the medium in which a substance diffuses.
		Diffusion coefficient may vary with location if the diffusing medium is not homogeneous.
	soilphysics.okstate.edu /toolkit/diffusion/model.htm (528 words)

	Diffusion equation - Wikipedia, the free encyclopedia
		The diffusion equation is a partial differential equation, which describes the density fluctuations in a material undergoing diffusion.
		The diffusion equation can be derived in a straightforward way from the continuity equation, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system.
		The diffusion equation can be obtained easily from this when combined with the phenomenological Fick's first law, which assumes that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
	en.wikipedia.org /wiki/Diffusion_equation (223 words)

	[No title]
		The conclusion I came to is that the diffusion equation should be superposed on or fitted to the wave equation.
		It is necessary to be explicit about the structure of the equation because the decay process being described is based on the idea that the decay process is intrinsic to the photon wave, not to its origin or to the motion of its origin.
		Light, it appears, has two diffusions going on at the same time--the very rapid diffusion of the wavefront of a disturbance at the speed of light in which the permittivity is equivalent to the diffusivity, and the very slow diffusion described here, in which the action quantum is equivalent to the diffusivity.
	www.eskimo.com /~mikel137/logarith.htm (1242 words)

	Heat equation - Wikipedia, the free encyclopedia
		Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object.
		Diffusion (of particles, heat, momentum...) describes the return to global thermodynamic equilibrium of an inhomogeneous system, and as such is a time-irreversible phenomenon, associated to an increase in the entropy of the universe: in the case of particle diffusion, if
		The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options.
	en.wikipedia.org /wiki/Heat_equation (2206 words)

	Lecture #10
		In some steady-state diffusion problems the concentration of the diffusant must be expressed in terms of a pressure drop across a wall or membrane, and concentration, per se, may not be measurable or known.
		Equation (8), an ordinary differential equation, is particularly useful when one wants to determine D(C) from a diffusion experiment conducted in a geometry with compatible boundary conditions, i.e.
		Matano (1932) described a useful geometry to apply the Boltzmann-transformed diffusion equation to experimental observations, for the purpose of obtaining diffusivity data.
	www.rpi.edu /dept/materials/COURSES/DIFFUSION/LECTURE10/lecture10.html (703 words)

	Drift-Diffusion Equation
		Equations (1) may be solved analytically for the case of steady-state transport in one dimension, provided that recombination and generation may be neglected.
		The carrier densities may be rewritten in terms of the quasi-Fermi levels, or, equivalently, one multiplies the drift-diffusion equation by an appropriate integrating factor.
		Equations (1) may also be used in numerical simulations, to evaluate the current density between discrete mesh points.
	www.utdallas.edu /dept/ee/frensley/technical/hetphys/node15.html (771 words)

	Diffusion of solute in soil media, D.L. Nofziger (Site not responding. Last check: 2007-10-20)
		The effective diffusion coefficient in the soil is calculated taking into account the water content and porosity of the soil and the sorption of the chemical on the soil solids.
		Diffusion coefficient: Diffusion coefficient is a parameter expressing the transfer rate of a substance by random molecular motion.
		It equals the diffusion coefficient of a chemical obtained under non-adsorbing conditions divided by a retardation factor of the adsorbing system.
	soilphysics.okstate.edu /software/Diffusion/document.html (1069 words)

	Derivation of the diffusion equation
		The Eddington or diffusion approximation characterizes the diffuse radiance as a sum of a diffuse radiant fluence
		Equation (4.14) represents the first two terms of the Taylor expansion for the diffuse radiance
		and integrating over all angles yields an energy flux equation for the diffuse radiance which states that the change in the diffuse fluence equals the diffuse flux lost to absorption plus that gained from collimated light.
	omlc.ogi.edu /pubs/prahl-pubs/prahl88/node38.html (745 words)

	Guide and Reference (Site not responding. Last check: 2007-10-20)
		The objective of the diffusion program is to solve for the temperature of a beam at any point and at any time, given an initial temperature distribution.
		Equation 2 is solved by representing the operators in a sine function basis and solving the resulting matrix equations.
		Equation 10 is the simplest form for the matrix elements of the diffusion operator, and these elements are calculated in the subroutine get_diffusion_matrix.
	www.navo.hpc.mil /usersupport/IBM/PESSL/pessl291.html (439 words)

	Non-linear Diffusion Techniques for Denoising
		The idea behind the use of the diffusion equation in image processing arose from the use of the Gaussian filter in multi-scale image analysis.
		Since the diffusivity is a scalar, we use the terminology from partial differential equations and refer to this case as isotropic non-linear diffusion.
		We reserve the term anisotropic for the case where the diffusivity is a tensor-valued function, varying with both the edge location and its orientation.
	www.llnl.gov /casc/sapphire/diffusion/diffusion.html (619 words)

	LS Note 221: Diffusion in Phase Space (Site not responding. Last check: 2007-10-20)
		To find the diffusion coefficient from first principles, we would need to study the changes in J produced by whatever collisions and other processes are causing the diffusion.
		This is the integral equation for the eigenfunction
		Equation (2.3) tells us, just as in Section 3, that in the steady state the currents are constant and by the boundary condition at J=0 the currents must be zero.
	www.aps.anl.gov /techpub/lsnotes/ls221/ls221.html (2301 words)

	[No title]
		The diffusivity can be inferred by measuring the second moment of the conditional probability distribution of the diffusing species.
		DIFFUSION AND NMR Basic Principles The dependence of diffusion coefficient on the NMR signal can be described from the experiment of a simple bipolar pulsed gradient experiment.
		DIFFUSION AND NMR Diffusion Spin Echo NMR Imaging In Chapter II, the effects of diffusion on the NMR signal have been discussed.
	www.ps.uci.edu /~markm/mri/diffusion_young.doc (2070 words)

	Chapter 8: Modeling molecular diffusion
		Diffusion is the process by which random Brownian movement of molecules or ions cause an average movement towards regions of lower concentration, which may result in the collapse of the concentration gradient.
		However, when diffusion is used as a transport mechanism to link cascading processes that are spatially separated or when diffusion is used in conjunction with processes that are triggered at certain concentration level thresholds, a detailed description of the geometry of the system may become very important.
		The apparent diffusion constant is defined as the diffusion constant for the effective 1D diffusion along the longitudinal axis of the dendritic shaft.
	www.tnb.ua.ac.be /publications/pub034/TNB_pub34.shtml (10884 words)

	Diffusion (Site not responding. Last check: 2007-10-20)
		Diffusion processes are ubiquitous in chemical engineering and are governed by the same equation as heat transfer [1].
		Potential flow problems are also described by the diffusion equation, for example, Darcy’s law problems, but the physical interpretation is different.
		This material property, denoted by D, describes the diffusive conductivity of a material, that is, the relation between the concentration gradient and the mass flux.
	www-math.cudenver.edu /~jmandel/doc/guide/guide63.htm (364 words)

	Dimensionless form of the diffusion equation
		This is useful for emphasizing the underlying physics of the derivation of the boundary conditions and the diffusion equation.
		The parameter h and the function Q(r) are defined by Equation (4.46).
		Rearranging Equation (4.18) to express the diffuse radiant flux in terms of the average diffuse radiance d yields the following equation
	omlc.ogi.edu /pubs/prahl-pubs/prahl88/node48.html (266 words)

	Numerical Methods Lecture Notes: pdes
		The system of linear equations described by (103) in combination with the boundary conditions may be solved in a variety of ways.
		The analysis for the diffusion equation in one or three dimensions may be computed in a similar manner.
		Now, since the diffusion equation is linear, and as our stability analysis of the previous section shows the conditions under which the solution for each Fourier mode is stable, we can see that the equation (128) applies equally for arbitrary initial conditions.
	www.damtp.cam.ac.uk /user/fdl/people/sd103/lectures/nummeth98/pdes.htm (2237 words)

	The Beltrami Flow
		In this form, the Beltrami flow equation is not a ``pure'' diffusion equation.
		Our aim is to decouple the set in equation (9) so that each row and each column of pixels can be handled separately.
		The reaction term is responsible for edge enhancement, while the diffusion term smooths the noise away from the edges.
	math.lbl.gov /~deschamp/tvcg_2003/node2.html (1029 words)

	FlexPDE Diffusion (Site not responding. Last check: 2007-10-20)
		This problem considers the thermally driven diffusion of a dopant into a solid from a constant masked source.
		Parameters have been chosen to be those typically encountered in semiconductor diffusion.
		At early times, the solution near the source can be compared to the analytic solution for 1D diffusion.
	www.pdesolutions.com /diffusion.html (79 words)

	Continuity equation
		A solution to these equations can be obtained by substituting the expression for the electron and hole current, (2.7.29) and (2.7.30).
		This then yields two partial differential equations as a function of the electron density, the hole density and the electric field.
		The diffusion equation will be used to calculate the diffusion current in p-n junctions and bipolar transistors.
	ece-www.colorado.edu /~bart/ecen3320/newbook/chapter2/ch2_9.htm (326 words)

	Diffusion
		In this equation [C(t, x+Dx) - C(t, x)]/ Dx is the empirically derived equation for the diffusive flux between two adjacent segments.
		D is the diffusion coefficient that characterizes the environment, the media; it tells us how fast diffusion can occur in this kind of media.
		We cannot use equation (2) to calculate both the value on the left hand side boundary C(t,0) and on the right hand side boundary C(t,N), where N is the number of the maximal segment that we consider.
	www.uvm.edu /giee/AV/CS/diffusion/index.html (448 words)

	Conversion of the Black-Scholes Equation to the Diffusion Equation
		We first bring the equation into the standard form of the diffusion equation, and then solve it using the Green's function for the diffusion equation on the initial condition at
		The first difference we notice from the canonical equation is that the coefficients depend on
		However, the equation is homogeneous or invariant under the scaling of
	www.physics.uci.edu /~silverma/bseqn/bs/node4.html (176 words)

	Diffusion Equation (Site not responding. Last check: 2007-10-20)
		Diffusion is an everyday experience; a hot cup of tea distributes its thermal energy or an uncorked perfume bottle distributes its scent throughout a room.
		Since nonunifrom distributions tend to distribute themselves in such a way as to produce uniformity, it is reasonable to assume that the flux density, J, is proportional to the concentration gradient.
		The above equation is the differential form of the conservation of mass.
	webphysics.davidson.edu /Faculty/wc/WaveHTML/node11.html (226 words)

	[No title]
		These two types of diffusion are : a} Constant source diffusion (pre-deposition) or b} Constant total dopant diffusion (drive-in) Constant Source Diffusion : During constant source diffusion there is a continuous supply of dopant to surface of the silicon.
		This keeps the surface concentration constant throughout the diffusion (usually at the solid solubility level of the dopant species in silicon at the process temperature).
		A related quantity is the gradient of the diffusion profile dc/dx.
	www.fortunecity.com /village/williams/252/diff.math.doc (928 words)

	The MATLAB Notebook v1.5.2
		diffusion equation), a partial differential equation that describes many physical processes including conductive heat flow or the diffusion of an impurity in a motionless fluid.
		x is the limiting "steady state" for this problem; it satisfies the boundary conditions and it yields 0 on both sides of the partial differential equation.
		Generally speaking, it is best to understand some of the theory of partial differential equations before attempting a numerical solution like we have done here.
	www.math.umd.edu /undergraduate/schol/matlab/pde.html (2143 words)

	:: Quantnotes.com :: Fundamentals ::
		The Black-Scholes equation resembles a linear parabolic equation similar to the diffusion equation.
		A general linear parabolic equation is of the form:
		For now it is just a matter of solving the diffusion equation by methods of Fourier Transform.
	www.quantnotes.com /fundamentals/options/solvingbs.htm (216 words)

	An example 1-d solution of the diffusion equation
		Let us now solve the diffusion equation in 1-d using the finite difference technique discussed above.
		Note that the above equation describes a Gaussian pulse which gradually decreases in height and broadens in width in such a manner that its area is conserved.
		Figure 71: Diffusive evolution of a 1-d Gaussian pulse.
	farside.ph.utexas.edu /teaching/329/lectures/node78.html (217 words)

	2.12 The diffusion equation (Site not responding. Last check: 2007-10-20)
		The continuity equation describes a basic concept, namely that a change in carrier density over time is due to a difference between the incoming and outgoing flux of carriers plus the generation and minus the recombination.
		2.11.2 The Diffusion equation in a quasi-neutral region
		In the quasi-neutral region, the current is due to diffusion only.
	ece-www.colorado.edu /~bart/book/diffeq.htm (152 words)

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