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Diffusion

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the generic concept of the time-dependent process. For other uses, see <a href="Diffusion_(disambiguation)" class="mw-disambig" title="Diffusion (disambiguation)">Diffusion (disambiguation)</a>.</div>

Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher <a href="Concentration" title="Concentration">concentration</a> to a region of lower concentration. Diffusion is driven by a gradient in <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> or <a href="Chemical_potential" title="Chemical potential">chemical potential</a>. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in <a href="Spinodal_decomposition" title="Spinodal decomposition">spinodal decomposition</a>. Diffusion is a <a href="Stochastic_process" title="Stochastic process">stochastic process</a> due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as <a href="Statistics" title="Statistics">statistics</a>, <a href="Probability_theory" title="Probability theory">probability theory</a>, <a href="Information_theory" title="Information theory">information theory</a>, <a href="Neural_networks" class="mw-redirect" title="Neural networks">neural networks</a>, <a href="Finance" title="Finance">finance</a>, and <a href="Marketing" title="Marketing">marketing</a>.

The concept of diffusion is widely used in many fields, including <a href="Physics" title="Physics">physics</a> (<a href="Molecular_diffusion" title="Molecular diffusion">particle diffusion</a>), <a href="Chemistry" title="Chemistry">chemistry</a>, <a href="Biology" title="Biology">biology</a>, <a href="Sociology" title="Sociology">sociology</a>, <a href="Economics" title="Economics">economics</a>, <a href="Statistics" title="Statistics">statistics</a>, <a href="Data_science" title="Data science">data science</a>, and <a href="Finance" title="Finance">finance</a> (diffusion of people, ideas, data and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection.

A <a href="Gradient" title="Gradient">gradient</a> is the change in the value of a quantity; for example, concentration, <a href="Pressure" title="Pressure">pressure</a>, or <a href="Temperature" title="Temperature">temperature</a> with the change in another variable, usually <a href="Distance" title="Distance">distance</a>. A change in concentration over a distance is called a <a href="Molecular_diffusion" title="Molecular diffusion">concentration gradient</a>, a change in pressure over a distance is called a <a href="Pressure_gradient" title="Pressure gradient">pressure gradient</a>, and a change in temperature over a distance is called a <a href="Temperature_gradient" title="Temperature gradient">temperature gradient</a>.

The word diffusion derives from the <a href="Latin" title="Latin">Latin</a> word, diffundere, which means "to spread out".

A distinguishing feature of diffusion is that it depends on particle <a href="Random_walk" title="Random walk">random walk</a>, and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of <a href="Advection" title="Advection">advection</a>.<a href="#cite_note-1">[1]</a> The term <a href="Convection" title="Convection">convection</a> is used to describe the combination of both <a href="Transport_phenomena" title="Transport phenomena">transport phenomena</a>.

If a diffusion process can be described by <a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws</a>, it is called a normal diffusion (or Fickian diffusion); Otherwise, it is called an <a href="Anomalous_diffusion" title="Anomalous diffusion">anomalous diffusion</a> (or non-Fickian diffusion).

When talking about the extent of diffusion, two length scales are used in two different scenarios (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is the <a href="Diffusion_coefficient" class="mw-redirect" title="Diffusion coefficient">diffusion coefficient</a>, having dimensions area / time):

<ol><li><a href="Brownian_motion" title="Brownian motion">Brownian motion</a> of an <a href="Impulse_response" title="Impulse response">impulsive</a> point source (for example, one single spray of perfume)—the square root of the <a href="Mean_squared_displacement" title="Mean squared displacement">mean squared displacement</a> from this point. In Fickian diffusion, this is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2nDt}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2nDt}}}</annotation>

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</math><img src="./ea8f794e1c49c674621d2d215146303cb7d5c113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.257ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2nDt}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is the <a href="Dimension" title="Dimension">dimension</a> of this Brownian motion;</li>

<li><a href="Fick's_laws_of_diffusion#Example_solutions_and_generalization" title="Fick's laws of diffusion">Constant concentration source</a> in one dimension—the diffusion length. In Fickian diffusion, this is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {Dt}}}">

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<annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {Dt}}}</annotation>

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</math><img src="./76beaf1fb4136b1aa5841c63ad61272a80d6690c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.862ex; height:3.009ex;" alt="{\displaystyle 2{\sqrt {Dt}}}" loading="lazy">.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Diffusion_vs._bulk_flow">Diffusion vs. bulk flow</h2></div>

"Bulk flow" is the movement/flow of an entire body due to a pressure gradient (for example, water coming out of a tap). "Diffusion" is the gradual movement/dispersion of concentration within a body with no net movement of matter. An example of a process where both <a href="Mass_flow_(life_sciences)" title="Mass flow (life sciences)">bulk motion</a> and diffusion occur is human breathing.<a href="#cite_note-2">[2]</a>

First, there is a "bulk flow" process. The <a href="Lungs" class="mw-redirect" title="Lungs">lungs</a> are located in the <a href="Thoracic_cavity" title="Thoracic cavity">thoracic cavity</a>, which expands as the first step in external respiration. This expansion leads to an increase in volume of the <a href="Pulmonary_alveolus" title="Pulmonary alveolus">alveoli</a> in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the <a href="Air" class="mw-redirect" title="Air">air</a> outside the body at relatively high pressure and the alveoli at relatively low pressure. The air moves down the pressure gradient through the airways of the lungs and into the alveoli until the pressure of the air and that in the alveoli are equal, that is, the movement of air by bulk flow stops once there is no longer a pressure gradient.

Second, there is a "diffusion" process. The air arriving in the alveoli has a higher concentration of oxygen than the "stale" air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in the alveoli and the blood in the <a href="Capillaries" class="mw-redirect" title="Capillaries">capillaries</a> that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that the concentration of <a href="Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a> in the alveoli decreases. This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli, as fresh air has a very low concentration of carbon dioxide compared to the <a href="Blood" title="Blood">blood</a> in the body.

Third, there is another "bulk flow" process. The pumping action of the <a href="Heart" title="Heart">heart</a> then transports the blood around the body. As the left ventricle of the heart contracts, the volume decreases, which increases the pressure in the ventricle. This creates a pressure gradient between the heart and the capillaries, and blood moves through <a href="Blood_vessel" title="Blood vessel">blood vessels</a> by bulk flow down the pressure gradient.

<div class="mw-heading mw-heading2"><h2 id="Diffusion_in_the_context_of_different_disciplines">Diffusion in the context of different disciplines</h2></div>

There are two ways to introduce the notion of diffusion: either a <a href="https://en.wiktionary.org/wiki/phenomenon" class="extiw external" title="wikt:phenomenon">phenomenological approach</a> starting with <a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a> and their mathematical consequences, or a physical and atomistic one, by considering the <a href="Random_walk" title="Random walk">random walk</a> of the diffusing particles.<a href="#cite_note-3">[3]</a>

In the phenomenological approach, diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion. According to Fick's laws, the diffusion <a href="Flux#Flux_as_flow_rate_per_unit_area" title="Flux">flux</a> is proportional to the negative <a href="Gradient" title="Gradient">gradient</a> of concentrations. It goes from regions of higher concentration to regions of lower concentration. Sometime later, various generalizations of Fick's laws were developed in the frame of <a href="Thermodynamics" title="Thermodynamics">thermodynamics</a> and <a href="Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">non-equilibrium thermodynamics</a>.<a href="#cite_note-4">[4]</a>

From the atomistic point of view, diffusion is considered as a result of the random walk of the diffusing particles. In <a href="Molecular_diffusion" title="Molecular diffusion">molecular diffusion</a>, the moving molecules in a gas, liquid, or solid are self-propelled by kinetic energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by <a href="Robert_Brown_(botanist%2C_born_1773)" title="Robert Brown (botanist, born 1773)">Robert Brown</a>, who found that minute particle suspended in a liquid medium and just large enough to be visible under an optical microscope exhibit a rapid and continually irregular motion of particles known as Brownian movement. The theory of the <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> and the atomistic backgrounds of diffusion were developed by <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>.<a href="#cite_note-5">[5]</a>

The concept of diffusion is typically applied to any subject matter involving random walks in <a href="Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">ensembles</a> of individuals.

In <a href="Chemistry" title="Chemistry">chemistry</a> and <a href="Materials_science" title="Materials science">materials science</a>, diffusion also refers to the movement of fluid molecules in porous solids.<a href="#cite_note-6">[6]</a> Different types of diffusion are distinguished in porous solids. <a href="Molecular_diffusion" title="Molecular diffusion">Molecular diffusion</a> occurs when the collision with another molecule is more likely than the collision with the pore walls. Under such conditions, the diffusivity is similar to that in a non-confined space and is proportional to the mean free path. <a href="Knudsen_diffusion" title="Knudsen diffusion">Knudsen diffusion</a> occurs when the pore diameter is comparable to or smaller than the mean free path of the molecule diffusing through the pore. Under this condition, the collision with the pore walls becomes gradually more likely and the diffusivity is lower. Finally there is configurational diffusion, which happens if the molecules have comparable size to that of the pore. Under this condition, the diffusivity is much lower compared to molecular diffusion and small differences in the kinetic diameter of the molecule cause large differences in <a href="Mass_diffusivity" title="Mass diffusivity">diffusivity</a>.

<a href="Biologist" title="Biologist">Biologists</a> often use the terms "net movement" or "net diffusion" to describe the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes so long as there is a higher concentration of oxygen outside the cell. However, because the movement of molecules is random, occasionally oxygen molecules move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the <a href="Probability" title="Probability">probability</a> that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) is into the cell. In other words, there is a net movement of oxygen molecules down the concentration gradient.

In <a href="Astronomy" title="Astronomy">astronomy</a>, atomic diffusion is used to model the <a href="Stellar_atmosphere" title="Stellar atmosphere">stellar atmospheres</a> of <a href="Chemically_peculiar_star" title="Chemically peculiar star">chemically peculiar stars</a>.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> Diffusion of the elements is critical in understanding the surface composition of degenerate <a href="White_dwarf" title="White dwarf">white dwarf</a> stars and their evolution over time.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="History_of_diffusion_in_physics">History of diffusion in physics</h2></div>

In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, <a href="Pliny_the_Elder" title="Pliny the Elder">Pliny the Elder</a> had previously described the <a href="Cementation_process" title="Cementation process">cementation process</a>, which produces steel from the element <a href="Iron" title="Iron">iron</a> (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of <a href="Stained_glass" title="Stained glass">stained glass</a> or <a href="Earthenware" title="Earthenware">earthenware</a> and <a href="Chinese_ceramics" title="Chinese ceramics">Chinese ceramics</a>.

In modern science, the first systematic experimental study of diffusion was performed by <a href="Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Thomas Graham</a>. He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833:<a href="#cite_note-10">[10]</a>

<blockquote>"...gases of different nature, when brought into contact, do not arrange themselves according to their density, the heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time." </blockquote>

The measurements of Graham contributed to <a href="James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> deriving, in 1867, the coefficient of diffusion for CO2 in the air. The error rate is less than 5%.

In 1855, <a href="Adolf_Fick" class="mw-redirect" title="Adolf Fick">Adolf Fick</a>, the 26-year-old anatomy demonstrator from Zürich, proposed <a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">his law of diffusion</a>. He used Graham's research, stating his goal as "the development of a fundamental law, for the operation of diffusion in a single element of space". He asserted a deep analogy between diffusion and conduction of heat or electricity, creating a formalism similar to <a href="Thermal_conduction" title="Thermal conduction">Fourier's law for heat conduction</a> (1822) and <a href="Ohm's_law" title="Ohm's law">Ohm's law</a> for electric current (1827).

<a href="Robert_Boyle" title="Robert Boyle">Robert Boyle</a> demonstrated diffusion in solids in the 17th century<a href="#cite_note-11">[11]</a> by penetration of zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century. <a href="William_Chandler_Roberts-Austen" title="William Chandler Roberts-Austen">William Chandler Roberts-Austen</a>, the well-known British metallurgist and former assistant of Thomas Graham studied systematically solid state diffusion on the example of gold in lead in 1896. :<a href="#cite_note-Mehrer2009-12">[12]</a>

"... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals."

</blockquote>

In 1858, <a href="Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> introduced the concept of the <a href="Mean_free_path" title="Mean free path">mean free path</a>. In the same year, <a href="James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> was developed by <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <a href="Marian_Smoluchowski" title="Marian Smoluchowski">Marian Smoluchowski</a> and <a href="Jean-Baptiste_Perrin" class="mw-redirect" title="Jean-Baptiste Perrin">Jean-Baptiste Perrin</a>. <a href="Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>, in the development of the atomistic backgrounds of the macroscopic <a href="Transport_phenomena" title="Transport phenomena">transport processes</a>, introduced the <a href="Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a>, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.<a href="#cite_note-ChapmanCowling-13">[13]</a>

In 1920–1921, <a href="George_de_Hevesy" title="George de Hevesy">George de Hevesy</a> measured <a href="Self-diffusion" title="Self-diffusion">self-diffusion</a> using <a href="Radioisotope" class="mw-redirect" title="Radioisotope">radioisotopes</a>. He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead.

<a href="Yakov_Frenkel" title="Yakov Frenkel">Yakov Frenkel</a> (sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and <a href="Interstitial_defect" title="Interstitial defect">interstitial</a> atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data.

Sometime later, <a href="Carl_Wagner" title="Carl Wagner">Carl Wagner</a> and <a href="Walter_H._Schottky" class="mw-redirect" title="Walter H. Schottky">Walter H. Schottky</a> developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.<a href="#cite_note-Mehrer2009-12">[12]</a>

<a href="Henry_Eyring_(chemist)" title="Henry Eyring (chemist)">Henry Eyring</a>, with co-authors, applied his theory of <a href="Transition_state_theory" title="Transition state theory">absolute reaction rates</a> to Frenkel's quasichemical model of diffusion.<a href="#cite_note-14">[14]</a> The analogy between <a href="Chemical_kinetics" title="Chemical kinetics">reaction kinetics</a> and diffusion leads to various nonlinear versions of Fick's law.<a href="#cite_note-GorbanMMNP2011-15">[15]</a>

<div class="mw-heading mw-heading2"><h2 id="Basic_models_of_diffusion">Basic models of diffusion</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_of_diffusion_flux">Definition of diffusion flux</h3></div>

Each model of diffusion expresses the diffusion flux with the use of concentrations, densities and their derivatives. Flux is a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation>

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</math><img src="./7686846b1a6b756cb514954000004ab5e7b2a5ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }" loading="lazy"> representing the quantity and direction of transfer. Given a small <a href="Area" title="Area">area</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta S}</annotation>

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</math><img src="./15a2efec755f08f95da4e8d1f7f2682861fb59be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.176ex;" alt="{\displaystyle \Delta S}" loading="lazy"> with normal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nu }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nu }}}</annotation>

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</math><img src="./57d030f5358ad62e959e7907ec1508746a563160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.413ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\nu }}}" loading="lazy">, the transfer of a <a href="Physical_quantity" title="Physical quantity">physical quantity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> through the area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta S}</annotation>

</semantics>

</math><img src="./15a2efec755f08f95da4e8d1f7f2682861fb59be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.176ex;" alt="{\displaystyle \Delta S}" loading="lazy"> per time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>

</semantics>

</math><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"> is

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}</annotation>

</semantics>

</math><img src="./128ef623007330633b78107463ea2dfcd7136425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.129ex; height:2.843ex;" alt="{\displaystyle \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}</annotation>

</semantics>

</math><img src="./c0cf65696f51b30c5c6459fecc4b514f122acb7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}" loading="lazy"> is the <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o(\cdots )}">

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<annotation encoding="application/x-tex">{\displaystyle o(\cdots )}</annotation>

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</math><img src="./e7f4091204b3105d76efe55de1b966e968e083d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.66ex; height:2.843ex;" alt="{\displaystyle o(\cdots )}" loading="lazy"> is the <a href="Little-o_notation" class="mw-redirect" title="Little-o notation">little-o notation</a>. If we use the notation of <a href="Vector_area" title="Vector area">vector area</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}</annotation>

</semantics>

</math><img src="./6b0a29b1a4cd2027f316f1b8eda89e3b1bd451b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.755ex; height:2.176ex;" alt="{\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}" loading="lazy"> then

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}</annotation>

</semantics>

</math><img src="./67eb487a1cf501db6387cbf586cbf29df6345c2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.301ex; height:2.843ex;" alt="{\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}" loading="lazy"></dd></dl>

The <a href="Dimensional_analysis" title="Dimensional analysis">dimension</a> of the diffusion flux is [flux] = [quantity]/([time]·[area]). The diffusing physical quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> may be the number of particles, mass, energy, electric charge, or any other scalar <a href="Extensive_quantity" class="mw-redirect" title="Extensive quantity">extensive quantity</a>. For its density, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, the diffusion equation has the form

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}</annotation>

</semantics>

</math><img src="./8121584b68d0e12291d92fb0ad2fc83cdf9aad4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.761ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> is intensity of any local source of this quantity (for example, the rate of a chemical reaction).

For the diffusion equation, the no-flux boundary conditions can be formulated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}</annotation>

</semantics>

</math><img src="./9825b82ee811b0ba1e906fede9b0aab4f797d9ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.176ex; height:2.843ex;" alt="{\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}" loading="lazy"> on the boundary, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nu }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nu }}}</annotation>

</semantics>

</math><img src="./57d030f5358ad62e959e7907ec1508746a563160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.413ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\nu }}}" loading="lazy"> is the normal to the boundary at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Normal_single_component_concentration_gradient">Normal single component concentration gradient</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div>

Fick's first law: The diffusion flux, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation>

</semantics>

</math><img src="./7686846b1a6b756cb514954000004ab5e7b2a5ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }" loading="lazy">, is proportional to the negative gradient of spatial concentration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(x,t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n(x,t)}</annotation>

</semantics>

</math><img src="./83e53b78791849dd11cf02b8d0f9af829fb7bbc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.407ex; height:2.843ex;" alt="{\displaystyle n(x,t)}" loading="lazy">:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =-D(x)\,\nabla n(x,t),}</annotation>

</semantics>

</math><img src="./e077043433b8454791868f07fa25257610f54f69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.728ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} =-D(x)\,\nabla n(x,t),}" loading="lazy"></dd></dl>

where D is the <a href="Mass_diffusivity" title="Mass diffusivity">diffusion coefficient</a>, which can be estimated for a given mixture using, for example, the empirical Vignes correlation model<a href="#cite_note-16">[16]</a> or the physically motivated entropy scaling.<a href="#cite_note-17">[17]</a> The corresponding <a href="Diffusion_equation" title="Diffusion equation">diffusion equation</a> (Fick's second law) is

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D(x)\,\nabla n(x,t))\,.}</annotation>

</semantics>

</math><img src="./86e9ef052a5407d0903c28b414a2cfa377553176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.912ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D(x)\,\nabla n(x,t))\,.}" loading="lazy"></dd></dl>

In case the diffusion coefficient is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, Fick's second law can be simplified to

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=D\,\Delta n(x,t)\ ,}</annotation>

</semantics>

</math><img src="./63835ecba3329656c052314b7c969562e345c3eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.542ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=D\,\Delta n(x,t)\ ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> is the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a>,

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}</annotation>

</semantics>

</math><img src="./adec6221c21e124e481d225ffce1561975e7069f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.052ex; height:7.009ex;" alt="{\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Multicomponent_diffusion_and_thermodiffusion">Multicomponent diffusion and thermodiffusion</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></div>

Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\nabla n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\nabla n}</annotation>

</semantics>

</math><img src="./b86681c20965fdd279bcab10e6e123a9b83b8a36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.139ex; height:2.343ex;" alt="{\displaystyle -\nabla n}" loading="lazy">.

In 1931, <a href="Lars_Onsager" title="Lars Onsager">Lars Onsager</a><a href="#cite_note-Onsager1931-18">[18]</a> included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For

multi-component transport,

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}</annotation>

</semantics>

</math><img src="./8536ffece049a8b71606cea568ed594be02762f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.949ex; height:5.843ex;" alt="{\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}}</annotation>

</semantics>

</math><img src="./e487b57ee92e16a526d0da3cc0cb73eccb5554ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.18ex; height:2.509ex;" alt="{\displaystyle \mathbf {J} _{i}}" loading="lazy"> is the flux of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th physical quantity (component), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation>

</semantics>

</math><img src="./ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">th <a href="Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">thermodynamic force</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{ij}}</annotation>

</semantics>

</math><img src="./a310c7a19bc362591bb6809bb95b7e67baa59965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.06ex; height:2.843ex;" alt="{\displaystyle L_{ij}}" loading="lazy"> is Onsager's matrix of kinetic <a href="Transport_coefficient" title="Transport coefficient">transport coefficients</a>.

The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the <a href="Entropy" title="Entropy">entropy</a> density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> (he used the term "force" in quotation marks or "driving force"):

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{i}}}\,,}</annotation>

</semantics>

</math><img src="./4da389d3c04f44f892763fbb650c0e5a16a42907.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.241ex; height:6.176ex;" alt="{\displaystyle X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{i}}}\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation>

</semantics>

</math><img src="./57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"> are the "thermodynamic coordinates".

For the heat and mass transfer one can take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}=u}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{0}=u}</annotation>

</semantics>

</math><img src="./d9095fc3d8cc58402ff93e16cbd8d294f94338c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.877ex; height:2.009ex;" alt="{\displaystyle n_{0}=u}" loading="lazy"> (the density of internal energy) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation>

</semantics>

</math><img src="./57f87f905ba5a4d8c691ccaecd65fc47bd007ba4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}" loading="lazy"> is the concentration of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th component. The corresponding driving forces are the space vectors

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}</annotation>

</semantics>

</math><img src="./b3b0840ce38c3ab2c02127d6648bb2395452e946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.804ex; height:5.176ex;" alt="{\displaystyle X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}" loading="lazy"> because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}">

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}</annotation>

</semantics>

</math><img src="./45a9ca71a6f5a299402ef594207736e64235ce20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.457ex; height:6.509ex;" alt="{\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}" loading="lazy"></dd></dl>

where T is the absolute temperature and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation>

</semantics>

</math><img src="./dea0a0293841cce9eef98b55e53a92b82ae59ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}" loading="lazy"> is the chemical potential of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">th component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.

For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium:

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\,\partial n_{k}}}\right|_{n=n^{*}}\nabla n_{k}\ ,}</annotation>

</semantics>

</math><img src="./0ceb3ac1de281751b802241370c33fc69d4bd764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.71ex; height:7.343ex;" alt="{\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\,\partial n_{k}}}\right|_{n=n^{*}}\nabla n_{k}\ ,}" loading="lazy"></dd></dl>

where the derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> are calculated at equilibrium <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle n^{*}}</annotation>

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</math><img src="./9615e0726d9d6cc58d117a844dca1ca84bb81f7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.343ex;" alt="{\displaystyle n^{*}}" loading="lazy">.

The matrix of the kinetic coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle L_{ij}}</annotation>

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</math><img src="./a310c7a19bc362591bb6809bb95b7e67baa59965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.06ex; height:2.843ex;" alt="{\displaystyle L_{ij}}" loading="lazy"> should be symmetric (<a href="Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a>) and <a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a> (<a href="Second_law_of_thermodynamics" title="Second law of thermodynamics">for the entropy growth</a>).

The transport equations are

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}\operatorname {div} X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\,\partial n_{k}}}\right|_{n=n^{*}}\right]\,\Delta n_{k}\ .}</annotation>

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</math><img src="./f61d376b495038f57128d2c6ea83f733b7ae0b83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.826ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial n_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}\operatorname {div} X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\,\partial n_{k}}}\right|_{n=n^{*}}\right]\,\Delta n_{k}\ .}" loading="lazy"></dd></dl>

Here, all the indexes i, j, k = 0, 1, 2, ... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{ik}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{ik}}</annotation>

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</math><img src="./ec9de6defea9f68c3dc1f888bb1023881b3e4123.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.58ex; height:2.509ex;" alt="{\displaystyle D_{ik}}" loading="lazy"> of the diffusion (i,k > 0), thermodiffusion (i > 0, k = 0 or k > 0, i = 0) and <a href="Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> (i = k = 0) coefficients.

Under <a href="Isothermal_process" title="Isothermal process">isothermal conditions</a> T = constant. The relevant <a href="Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a> is the free energy (or the <a href="Free_entropy" title="Free entropy">free entropy</a>). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(1/T)\,\nabla \mu _{j}}">

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<annotation encoding="application/x-tex">{\displaystyle -(1/T)\,\nabla \mu _{j}}</annotation>

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</math><img src="./2ccce0811144693833fd033a88dad7078d20c6a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.213ex; height:3.009ex;" alt="{\displaystyle -(1/T)\,\nabla \mu _{j}}" loading="lazy">, and the matrix of diffusion coefficients is

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<annotation encoding="application/x-tex">{\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right|_{n=n^{*}}}</annotation>

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</math><img src="./4cec19e6740db6f14803d8a1b700dda3e25c1a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.642ex; height:7.176ex;" alt="{\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right|_{n=n^{*}}}" loading="lazy"></dd></dl>

(i,k > 0).

There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{j}L_{ij}X_{j}}">

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</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{j}L_{ij}X_{j}}</annotation>

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</math><img src="./cc6370157ef78650d7f028264530ab8a28ae6802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.645ex; height:3.343ex;" alt="{\textstyle \sum _{j}L_{ij}X_{j}}" loading="lazy"> can be measured. For example, in the original work of Onsager<a href="#cite_note-Onsager1931-18">[18]</a> the thermodynamic forces include additional multiplier T, whereas in the <a href="Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a><a href="#cite_note-19">[19]</a> this multiplier is omitted but the sign of the thermodynamic forces is opposite. All these changes are supplemented by the corresponding changes in the coefficients and do not affect the measurable quantities.

<div class="mw-heading mw-heading4"><h4 id="Nondiagonal_diffusion_must_be_nonlinear">Nondiagonal diffusion must be nonlinear</h4></div>

The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}</annotation>

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</math><img src="./81e382a1947aa1dc8735d23b59404e384e29367b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.089ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}" loading="lazy"></dd></dl>

If the matrix of diffusion coefficients is diagonal, then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is non-diagonal, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{12}\neq 0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle D_{12}\neq 0}</annotation>

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</math><img src="./a36dd99272e3a8ccd32ec7c26d33ff92009f1e97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.061ex; height:2.676ex;" alt="{\displaystyle D_{12}\neq 0}" loading="lazy">, and consider the state with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}=\cdots =c_{n}=0}">

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<annotation encoding="application/x-tex">{\displaystyle c_{2}=\cdots =c_{n}=0}</annotation>

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</math><img src="./56bec79d38fa6f8540601b5e1d3aa9a295b907c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.467ex; height:2.509ex;" alt="{\displaystyle c_{2}=\cdots =c_{n}=0}" loading="lazy">. At this state, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}</annotation>

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</math><img src="./38d40e9fcdf6d5ab081f5f6bf200bf6944cc59fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.982ex; height:2.843ex;" alt="{\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{12}\,\Delta c_{1}(x)<0}">

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<annotation encoding="application/x-tex">{\displaystyle D_{12}\,\Delta c_{1}(x)<0}</annotation>

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</math><img src="./ee9ccf9ce4d7b3f01f4e5c74368f9049cd83ffbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.584ex; height:2.843ex;" alt="{\displaystyle D_{12}\,\Delta c_{1}(x)<0}" loading="lazy"> at some points, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{2}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{2}(x)}</annotation>

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</math><img src="./7750489f279606a91bf8b550c637b087efc6b822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.2ex; height:2.843ex;" alt="{\displaystyle c_{2}(x)}" loading="lazy"> becomes negative at these points in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. Non-diagonal equations of multicomponent diffusion must be non-linear.<a href="#cite_note-GorbanMMNP2011-15">[15]</a>

<div class="mw-heading mw-heading3"><h3 id="Applied_forces">Applied forces</h3></div>

The <a href="Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a> connects the diffusion coefficient and the mobility (the ratio of the particle's terminal <a href="Drift_velocity" title="Drift velocity">drift velocity</a> to an applied <a href="Force" title="Force">force</a>).<a href="#cite_note-20">[20]</a> For charged particles:

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<annotation encoding="application/x-tex">{\displaystyle D={\frac {\mu \,k_{\text{B}}T}{q}},}</annotation>

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</math><img src="./666dd11f3911d5e87d853377d393cc02a595db3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.538ex; height:5.843ex;" alt="{\displaystyle D={\frac {\mu \,k_{\text{B}}T}{q}},}" loading="lazy"></dd></dl>

where D is the <a href="Fick's_law_of_diffusion" class="mw-redirect" title="Fick's law of diffusion">diffusion constant</a>, μ is the "mobility", kB is the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, T is the <a href="Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a>, and q is the <a href="Elementary_charge" title="Elementary charge">elementary charge</a>, that is, the charge of one electron.

Below, to combine in the same formula the chemical potential μ and the mobility, we use for mobility the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Diffusion_across_a_membrane">Diffusion across a membrane</h3></div>

The mobility-based approach was further applied by T. Teorell.<a href="#cite_note-21">[21]</a> In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula:

<dl><dd>the flux is equal to mobility × concentration × force per gram-ion.</dd></dl>

This is the so-called Teorell formula. The term "gram-ion" ("gram-particle") is used for a quantity of a substance that contains the <a href="Avogadro_constant" title="Avogadro constant">Avogadro number</a> of ions (particles). The common modern term is <a href="Mole_(unit)" title="Mole (unit)">mole</a>.

The force under isothermal conditions consists of two parts:

<ol><li>Diffusion force caused by concentration gradient: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}">

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<annotation encoding="application/x-tex">{\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}</annotation>

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</math><img src="./16ee12d5ba735b50f1ef17e9d113c18d58015fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.127ex; height:5.176ex;" alt="{\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}" loading="lazy">.</li>

<li>Electrostatic force caused by electric potential gradient: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\,\nabla \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle q\,\nabla \varphi }</annotation>

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</math><img src="./320618130d088ba650502730a8b41fada726e296.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.913ex; height:2.676ex;" alt="{\displaystyle q\,\nabla \varphi }" loading="lazy">.</li></ol>

Here R is the <a href="Gas_constant" title="Gas constant">gas constant</a>, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript "eq", q is the charge and φ is the electric potential.

The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein–Teorell approach, if for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule.

The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is<a href="#cite_note-GorbanMMNP2011-15">[15]</a>

</mrow>

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<mtext>external force per mole</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}</annotation>

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</math><img src="./5e2e9d45e2b2f4e941883907d1062a9902443872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.225ex; height:6.176ex;" alt="{\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}" loading="lazy"></dd></dl>

where μ is the <a href="Chemical_potential" title="Chemical potential">chemical potential</a>, μ0 is the standard value of the chemical potential.

The expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}</annotation>

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</math><img src="./ebf9dd6bce92dc028a635d05d2443ca38096b561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.836ex; height:6.176ex;" alt="{\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}" loading="lazy"> is the so-called <a href="Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a>. It measures the "effective concentration" of a species in a non-ideal mixture. In this notation, the Teorell formula for the flux has a very simple form<a href="#cite_note-GorbanMMNP2011-15">[15]</a>

</mrow>

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<mtext>external force per mole</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}</annotation>

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</math><img src="./2aac0ac115a8bf9d09a24b4f9dc68f639ce057a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.56ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}" loading="lazy"></dd></dl>

The standard derivation of the activity includes a normalization factor and for small concentrations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}">

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<annotation encoding="application/x-tex">{\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}</annotation>

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</math><img src="./be70d2c715bd1f9a904bcc9b3797094d5360100d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.031ex; height:3.009ex;" alt="{\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{\ominus }}">

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<annotation encoding="application/x-tex">{\displaystyle n^{\ominus }}</annotation>

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</math><img src="./0f8aee2b1a42d52398634a97fc56108b2cc88e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.906ex; height:2.509ex;" alt="{\displaystyle n^{\ominus }}" loading="lazy"> is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n/n^{\ominus }}">

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle n/n^{\ominus }}</annotation>

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</math><img src="./bc988f8e1d07ca158fd134d7b8aa9b85c18ff714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:3.009ex;" alt="{\displaystyle n/n^{\ominus }}" loading="lazy">:

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<mtext>external force per mole</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\text{external force per mole}}))].}</annotation>

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</math><img src="./3f8ae11dd009457b8fd39d1a583ed5d4b3e30ab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.706ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\text{external force per mole}}))].}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Ballistic_time_scale">Ballistic time scale</h3></div>

The Einstein model neglects the inertia of the diffusing partial. The alternative

<a href="Langevin_equation" title="Langevin equation">Langevin equation</a> starts with Newton's second law of motion:<a href="#cite_note-Bian_SoftMatt-22">[22]</a>

<mfrac>

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<mrow>

<msup>

</mrow>

</msup>

</mrow>

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</mrow>

<mfrac>

</mfrac>

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<mfrac>

<mrow>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}</annotation>

</semantics>

</math><img src="./9f4d030b5fedbb12b8da476401ff31a6d10f77c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.235ex; height:6.176ex;" alt="{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}" loading="lazy"></dd></dl>

where

<ul><li>x is the position.</li>

<li>μ is the mobility of the particle in the fluid or gas, which can be calculated using the <a href="Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a>.</li>

<li>m is the mass of the particle.</li>

<li>F is the random force applied to the particle.</li>

Solving this equation, one obtained the time-dependent diffusion constant in the long-time limit and when the particle is significantly denser than the surrounding fluid,<a href="#cite_note-Bian_SoftMatt-22">[22]</a>

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<annotation encoding="application/x-tex">{\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}</annotation>

</semantics>

</math><img src="./e93db9ffc3b562d266044a6ffd6b2502c854291a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.239ex; height:3.343ex;" alt="{\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}" loading="lazy"></dd></dl>

where

<ul><li>kB is the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>;</li>

<li>T is the <a href="Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a>.</li>

<li>μ is the mobility of the particle in the fluid or gas, which can be calculated using the <a href="Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a>.</li>

<li>m is the mass of the particle.</li>

At long time scales, Einstein's result is recovered, but short time scales, the ballistic regime are also explained. Moreover, unlike the Einstein approach, a velocity can be defined, leading to the <a href="Fluctuation-dissipation_theorem" class="mw-redirect" title="Fluctuation-dissipation theorem">Fluctuation-dissipation theorem</a>, connecting the competition between friction and random forces in defining the temperature.<a href="#cite_note-Bian_SoftMatt-22">[22]</a>: 3.2

<div class="mw-heading mw-heading3"><h3 id="Jumps_on_the_surface_and_in_solids">Jumps on the surface and in solids</h3></div>

<a href="Surface_diffusion" title="Surface diffusion">Diffusion of reagents on the surface</a> of a <a href="Catalyst" class="mw-redirect" title="Catalyst">catalyst</a> may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.

The system includes several reagents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},A_{2},\ldots ,A_{m}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},\ldots ,A_{m}}</annotation>

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</math><img src="./53bd9d8024bb7542f39ed6446c586b8ee2da8404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.225ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},\ldots ,A_{m}}" loading="lazy"> on the surface. Their surface concentrations are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},c_{2},\ldots ,c_{m}.}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1},c_{2},\ldots ,c_{m}.}</annotation>

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</math><img src="./b092ce106163df27010a07e86eb61ab78af862e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.663ex; height:2.009ex;" alt="{\displaystyle c_{1},c_{2},\ldots ,c_{m}.}" loading="lazy"> The surface is a lattice of the adsorption places. Each

reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free places is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=c_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle z=c_{0}}</annotation>

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</math><img src="./59b0bf39018e7d0e04947194fc5195aae843336e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.248ex; height:2.009ex;" alt="{\displaystyle z=c_{0}}" loading="lazy">. The sum of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation>

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</math><img src="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy"> (including free places) is constant, the density of adsorption places b.

The jump model gives for the diffusion flux of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation>

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</math><img src="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> (i = 1, ..., n):

<msub>

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</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=-D_{i}[z\,\nabla c_{i}-c_{i}\nabla z]\,.}</annotation>

</semantics>

</math><img src="./e62d2876591f6c0c24854dc77bd002742a487757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.027ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} _{i}=-D_{i}[z\,\nabla c_{i}-c_{i}\nabla z]\,.}" loading="lazy"></dd></dl>

The corresponding diffusion equation is:<a href="#cite_note-GorbanMMNP2011-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}[z\,\Delta c_{i}-c_{i}\,\Delta z]\,.}</annotation>

</semantics>

</math><img src="./59acf7a4d07ec81e21aec16a7dd999c091b60b79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.415ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}[z\,\Delta c_{i}-c_{i}\,\Delta z]\,.}" loading="lazy"></dd></dl>

Due to the conservation law, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}</annotation>

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</math><img src="./f20c8bb1cb3749064c232c430293810b2f0fa0c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.607ex; height:6.843ex;" alt="{\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}" loading="lazy"> and we

have the system of m diffusion equations. For one component we get Fick's law and linear equations because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}</annotation>

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</math><img src="./b2d67337c71b81e1cb225e5fe89e7a8a530b24af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.234ex; height:2.843ex;" alt="{\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}" loading="lazy">. For two and more components the equations are nonlinear.

If all particles can exchange their positions with their closest neighbours then a simple generalization gives

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\,\nabla c_{i}-c_{i}\,\nabla c_{j}]}</annotation>

</semantics>

</math><img src="./aba83bc12bd5bab3419c70e17305975783df881d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.844ex; height:5.843ex;" alt="{\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\,\nabla c_{i}-c_{i}\,\nabla c_{j}]}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\,\Delta c_{i}-c_{i}\,\Delta c_{j}]}</annotation>

</semantics>

</math><img src="./63b9c514f4d2f44400b6315598831cacc5edaee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.429ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\,\Delta c_{i}-c_{i}\,\Delta c_{j}]}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{ij}=D_{ji}\geq 0}">

<msub>

</mrow>

</msub>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{ij}=D_{ji}\geq 0}</annotation>

</semantics>

</math><img src="./4fa6755d8cbb508049601a971b6711f5f411601c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.162ex; height:2.843ex;" alt="{\displaystyle D_{ij}=D_{ji}\geq 0}" loading="lazy"> is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation>

</semantics>

</math><img src="./1882ba8f1dc60f0c68a642abb5af093c73910921.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}" loading="lazy">.

Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.

<div class="mw-heading mw-heading3"><h3 id="Porous_media">Porous media</h3></div>

For diffusion in porous media the basic equations are (if Φ is constant):<a href="#cite_note-23">[23]</a>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =-\phi D\,\nabla n^{m}}</annotation>

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</math><img src="./b3fa5a3ffd20c85fa14169bee4378e8ea92b3f48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.99ex; height:2.676ex;" alt="{\displaystyle \mathbf {J} =-\phi D\,\nabla n^{m}}" loading="lazy"></dd>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}</annotation>

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</math><img src="./7b7dc637c003fe9d293aa9e6fac9748d978f516a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.998ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}" loading="lazy"></dd></dl>

where D is the diffusion coefficient, Φ is porosity, n is the concentration, m > 0 (usually m > 1, the case m = 1 corresponds to Fick's law).

Care must be taken to properly account for the porosity (Φ) of the porous medium in both the flux terms and the accumulation terms.<a href="#cite_note-24">[24]</a> For example, as the porosity goes to zero, the molar flux in the porous medium goes to zero for a given concentration gradient. Upon applying the divergence of the flux, the porosity terms cancel out and the second equation above is formed.

For diffusion of gases in porous media this equation is the formalization of <a href="Darcy's_law" title="Darcy's law">Darcy's law</a>: the <a href="Volumetric_flux" title="Volumetric flux">volumetric flux</a> of a gas in the porous media is

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<annotation encoding="application/x-tex">{\displaystyle q=-{\frac {k}{\mu }}\,\nabla p}</annotation>

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</math><img src="./317c8e7284e54ff00f6d5abacd717fa12da05029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.706ex; height:5.843ex;" alt="{\displaystyle q=-{\frac {k}{\mu }}\,\nabla p}" loading="lazy"></dd></dl>

where k is the <a href="Permeation" title="Permeation">permeability</a> of the medium, μ is the <a href="Viscosity" title="Viscosity">viscosity</a> and p is the pressure.

The advective molar flux is given as

J = nq

and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\sim n^{\gamma }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle p\sim n^{\gamma }}</annotation>

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</math><img src="./d5f663de35f7e79b9471d9d5a10cf8409a4956a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.877ex; height:2.676ex;" alt="{\displaystyle p\sim n^{\gamma }}" loading="lazy"> Darcy's law gives the equation of diffusion in porous media with m = γ + 1.

In porous media, the average linear velocity (ν), is related to the volumetric flux as:

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<annotation encoding="application/x-tex">{\displaystyle \upsilon =q/\phi }</annotation>

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</math><img src="./4e3b633c1887f8d521153c3aab108b81b75a819f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.971ex; height:2.843ex;" alt="{\displaystyle \upsilon =q/\phi }" loading="lazy">

Combining the advective molar flux with the diffusive flux gives the advection dispersion equation

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}</annotation>

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</math><img src="./6fc7493e64e704687a9390a12557545fed63defc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.949ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}" loading="lazy">

For underground water infiltration, the <a href="Boussinesq_approximation_(buoyancy)" title="Boussinesq approximation (buoyancy)">Boussinesq approximation</a> gives the same equation with m = 2.

For plasma with the high level of radiation, the <a href="Yakov_Borisovich_Zel'dovich" class="mw-redirect" title="Yakov Borisovich Zel'dovich">Zeldovich</a>–Raizer equation gives m > 4 for the heat transfer.

<div class="mw-heading mw-heading2"><h2 id="Diffusion_in_physics">Diffusion in physics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Diffusion_coefficient_in_kinetic_theory_of_gases">Diffusion coefficient in kinetic theory of gases</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Kinetic_theory_of_gases#Diffusion_coefficient_and_diffusion_flux" title="Kinetic theory of gases">Kinetic theory of gases § Diffusion coefficient and diffusion flux</a></div>

The diffusion coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is the coefficient in the <a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">Fick's first law</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D\,\partial n/\partial x}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle J=-D\,\partial n/\partial x}</annotation>

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</math><img src="./389e8c8237d7c26d83c9b4cd3aba65df2b0ca1e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.212ex; height:2.843ex;" alt="{\displaystyle J=-D\,\partial n/\partial x}" loading="lazy">, where J is the diffusion flux (<a href="Amount_of_substance" title="Amount of substance">amount of substance</a>) per unit area per unit time, n (for ideal mixtures) is the concentration, x is the position [length].

Consider two gases with molecules of the same diameter d and mass m (<a href="Self-diffusion" title="Self-diffusion">self-diffusion</a>). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient

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<annotation encoding="application/x-tex">{\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}</annotation>

</semantics>

</math><img src="./44b8ae29eec5ca6b6acd318062260c58b6cdfab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.482ex; height:7.676ex;" alt="{\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}" loading="lazy"></dd></dl>

where kB is the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, T is the <a href="Temperature" title="Temperature">temperature</a>, P is the <a href="Pressure" title="Pressure">pressure</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

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<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

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</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> is the <a href="Mean_free_path" title="Mean free path">mean free path</a>, and vT is the mean thermal speed:

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<annotation encoding="application/x-tex">{\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}</annotation>

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</math><img src="./2517767774f635a8f988833518c3b57530e634f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.924ex; height:6.676ex;" alt="{\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}" loading="lazy"></dd></dl>

We can see that the diffusion coefficient in the mean free path approximation grows with T as T3/2 and decreases with P as 1/P. If we use for P the <a href="Ideal_gas_law" title="Ideal gas law">ideal gas law</a> P = RnT with the total concentration n, then we can see that for given concentration n the diffusion coefficient grows with T as T1/2 and for given temperature it decreases with the total concentration as 1/n.

For two different gases, A and B, with molecular masses mA, mB and molecular diameters dA, dB, the mean free path estimate of the diffusion coefficient of A in B and B in A is:

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<annotation encoding="application/x-tex">{\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}</annotation>

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</math><img src="./24885a7b46490e7ba358100addd7f4ff2737f4a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.117ex; height:8.009ex;" alt="{\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_theory_of_diffusion_in_gases_based_on_Boltzmann's_equation">The theory of diffusion in gases based on Boltzmann's equation</h3></div>

In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}(x,c,t)}">

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</math><img src="./e07158d06bb6c24df9108c0c0bab3a9cf0f5f3ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.992ex; height:2.843ex;" alt="{\displaystyle f_{i}(x,c,t)}" loading="lazy">, where t is the time moment, x is position and c is velocity of molecule of the ith component of the mixture. Each component has its mean velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}">

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In the <a href="Chapman%E2%80%93Enskog_theory" title="Chapman–Enskog theory">Chapman–Enskog</a> approximation, all the distribution functions are expressed through the densities of the conserved quantities:<a href="#cite_note-ChapmanCowling-13">[13]</a>

<ul><li>individual concentrations of particles, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}">

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<li>density of momentum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}">

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<li>density of kinetic energy <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}</annotation>

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The kinetic temperature T and pressure P are defined in 3D space as

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<annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}</annotation>

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</math><img src="./3110a480cc2ed2918f0cd0244666c7d5270de029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.33ex; height:6.343ex;" alt="{\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n=\sum _{i}n_{i}}">

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For two gases, the difference between velocities, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}-C_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1}-C_{2}}</annotation>

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</math><img src="./3ca5f4da98421fcd9a64a93e80aea6dc87e11638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.273ex; height:2.509ex;" alt="{\displaystyle C_{1}-C_{2}}" loading="lazy"> is given by the expression:<a href="#cite_note-ChapmanCowling-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}</annotation>

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</math><img src="./7ad79bb78267c350d78ae002648e97b8b376e686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:104.631ex; height:6.509ex;" alt="{\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}}">

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</math><img src="./58133b8078ef48c84ccc76dc178c14614a13d455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.601ex; height:2.509ex;" alt="{\displaystyle k_{T}}" loading="lazy"> is the thermodiffusion ratio.

The coefficient D12 is positive. This is the diffusion coefficient. Four terms in the formula for C1−C2 describe four main effects in the diffusion of gases:

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<annotation encoding="application/x-tex">{\displaystyle \nabla \,\left({\frac {n_{1}}{n}}\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}</annotation>

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</math><img src="./04f6eec1b3de6082e4e673390831337171aba06f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.649ex; height:6.509ex;" alt="{\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}" loading="lazy"> describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is barodiffusion;</li>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}</annotation>

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</math><img src="./7613b9c6b92a333c2df2dd93586c77b10bc829d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.066ex; height:5.509ex;" alt="{\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}" loading="lazy"> describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient.</li>

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<annotation encoding="application/x-tex">{\displaystyle k_{T}{\frac {1}{T}}\nabla T}</annotation>

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</math><img src="./4d512540d66035966ea079093816eebf95b62102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.645ex; height:5.176ex;" alt="{\displaystyle k_{T}{\frac {1}{T}}\nabla T}" loading="lazy"> describes <a href="Thermodiffusion" class="mw-redirect" title="Thermodiffusion">thermodiffusion</a>, the diffusion flux caused by the temperature gradient.</li></ol>

All these effects are called diffusion because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a bulk transport and differ from advection or convection.

In the first approximation,<a href="#cite_note-ChapmanCowling-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}}</annotation>

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</math> for rigid spheres;</li>

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<annotation encoding="application/x-tex">{\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \kappa _{12}r^{-\nu }.}</annotation>

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</math><img src="./e70da5e66512b6ce6005c4d8c50730a62bd31d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.293ex; height:2.843ex;" alt="{\displaystyle \kappa _{12}r^{-\nu }.}" loading="lazy"></li></ul>

The number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}({\nu })}">

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<annotation encoding="application/x-tex">{\displaystyle A_{1}({\nu })}</annotation>

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</math><img src="./c936753fcd4212799e795205102391c4e7c1b475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.839ex; height:2.843ex;" alt="{\displaystyle A_{1}({\nu })}" loading="lazy"> is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book<a href="#cite_note-ChapmanCowling-13">[13]</a>)

We can see that the dependence on T for the rigid spheres is the same as for the simple mean free path theory but for the power repulsion laws the exponent is different. Dependence on a total concentration n for a given temperature has always the same character, 1/n.

In applications to gas dynamics, the diffusion flux and the bulk flow should be joined in one system of transport equations. The bulk flow describes the mass transfer. Its velocity V is the mass average velocity. It is defined through the momentum density and the mass concentrations:

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}</annotation>

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</math><img src="./f1338f199241559cc69498082d839d657e12b14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.859ex; height:6.176ex;" alt="{\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}=m_{i}n_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{i}=m_{i}n_{i}}</annotation>

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</math><img src="./d2b99a8f27d66a325df4109d279429c80612512e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.135ex; height:2.176ex;" alt="{\displaystyle \rho _{i}=m_{i}n_{i}}" loading="lazy"> is the mass concentration of the ith species, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho =\sum _{i}\rho _{i}}">

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<annotation encoding="application/x-tex">{\textstyle \rho =\sum _{i}\rho _{i}}</annotation>

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</math><img src="./f9e896c6d06097eee40e53c24b97734f8aec4359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.943ex; height:3.009ex;" alt="{\textstyle \rho =\sum _{i}\rho _{i}}" loading="lazy"> is the mass density.

By definition, the diffusion velocity of the ith component is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}=C_{i}-V}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle v_{i}=C_{i}-V}</annotation>

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</math><img src="./5334ac70a0df91ac78d451d57312273ab3df86c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.115ex; height:2.509ex;" alt="{\displaystyle v_{i}=C_{i}-V}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}\rho _{i}v_{i}=0}">

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<annotation encoding="application/x-tex">{\textstyle \sum _{i}\rho _{i}v_{i}=0}</annotation>

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</math><img src="./fca540e312885c99be5304d1c3bbab7872209b44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.83ex; height:3.009ex;" alt="{\textstyle \sum _{i}\rho _{i}v_{i}=0}" loading="lazy">.

The mass transfer of the ith component is described by the <a href="Continuity_equation" title="Continuity equation">continuity equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}</annotation>

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</math><img src="./13efa65115eb15709d0f0e07e788afc526ef2f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.171ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation>

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</math><img src="./7301a4cfd04d4f5db4549fdf23746a0d2ce9f387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}" loading="lazy"> is the net mass production rate in chemical reactions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}W_{i}=0}">

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<annotation encoding="application/x-tex">{\textstyle \sum _{i}W_{i}=0}</annotation>

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</math><img src="./0fd533319d12f9950eeb4d6040d9ba0db1680471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.895ex; height:3.009ex;" alt="{\textstyle \sum _{i}W_{i}=0}" loading="lazy">.

In these equations, the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (\rho _{i}V)}">

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<annotation encoding="application/x-tex">{\displaystyle \nabla (\rho _{i}V)}</annotation>

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</math><img src="./9678db80562a408fa692f3ae43db50ce4bb93ab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.534ex; height:2.843ex;" alt="{\displaystyle \nabla (\rho _{i}V)}" loading="lazy"> describes advection of the ith component and the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (\rho _{i}v_{i})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \nabla (\rho _{i}v_{i})}</annotation>

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</math><img src="./494ed640f1181691c279b703690d7865fa5c3aab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.674ex; height:2.843ex;" alt="{\displaystyle \nabla (\rho _{i}v_{i})}" loading="lazy"> represents diffusion of this component.

In 1948, <a href="Wendell_H._Furry" title="Wendell H. Furry">Wendell H. Furry</a> proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.<a href="#cite_note-25">[25]</a> For the diffusion velocities in multicomponent gases (N components) they used

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<annotation encoding="application/x-tex">{\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}</annotation>

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</math><img src="./7aeac7715f2a0564fbd425e275c3ca71af7fff16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.39ex; height:7.676ex;" alt="{\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}</annotation>

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</math><img src="./6bd0fa2de03db870590d4864c0121ff0be93baac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.173ex; height:3.176ex;" alt="{\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}</annotation>

</semantics>

</math><img src="./ff8231f63097a42d71a702dd5c77aa1a0bdffe2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.171ex; height:7.509ex;" alt="{\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}" loading="lazy"></dd></dl>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{ij}}</annotation>

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</math><img src="./88badd2360f6d55865a1e8f1b4e3994451bcc075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.401ex; height:2.843ex;" alt="{\displaystyle D_{ij}}" loading="lazy"> is the diffusion coefficient matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}^{(T)}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{i}^{(T)}}</annotation>

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</math><img src="./a08d19cbfd375c42bbc48bddecd4b0a240edbd98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.593ex; height:3.676ex;" alt="{\displaystyle D_{i}^{(T)}}" loading="lazy"> is the thermal diffusion coefficient, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation>

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</math><img src="./65da883ca3d16b461e46c94777b0d9c4aa010e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}" loading="lazy"> is the body force per unit mass acting on the ith species, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}=P_{i}/P}">

<msub>

</mrow>

</msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}=P_{i}/P}</annotation>

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</math><img src="./4c269a5ba2076f26f51956549e257425ad91d212.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.022ex; height:2.843ex;" alt="{\displaystyle X_{i}=P_{i}/P}" loading="lazy"> is the partial pressure fraction of the ith species (and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation>

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</math><img src="./3ba1396129f7be3c7f828a571b6649e6807d10d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}" loading="lazy"> is the partial pressure), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}=\rho _{i}/\rho }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Y_{i}=\rho _{i}/\rho }</annotation>

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</math><img src="./8af34b032213668c3113d079d6f71e024e0f6635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.615ex; height:2.843ex;" alt="{\displaystyle Y_{i}=\rho _{i}/\rho }" loading="lazy"> is the mass fraction of the ith species, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}">

</mrow>

</munder>

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</munder>

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</mrow>

<annotation encoding="application/x-tex">{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}</annotation>

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</math><img src="./18c246dd5e2ad7eb4b6e1c93392c40bd402f26f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.162ex; height:3.009ex;" alt="{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Diffusion_of_electrons_in_solids">Diffusion of electrons in solids</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Diffusion_current" title="Diffusion current">Diffusion current</a></div>

When the density of electrons in solids is not in equilibrium, diffusion of electrons occurs. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light shines on one end (see right figure), electrons diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as <a href="Diffusion_current" title="Diffusion current">diffusion current</a>.

Diffusion current can also be described by <a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">Fick's first law</a>

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<annotation encoding="application/x-tex">{\displaystyle J=-D\,\partial n/\partial x\,,}</annotation>

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</math><img src="./f7686fa789286940651b862b24dda1dcc13250e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.246ex; height:2.843ex;" alt="{\displaystyle J=-D\,\partial n/\partial x\,,}" loading="lazy"></dd></dl>

where J is the diffusion current density (<a href="Amount_of_substance" title="Amount of substance">amount of substance</a>) per unit area per unit time, n (for ideal mixtures) is the electron density, x is the position [length].

<div class="mw-heading mw-heading3"><h3 id="Diffusion_in_geophysics">Diffusion in geophysics</h3></div>

Analytical and numerical models that solve the diffusion equation for different initial and boundary conditions have been popular for studying a wide variety of changes to the Earth's surface. Diffusion has been used extensively in erosion studies of hillslope retreat, bluff erosion, fault scarp degradation, wave-cut terrace/shoreline retreat, alluvial channel incision, coastal shelf retreat, and delta progradation.<a href="#cite_note-26">[26]</a> Although the Earth's surface is not literally diffusing in many of these cases, the process of diffusion effectively mimics the holistic changes that occur over decades to millennia. Diffusion models may also be used to solve inverse boundary value problems in which some information about the depositional environment is known from paleoenvironmental reconstruction and the diffusion equation is used to figure out the sediment influx and time series of landform changes.<a href="#cite_note-27">[27]</a>

<div class="mw-heading mw-heading3"><h3 id="Dialysis">Dialysis</h3></div>

Dialysis works on the principles of the diffusion of solutes and <a href="Ultrafiltration" title="Ultrafiltration">ultrafiltration</a> of fluid across a <a href="Semi-permeable_membrane" class="mw-redirect" title="Semi-permeable membrane">semi-permeable membrane</a>. Diffusion is a property of substances in water; substances in water tend to move from an area of high concentration to an area of low concentration.<a href="#cite_note-Mosby-28">[28]</a> Blood flows by one side of a semi-permeable membrane, and a dialysate, or special dialysis fluid, flows by the opposite side. A semipermeable membrane is a thin layer of material that contains holes of various sizes, or pores. Smaller solutes and fluid pass through the membrane, but the membrane blocks the passage of larger substances (for example, red blood cells and large proteins). This replicates the filtering process that takes place in the kidneys when the blood enters the kidneys and the larger substances are separated from the smaller ones in the <a href="Glomerulus_(kidney)" title="Glomerulus (kidney)">glomerulus</a>.<a href="#cite_note-Mosby-28">[28]</a>

<div class="mw-heading mw-heading2"><h2 id="Random_walk_(random_motion)">Random walk (random motion)</h2></div>

One common misconception is that individual atoms, ions or molecules move randomly, which they do not. In the animation on the right, the ion in the left panel appears to have "random" motion in the absence of other ions. As the right panel shows, however, this motion is not random but is the result of "collisions" with other ions. As such, the movement of a single atom, ion, or molecule within a mixture just appears random when viewed in isolation. The movement of a substance within a mixture by "random walk" is governed by the kinetic energy within the system that can be affected by changes in concentration, pressure or temperature. (This is a classical description. At smaller scales, quantum effects will be non-negligible, in general. Thus, the study of the movement of a single atom becomes more subtle since particles at such small scales are described by probability amplitudes rather than deterministic measures of position and velocity.)

<div class="mw-heading mw-heading3"><h3 id="Separation_of_diffusion_from_convection_in_gases">Separation of diffusion from convection in gases</h3></div>

While Brownian motion of multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task.

Under normal conditions, molecular diffusion dominates only at lengths in the nanometre-to-millimetre range. On larger length scales, transport in liquids and gases is normally due to another <a href="Transport_phenomena" title="Transport phenomena">transport phenomenon</a>, <a href="Convection" title="Convection">convection</a>. To separate diffusion in these cases, special efforts are needed.

In contrast, <a href="Heat_conduction" class="mw-redirect" title="Heat conduction">heat conduction</a> through solid media is an everyday occurrence (for example, a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.

<div class="mw-heading mw-heading3"><h3 id="Other_types_of_diffusion">Other types of diffusion</h3></div>

<ul><li><a href="Anisotropic_diffusion" title="Anisotropic diffusion">Anisotropic diffusion</a>, also known as the Perona–Malik equation, enhances high gradients</li>

<li><a href="Atomic_diffusion" title="Atomic diffusion">Atomic diffusion</a>, in solids</li>

<li><a href="Bohm_diffusion" title="Bohm diffusion">Bohm diffusion</a>, spread of plasma across magnetic fields</li>

<li><a href="Eddy_diffusion" title="Eddy diffusion">Eddy diffusion</a>, in coarse-grained description of turbulent flow</li>

<li><a href="Effusion" title="Effusion">Effusion</a> of a gas through small holes</li>

<li><a href="Electronics" title="Electronics">Electronic</a> diffusion, resulting in an <a href="Current_(electricity)" class="mw-redirect" title="Current (electricity)">electric current</a> called the <a href="Diffusion_current" title="Diffusion current">diffusion current</a></li>

<li><a href="Facilitated_diffusion" title="Facilitated diffusion">Facilitated diffusion</a>, present in some organisms</li>

<li><a href="Gaseous_diffusion" title="Gaseous diffusion">Gaseous diffusion</a>, used for <a href="Isotope_separation" title="Isotope separation">isotope separation</a></li>

<li><a href="Heat_equation" title="Heat equation">Heat equation</a>, diffusion of thermal energy</li>

<li><a href="It%C5%8D_diffusion" class="mw-redirect" title="Itō diffusion">Itō diffusion</a>, mathematisation of Brownian motion, continuous stochastic process.</li>

<li><a href="Knudsen_diffusion" title="Knudsen diffusion">Knudsen diffusion</a> of gas in long pores with frequent wall collisions</li>

<li><a href="L%C3%A9vy_flight" title="Lévy flight">Lévy flight</a></li>

<li><a href="Molecular_diffusion" title="Molecular diffusion">Molecular diffusion</a>, diffusion of molecules from more dense to less dense areas</li>

<li><a href="Momentum_diffusion" title="Momentum diffusion">Momentum diffusion</a> ex. the diffusion of the <a href="Hydrodynamic" class="mw-redirect" title="Hydrodynamic">hydrodynamic</a> velocity field</li>

<li><a href="Photon_diffusion" title="Photon diffusion">Photon diffusion</a></li>

<li><a href="Plasma_diffusion" title="Plasma diffusion">Plasma diffusion</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>,<a href="#cite_note-29">[29]</a> model for diffusion</li>

<li><a href="Reverse_diffusion" title="Reverse diffusion">Reverse diffusion</a>, against the concentration gradient, in phase separation</li>

<li><a href="Rotational_diffusion" title="Rotational diffusion">Rotational diffusion</a>, random reorientation of molecules</li>

<li><a href="Spin_diffusion" title="Spin diffusion">Spin diffusion</a>, diffusion of <a href="Spin_magnetic_moment" class="mw-redirect" title="Spin magnetic moment">spin magnetic moments</a> in solids</li>

<li><a href="Surface_diffusion" title="Surface diffusion">Surface diffusion</a>, diffusion of adparticles on a surface</li>

<li><a href="Taxis" title="Taxis">Taxis</a> is an animal's directional movement activity in response to a stimulus

<ul><li><a href="Kinesis_(biology)" title="Kinesis (biology)">Kinesis</a> is an animal's non-directional movement activity in response to a stimulus</li></ul></li>

<li><a href="Trans-cultural_diffusion" class="mw-redirect" title="Trans-cultural diffusion">Trans-cultural diffusion</a>, diffusion of cultural traits across geographical area</li>

<li><a href="Turbulent_diffusion" title="Turbulent diffusion">Turbulent diffusion</a>, transport of mass, heat, or momentum within a turbulent fluid</li></ul>

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<ul><li><a href="Anomalous_diffusion" title="Anomalous diffusion">Anomalous diffusion</a> – Diffusion process with a non-linear relationship to time</li>

<li><a href="Convection%E2%80%93diffusion_equation" title="Convection–diffusion equation">Convection–diffusion equation</a> – Combination of the diffusion and convection (advection) equations</li>

<li><a href="Diffusion-limited_aggregation" title="Diffusion-limited aggregation">Diffusion-limited aggregation</a> – Process of particles clustering together</li>

<li><a href="Darken's_equations" title="Darken's equations">Darken's equations</a></li>

<li><a href="Isobaric_counterdiffusion" title="Isobaric counterdiffusion">Isobaric counterdiffusion</a> – Gaseous diffusion through body tissue at constant total pressure</li>

<li><a href="Sorption" title="Sorption">Sorption</a> – Physical or chemical process by which one substance becomes attached to another</li>

<li><a href="Osmosis" title="Osmosis">Osmosis</a> – Movement of molecules to lower concentration</li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation theory</a> – Mathematical theory on behavior of connected clusters in a random graph</li>

<li><a href="Social_Networks" class="mw-redirect" title="Social Networks">Social Networks</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Bian_SoftMatt-22">^ <a href="#cite_ref-Bian_SoftMatt_22-0">a</a> <a href="#cite_ref-Bian_SoftMatt_22-1">b</a> <a href="#cite_ref-Bian_SoftMatt_22-2">c</a> <cite id="CITEREFBian,_XinKim,_ChanghoKarniadakis,_George_Em2016" class="citation journal cs1">Bian, Xin; Kim, Changho; Karniadakis, George Em (2016-08-14). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5476231">"111 years of Brownian motion"</a>. Soft Matter. 12 (30): 6331–6346. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016SMat...12.6331B">2016SMat...12.6331B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2Fc6sm01153e">10.1039/c6sm01153e</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5476231">5476231</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/27396746">27396746</a>.</cite>

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<li id="cite_note-23"><a href="#cite_ref-23">^</a> J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0198569033</bdi>.

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<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFS._H._Lam2006" class="citation journal cs1">S. H. Lam (2006). <a rel="nofollow" class="external text" href="http://www.princeton.edu/~lam/MultiComp.pdf">"Multicomponent diffusion revisited"</a> (PDF). Physics of Fluids. 18 (7) 073101: 073101–073101–8. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006PhFl...18g3101L">2006PhFl...18g3101L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.2221312">10.1063/1.2221312</a>.</cite>

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<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFPasternackBrushHilgartner2001" class="citation journal cs1">Pasternack, Gregory B.; Brush, Grace S.; Hilgartner, William B. (2001-04-01). "Impact of historic land-use change on sediment delivery to a Chesapeake Bay subestuarine delta". <a href="Earth_Surface_Processes_and_Landforms" title="Earth Surface Processes and Landforms">Earth Surface Processes and Landforms</a>. 26 (4): 409–27. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001ESPL...26..409P">2001ESPL...26..409P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fesp.189">10.1002/esp.189</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1096-9837">1096-9837</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:129080402">129080402</a>.</cite>

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<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFGregory_B._Pasternack" class="citation web cs1">Gregory B. Pasternack. <a rel="nofollow" class="external text" href="http://pasternack.ucdavis.edu/research/projects/tidal-freshwater-deltas/tfd-modeling/">"Watershed Hydrology, Geomorphology, and Ecohydraulics :: TFD Modeling"</a>. pasternack.ucdavis.edu. Retrieved 2017-06-12.</cite>

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