Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher

_{2} in the air. The error rate is less than 5%.
In 1855,

_{B} is _{0} is the standard value of the chemical potential.
The expression $a\; =\; \backslash exp\backslash left(\backslash frac\backslash right)$ is the so-called

_{B} is

_{B} is the Boltzmann constant, ''T'' is the _{T}'' is the mean thermal speed:
:$\backslash ell\; =\; \backslash frac\backslash ,\; ,\; \backslash ;\backslash ;\backslash ;\; v\_T=\backslash sqrt\backslash ,\; .$
We can see that the diffusion coefficient in the mean free path approximation grows with ''T'' as ''T''^{3/2} and decreases with ''P'' as 1/''P''. If we use for ''P'' the ideal gas law ''P'' = ''RnT'' with the total concentration ''n'', then we can see that for given concentration ''n'' the diffusion coefficient grows with ''T'' as ''T''^{1/2} and for given temperature it decreases with the total concentration as 1/''n''.
For two different gases, A and B, with molecular masses ''m''_{A}, ''m''_{B} and molecular diameters ''d''_{A}, ''d''_{B}, the mean free path estimate of the diffusion coefficient of A in B and B in A is:
: $D\_=\backslash frac\backslash sqrt\backslash sqrt\backslash frac\backslash ,\; ,$

_{i}'' is the ''i''th particle mass),
* density of kinetic energy $$\backslash sum\_i\; \backslash left(\; n\_i\backslash frac\; +\; \backslash int\_c\; \backslash frac\; f\_i(x,c,t)\backslash ,\; dc\; \backslash right).$$
The kinetic temperature ''T'' and pressure ''P'' are defined in 3D space as
:$\backslash frack\_\; T=\backslash frac\; \backslash int\_c\; \backslash frac\; f\_i(x,c,t)\backslash ,\; dc;\; \backslash quad\; P=k\_nT,$
where $n=\backslash sum\_i\; n\_i$ is the total density.
For two gases, the difference between velocities, $C\_1-C\_2$ is given by the expression:
: $C\_1-C\_2=-\backslash fracD\_\backslash left\backslash ,$
where $F\_i$ is the force applied to the molecules of the ''i''th component and $k\_T$ is the thermodiffusion ratio.
The coefficient ''D''_{12} is positive. This is the diffusion coefficient. Four terms in the formula for ''C''_{1}−''C''_{2} describe four main effects in the diffusion of gases:
# $\backslash nabla\; \backslash ,\backslash left(\backslash frac\backslash right)$ describes the flux of the first component from the areas with the high ratio ''n''_{1}/''n'' to the areas with lower values of this ratio (and, analogously the flux of the second component from high ''n''_{2}/''n'' to low ''n''_{2}/''n'' because ''n''_{2}/''n'' = 1 – ''n''_{1}/''n'');
# $\backslash frac\backslash nabla\; P$ 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;
# $\backslash frac(F\_1-F\_2)$ 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.
# $k\_T\; \backslash frac\backslash nabla\; T$ describes thermodiffusion, the diffusion flux caused by the temperature gradient.
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,
* $$D\_=\backslash frac\backslash left[\backslash frac\; \backslash right]^$$ for rigid spheres;
* $$D\_=\backslash frac\; \backslash left[\backslash frac\backslash right]^\; \backslash left(\backslash frac\; \backslash right)^$$ for repulsing force $\backslash kappa\_r^.$
The number $A\_1()$ is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book)
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:
:$V=\backslash frac\; \backslash rho\; \backslash ,\; .$
where $\backslash rho\_i\; =m\_i\; n\_i$ is the mass concentration of the ''i''th species, $\backslash rho=\backslash sum\_i\; \backslash rho\_i$ is the mass density.
By definition, the diffusion velocity of the ''i''th component is $v\_i=C\_i-V$, $\backslash sum\_i\; \backslash rho\_i\; v\_i=0$.
The mass transfer of the ''i''th component is described by the continuity equation
:$\backslash frac+\backslash nabla(\backslash rho\_i\; V)\; +\; \backslash nabla\; (\backslash rho\_i\; v\_i)\; =\; W\_i\; \backslash ,\; ,$
where $W\_i$ is the net mass production rate in chemical reactions, $\backslash sum\_i\; W\_i=\; 0$.
In these equations, the term $\backslash nabla(\backslash rho\_i\; V)$ describes advection of the ''i''th component and the term $\backslash nabla\; (\backslash rho\_i\; v\_i)$ represents diffusion of this component.
In 1948, Wendell H. Furry 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. For the diffusion velocities in multicomponent gases (''N'' components) they used
:$v\_i=-\backslash left(\backslash sum\_^N\; D\_\; \backslash mathbf\_j\; +\; D\_i^\; \backslash ,\; \backslash nabla\; (\backslash ln\; T)\; \backslash right)\backslash ,\; ;$
:$\backslash mathbf\_j=\backslash nabla\; X\_j\; +\; (X\_j-Y\_j)\backslash ,\backslash nabla\; (\backslash ln\; P)\; +\; \backslash mathbf\_j\backslash ,\; ;$
:$\backslash mathbf\_j=\backslash frac\; \backslash left(\; Y\_j\; \backslash sum\_^N\; Y\_k\; (f\_k-f\_j)\; \backslash right)\backslash ,\; .$
Here, $D\_$ is the diffusion coefficient matrix, $D\_i^$ is the thermal diffusion coefficient, $f\_i$ is the body force per unit mass acting on the ''i''th species, $X\_i=P\_i/P$ is the partial pressure fraction of the ''i''th species (and $P\_i$ is the partial pressure), $Y\_i=\backslash rho\_i/\backslash rho$ is the mass fraction of the ''i''th species, and $\backslash sum\_i\; X\_i=\backslash sum\_i\; Y\_i=1.$

concentration
In chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during ...

to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...

or chemical potential
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed ...

. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition
Spinodal decomposition occurs when one thermodynamic phase spontaneously (i.e., without nucleation
Nucleation is the first step in the formation of either a new thermodynamic phase or a new structure via self-assembly
File:Iron oxide nanocube.jpg ...

.
The concept of diffusion is widely used in many fields, including physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

( particle diffusion), chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other .
...

, biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

, sociology
Sociology is a social science
Social science is the branch
The branches and leaves of a tree.
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the scie ...

, economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

, and finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available which could ...

(diffusion of people, ideas, 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 gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

is the change in the value of a quantity, for example, concentration, pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

, or temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

with the change in another variable, usually distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

. A change in concentration over a distance is called a concentration gradient
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...

, a change in pressure over a distance is called a pressure gradientIn atmospheric science, the pressure gradient (typically of air but more generally of any fluid
In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluid ...

, and a change in temperature over a distance is called a temperature gradient
A temperature gradient is a physical quantity
A physical quantity is a physical property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the alge ...

.
The word ''diffusion'' derives from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

word, ''diffundere'', which means "to spread out."
A distinguishing feature of diffusion is that it depends on particle random walk
In , a random walk is a , known as a stochastic or , that describes a path that consists of a succession of steps on some such as the s.
An elementary example of a random walk is the random walk on the integer number line, \mathbb Z, which ...

, and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of advection
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a fl ...

. The term convection
Convection is single or multiphase fluid flow that occurs Spontaneous process, spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When t ...

is used to describe the combination of both transport phenomena
In engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encomp ...

.
If a diffusion process can be described by Fick's laws, it's called a normal diffusion (or Fickian diffusion); Otherwise, it's called an anomalous diffusion
Anomalous diffusion is a diffusion
File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The bar ...

(or non-Fickian diffusion).
When talking about the extent of diffusion, two length scales are used in two different scenarios:
# Brownian motion
File:Brownian motion large.gif, This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random dire ...

of an point source (for example, one single spray of perfume)—the square root of the mean squared displacement
In statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural ...

from this point. In Fickian diffusion, this is $\backslash sqrt$, where $n$ is the dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of this Brownian motion;
# Constant concentration source in one dimension—the diffusion length. In Fickian diffusion, this is $2\backslash sqrt$.
Diffusion vs. bulk flow

"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, due to a concentration gradient, with no net movement of matter. An example of a process where both bulk motion and diffusion occur is human breathing. First, there is a "bulk flow" process. Thelungs
The lungs are the primary organs of the respiratory system
The respiratory system (also respiratory apparatus, ventilatory system) is a biological system consisting of specific organs and structures used for gas exchange in animal
...

are located in the thoracic cavity
250px, The picture displays the Mediastinum on sagittal plane, Thoracic diaphragm">sagittal_plane.html" ;"title="Mediastinum on sagittal plane">Mediastinum on sagittal plane, Thoracic diaphragm at the bottom, the heart (Cor), behind Sternum and ...

, which expands as the first step in external respiration. This expansion leads to an increase in volume of the alveoli
Alveolus (pl. alveoli, adj. alveolar) is a general anatomical term for a concave cavity or pit.
Alveolus may refer to:
In anatomy and zoology in general
* Pulmonary alveolus, an air sac in the lungs
** Alveolar cell or pneumocyte
** Alveolar duct
...

in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air
File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie represents trace gases that together compose about 0.043391% of the atmosphere (0.04402961% at April 2019 concentration ). Number ...

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 capillaries
A capillary is a small blood vessel from 5 to 10 micrometres (μm) in diameter, and having a wall one endothelial cell thick. They are the smallest blood vessels in the body: they convey blood between the arterioles and venules. These microvessel ...

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 carbon dioxide
Carbon dioxide (chemical formula
A chemical formula is a way of presenting information about the chemical proportions of s that constitute a particular or molecule, using symbols, numbers, and sometimes also other symbols, such as pare ...

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 blood
Blood is a body fluid
Body fluids, bodily fluids, or biofluids are liquid
A liquid is a nearly incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) refers t ...

in the body.
Third, there is another "bulk flow" process. The pumping action of the heart
The heart is a cardiac muscle, muscular Organ (biology), organ in most animals, which pumps blood through the blood vessels of the circulatory system. The pumped blood carries oxygen and nutrients to the body, while carrying metabolic waste ...

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 blood vessel
The blood vessels are the components of the circulatory system
The circulatory system, also called the cardiovascular system or the vascular system, is an organ system that permits blood to circulate and transport nutrients (such as amino ...

s by bulk flow down the pressure gradient.
Diffusion in the context of different disciplines

The concept of diffusion is widely used in:physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

( particle diffusion), chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other .
...

, biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

, sociology
Sociology is a social science
Social science is the branch
The branches and leaves of a tree.
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the scie ...

, economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

, and finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available which could ...

(diffusion of people, ideas and of price values). However, in each case the substance or collection undergoing diffusion is "spreading out" from a point or location at which there is a higher concentration of that substance or collection.
There are two ways to introduce the notion of ''diffusion'': either a phenomenological approach starting with Fick's laws of diffusion
Fick's laws of diffusion describe diffusion
File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the rig ...

and their mathematical consequences, or a physical and atomistic one, by considering the ''random walk
In , a random walk is a , known as a stochastic or , that describes a path that consists of a succession of steps on some such as the s.
An elementary example of a random walk is the random walk on the integer number line, \mathbb Z, which ...

of the diffusing particles''.
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 flux
of \mathbf(\mathbf) with the unit normal vector \mathbf(\mathbf) ''(blue arrows)'' at the point \mathbf multiplied by the area dS. The sum of \mathbf\cdot\mathbf dS for each patch on the surface is the flux through the surface
Flux describes ...

is proportional to the negative gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

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 thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

and non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an extr ...

.
From the ''atomistic point of view'', diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperature
Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present ...

, the moving molecules are self-propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown, 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 Brownian motion
File:Brownian motion large.gif, This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random dire ...

and the atomistic backgrounds of diffusion were developed by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity
The theo ...

.
The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals.
In chemistry
Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other .
...

and materials science
The interdisciplinary
Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic discipline
An academic discipline or academic field is a subdivision of knowledge that is Education, taught and resea ...

, diffusion refers to the movement of fluid molecules in porous solids. Molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperature
Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present ...

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. Knudsen diffusion
In physics, Knudsen diffusion, named after Martin Knudsen, is a means of diffusion
File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solution, solute molecules on the left sid ...

, which 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 diffusivity.
Biologist
A biologist is a professional who has specialized knowledge in the field of biology, understanding the underlying mechanisms that govern the functioning of biological systems within fields such as health, technology and the Biophysical environm ...

s 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 probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

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.
History of diffusion in physics

In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example,Pliny the Elder #REDIRECT Pliny the Elder#REDIRECT Pliny the Elder
Gaius Plinius Secundus (AD 23/2479), called Pliny the Elder (), was a Roman author, a naturalist
Natural history is a domain of inquiry involving organisms, including animals, fungus, fungi, ...

had previously described the cementation process
The cementation process is an obsolete technology
Technology ("science of craft", from Ancient Greek, Greek , ''techne'', "art, skill, cunning of hand"; and , ''wikt:-logia, -logia'') is the sum of Art techniques and materials, techniques, s ...

, which produces steel from the element iron (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of stained glass
File:Oostende Sint-Petrus-en-Pauluskerk Rosette.jpg, 300px, Outside-view of a stained glass of the Sint-Petrus-en-Pauluskerk from Ostend (Belgium), built between 1899 and 1908
The term stained glass refers to coloured glass as a material and to ...

or earthenware
Earthenware is glazed or unglazed nonvitreous pottery
Pottery is the process and the products of forming vessels and other objects with clay and other ceramic materials, which are fired at high temperatures to give them a hard, durable form ...

and Chinese ceramics
Chinese ceramics show a continuous development since pre-dynastic times and are one of the most significant forms of Chinese art and ceramic
A ceramic is any of the various hard, brittle, heat-resistant and corrosion-resistant materials made ...

.
In modern science, the first systematic experimental study of diffusion was performed by Thomas Graham. He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833:
"...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."The measurements of Graham contributed to

James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest.
In classica ...

deriving, in 1867, the coefficient of diffusion for COAdolf Fick
Adolf Eugen Fick (3 September 1829 – 21 August 1901) was a German-born physician
A physician (American English), medical practitioner (English in the Commonwealth of Nations, Commonwealth English), medical doctor, or simply doctor, is ...

, the 26-year-old anatomy demonstrator from Zürich, proposed his law of diffusion. 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 (1822) and Ohm's law
Ohm's law states that the electric current, current through a Electrical conductor, conductor between two points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of proporti ...

for electric current (1827).
Robert Boyle
Robert Boyle (; 25 January 1627 – 31 December 1691) was an Anglo-Irish natural philosopher, chemist, physicist, and inventor. Boyle is largely regarded today as the first modern chemist, and therefore one of the founders of modern che ...

demonstrated diffusion in solids in the 17th century by penetration of zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century. William Chandler Roberts-Austen
Sir William Chandler Roberts-Austen (3 March 1843, Kennington
Kennington is a district in south London, England. It is mainly within the London Borough of Lambeth, running along the boundary with the London Borough of Southwark, a boundary wh ...

, 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. :
"... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals."In 1858,

Rudolf Clausius
Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist
A physicist is a scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of sci ...

introduced the concept of the mean free path
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

. In the same year, James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist
A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest.
In classica ...

developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion
File:Brownian motion large.gif, This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random dire ...

was developed by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity
The theo ...

, Marian Smoluchowski
Marian Smoluchowski (; 28 May 1872 – 5 September 1917) was a Polish physicist who worked in the Polish territories of the Austro-Hungarian Empire. He was a pioneer of statistical physics, and an avid mountaineer.
Life
Born into an upper ...

and Jean-Baptiste Perrin
Jean Baptiste Perrin (30 September 1870 – 17 April 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids, verified Albert Einstein’s explanation of this phenomenon and thereby confirme ...

. Ludwig Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austria
Austria, officially the Republic of Austria, is a landlocked country in the southern part of Central Europe, located on the Eastern Alps. It is compo ...

, in the development of the atomistic backgrounds of the macroscopic transport processes, introduced the Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system
A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, w ...

, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.S. Chapman, T. G. Cowling (1970) ''The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', Cambridge University Press (3rd edition), .
In 1920–1921, George de Hevesy
George Charles de Hevesy ( hu, Hevesy György Károly, german: Georg Karl von Hevesy; 1 August 1885 – 5 July 1966) was a HungarianHungarian may refer to:
* Hungary, a country in Central Europe
* Kingdom of Hungary, state of Hungary, existin ...

measured self-diffusionAccording to IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations that represents chemists in individual countries. It is a member of the International Science Cou ...

using radioisotope
A radionuclide (radioactive nuclide, radioisotope or radioactive isotope) is a nuclide
A nuclide (or nucleide, from nucleus, also known as nuclear species) is a class of atoms characterized by their number of proton
A proton is a subatomic par ...

s. He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead.
Yakov Frenkel
__NOTOC__
Yakov Il'ich Frenkel (russian: Яков Ильич Френкель) (10 February 1894 – 23 January 1952) was a Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a socialist state
A ...

(sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and interstitial
An interstitial space or interstice is a space between structures or objects.
In particular, interstitial may refer to:
Biology
* Interstitial cell tumor
* Interstitial cell, any cell that lies between other cells
* Interstitial collagenase, e ...

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, Carl Wagner
Carl Wilhelm Wagner (May 25, 1901 – December 10, 1977) was a German Physical chemistry, Physical chemist. He is best known for his pioneering work on Solid-state chemistry, where his work on oxidation rate theory, counter diffusion of ions a ...

and Walter H. Schottky
Walter Hans Schottky (23 July 1886 – 4 March 1976) was a German physicist who played a major early role in developing the theory of electron and ion emission phenomena, invented the screen-grid vacuum tube
A vacuum tube, an electron t ...

developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.
Henry Eyring, with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion. The analogy between reaction kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with thermodynamics, which deals with the direction in which a pro ...

and diffusion leads to various nonlinear versions of Fick's law.
Basic models of diffusion

Diffusion flux

Each model of diffusion expresses the diffusion flux with the use of concentrations, densities and their derivatives. Flux is a vector $\backslash mathbf$ representing the quantity and direction of transfer. Given a smallarea
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

$\backslash Delta\; S$ with normal $\backslash boldsymbol$, the transfer of a physical quantity
A physical quantity is a physical property of a material or system that can be Quantification (science), quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ''numerical value'' ...

$N$ through the area $\backslash Delta\; S$ per time $\backslash Delta\; t$ is
:$\backslash Delta\; N\; =\; (\backslash mathbf,\backslash boldsymbol)\; \backslash ,\backslash Delta\; S\; \backslash ,\backslash Delta\; t\; +o(\backslash Delta\; S\; \backslash ,\backslash Delta\; t)\backslash ,\; ,$
where $(\backslash mathbf,\backslash boldsymbol)$ is the inner product
In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...

and $o(\backslash cdots)$ is the little-o notation
Big O notation is a mathematical notation that describes the limiting behavior of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be ...

. If we use the notation of vector areaIn 3-dimensional geometry, for a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area:
:\mathbf = \mathbfS
For an orientable
is non-orientable
In mathematics, orientability ...

$\backslash Delta\; \backslash mathbf=\backslash boldsymbol\; \backslash ,\; \backslash Delta\; S$ then
:$\backslash Delta\; N\; =\; (\backslash mathbf,\; \backslash Delta\; \backslash mathbf)\; \backslash ,\; \backslash Delta\; t\; +o(\backslash Delta\; \backslash mathbf\; \backslash ,\backslash Delta\; t)\backslash ,\; .$
The dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of the diffusion flux is lux
The lux (symbol: lx) is the SI derived unit
SI derived units are units of measurement derived from the
seven SI base unit, base units specified by the International System of Units (SI). They are either dimensionless quantity, dimensionless or ...

nbsp;= uantity(ime
Ime is a village in Norway.
IME or ime may refer to:
Organizations
* Institution of Mechanical Engineers
The Institution of Mechanical Engineers (IMechE) is an independent professional association and learned society headquartered in London, ...

rea
REA or Rea may refer to:
Places
* Rea, Lombardy:''See also Rhea (disambiguation) or Rea (disambiguation), REA
Rea is a ''comune'' (municipality) in the Province of Pavia in the Italy, Italian region Lombardy, located about 40 km south of Mi ...

. The diffusing physical quantity $N$ may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity
Extensive may refer to:
* Extensive property
* Extensive function
* Extensional
See also
* Extension (disambiguation)
{{Dab ...

. For its density, $n$, the diffusion equation has the form
:$\backslash frac=\; -\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; +W\; \backslash ,\; ,$
where $W$ 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 $(\backslash mathbf(x),\backslash boldsymbol(x))=0$ on the boundary, where $\backslash boldsymbol$ is the normal to the boundary at point $x$.
Fick's law and equations

Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient: :$\backslash mathbf=-D\; \backslash ,\backslash nabla\; n\; \backslash \; ,\; \backslash ;\backslash ;\; J\_i=-D\; \backslash frac\; \backslash \; .$ The corresponding diffusion equation (Fick's second law) is :$\backslash frac=\backslash nabla\backslash cdot(\; D\; \backslash ,\backslash nabla\; n(x,t))=D\; \backslash ,\; \backslash Delta\; n(x,t)\backslash \; ,$ where $\backslash Delta$ is theLaplace operator
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

,
:$\backslash Delta\; n(x,t)\; =\; \backslash sum\_i\; \backslash frac\; \backslash \; .$
Onsager's equations for multicomponent diffusion and thermodiffusion

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, $-\backslash nabla\; n$. In 1931,Lars Onsager
Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native ...

included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For
multi-component transport,
:$\backslash mathbf\_i=\backslash sum\_j\; L\_\; X\_j\; \backslash ,\; ,$
where $\backslash mathbf\_i$ is the flux of the ''i''th physical quantity (component) and $X\_j$ is the ''j''th thermodynamic force.
The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy
Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamic ...

density $s$ (he used the term "force" in quotation marks or "driving force"):
:$X\_i=\; \backslash nabla\; \backslash frac\; \backslash ,\; ,$
where $n\_i$ are the "thermodynamic coordinates".
For the heat and mass transfer one can take $n\_0=u$ (the density of internal energy) and $n\_i$ is the concentration of the $i$th component. The corresponding driving forces are the space vectors
: $X\_0=\; \backslash nabla\; \backslash frac\backslash \; ,\; \backslash ;\backslash ;\backslash ;\; X\_i=\; -\; \backslash nabla\; \backslash frac\; \backslash ;\; (i\; >0)\; ,$ because $\backslash mathrms\; =\; \backslash frac\; \backslash ,\backslash mathrmu-\backslash sum\_\; \backslash frac\; \backslash ,\; n\_i$
where ''T'' is the absolute temperature and $\backslash mu\_i$ is the chemical potential of the $i$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:
:$X\_i=\; \backslash sum\_\; \backslash left.\backslash frac\backslash \_\; \backslash nabla\; n\_k\; \backslash \; ,$
where the derivatives of $s$ are calculated at equilibrium $n^*$.
The matrix of the ''kinetic coefficients'' $L\_$ should be symmetric (Onsager reciprocal relations
In thermodynamics, the Onsager reciprocal relations express the equality of certain ratios between flux, flows and forces in thermodynamic systems out of equilibrium (thermo), equilibrium, but where a notion of local thermodynamic equilibrium, loc ...

) and positive definiteIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

( for the entropy growth).
The transport equations are
:$\backslash frac=\; -\; \backslash operatorname\; \backslash mathbf\_i\; =-\; \backslash sum\_\; L\_\backslash operatorname\; X\_j\; =\; \backslash sum\_\; \backslash left;\; href="/html/ALL/s/\backslash sum\_\_L\_\_\backslash left.\backslash frac\backslash right.html"\; ;"title="\backslash sum\_\; L\_\; \backslash left.\backslash frac\backslash right">\_\backslash right$
Here, all the indexes are related to the internal energy (0) and various components. The expression in the square brackets is the matrix $D\_$ of the diffusion (''i'',''k'' > 0), thermodiffusion (''i'' > 0, ''k'' = 0 or ''k'' > 0, ''i'' = 0) and thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...

() coefficients.
Under ''T'' = constant. The relevant thermodynamic potential is the free energy (or the free entropy
A thermodynamic
Thermodynamics is a branch of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, a ...

). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, $-(1/T)\backslash ,\backslash nabla\backslash mu\_j$, and the matrix of diffusion coefficients is
:$D\_=\backslash frac\backslash sum\_\; L\_\; \backslash left.\backslash frac\; \backslash \_$
(''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 $\backslash sum\_j\; L\_X\_j$ can be measured. For example, in the original work of Onsager the thermodynamic forces include additional multiplier ''T'', whereas in the Course of Theoretical Physics
The ''Course of Theoretical Physics'' is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s.
It is said that Landa ...

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.
Nondiagonal diffusion must be nonlinear

The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form :$\backslash frac\; =\; \backslash sum\_j\; D\_\; \backslash ,\; \backslash Delta\; c\_j.$ 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, $D\_\; \backslash neq\; 0$, and consider the state with $c\_2\; =\; \backslash cdots\; =\; c\_n\; =\; 0$. At this state, $\backslash partial\; c\_2\; /\; \backslash partial\; t\; =\; D\_\; \backslash ,\; \backslash Delta\; c\_1$. If $D\_\; \backslash ,\; \backslash Delta\; c\_1(x)\; <\; 0$ at some points, then $c\_2(x)$ 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.Einstein's mobility and Teorell formula

TheEinstein relation (kinetic theory)
In physics (specifically, the kinetic theory of gases) the Einstein relation is a previously unexpected connection revealed independently by William Sutherland (physicist), William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchow ...

connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocityIn physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...

to an applied force
In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

)
:$D\; =\; \backslash frac,$
where ''D'' is the diffusion constant, ''μ'' is the "mobility", ''k''Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...

, ''T'' is the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...

, and ''q'' is the elementary charge
The elementary charge, usually denoted by or sometimes e is the electric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positiv ...

, 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 $\backslash mathfrak$.
The mobility-based approach was further applied by T. Teorell. In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula:
:the flux is equal to mobility × concentration × force per gram-ion.
This is the so-called ''Teorell formula''. The term "gram-ion" ("gram-particle") is used for a quantity of a substance that contains Avogadro's number
The Avogadro constant (''N''A or ''L'') is the proportionality factor that relates the number of constituent particles (usually molecule
File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which ...

of ions (particles). The common modern term is mole
Mole (or Molé) may refer to:
Animals
* Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America
* Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpidae ...

.
The force under isothermal conditions consists of two parts:
# Diffusion force caused by concentration gradient: $-RT\; \backslash frac\; \backslash ,\; \backslash nabla\; n\; =\; -RT\; \backslash ,\; \backslash nabla\; (\backslash ln(n/n^\backslash text))$.
# Electrostatic force caused by electric potential gradient: $q\; \backslash ,\; \backslash nabla\; \backslash varphi$.
Here ''R'' is the gas constant, ''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
:$\backslash mathbf\; =\; \backslash mathfrak\; \backslash exp\backslash left(\backslash frac\backslash right)(-\backslash nabla\; \backslash mu\; +\; (\backslash text)),$
where ''μ'' is the chemical potential
In thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed ...

, ''μ''activity
Activity may refer to:
* Action (philosophy), in general
* Human activity: human behavior, in sociology behavior may refer to all basic human actions, economics may study human economic activities and along with cybernetics and psychology may stud ...

. 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
:$\backslash mathbf\; =\; \backslash mathfrak\; a\; (-\backslash nabla\; \backslash mu\; +\; (\backslash text)).$
The standard derivation of the activity includes a normalization factor and for small concentrations $a\; =\; n/n^\backslash ominus\; +\; o(n/n^\backslash ominus)$, where $n^\backslash ominus$ is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity $n/n^\backslash ominus$:
:$\backslash frac\; =\; \backslash nabla\; \backslash cdot;\; href="/html/ALL/s/mathfrak\_a\_(\backslash nabla\_\backslash mu\_-\_(\backslash text)).html"\; ;"title="mathfrak\; a\; (\backslash nabla\; \backslash mu\; -\; (\backslash text))">mathfrak\; a\; (\backslash nabla\; \backslash mu\; -\; (\backslash text))$
Fluctuation-dissipation theorem

Fluctuation-dissipation theorem based on the Langevin equation is developed to extend the Einstein model to the ballistic time scale. According to Langevin, the equation is based on Newton's second law of motion as :$m\; \backslash frac\; =\; -\backslash frac\backslash frac\; +\; F(t)$ where * ''x'' is the position. * ''μ'' is the mobility of the particle in the fluid or gas, which can be calculated using theEinstein relation (kinetic theory)
In physics (specifically, the kinetic theory of gases) the Einstein relation is a previously unexpected connection revealed independently by William Sutherland (physicist), William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchow ...

.
* ''m'' is the mass of the particle.
* ''F'' is the random force applied to the particle.
* ''t'' is time.
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,
:$D(t)\; =\; \backslash mu\; \backslash ,\; k\_\; T(1-e^)$
where
* ''k''Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...

;
* ''T'' is the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...

.
* ''μ'' is the mobility of the particle in the fluid or gas, which can be calculated using the Einstein relation (kinetic theory)
In physics (specifically, the kinetic theory of gases) the Einstein relation is a previously unexpected connection revealed independently by William Sutherland (physicist), William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchow ...

.
* ''m'' is the mass of the particle.
* ''t'' is time.
Teorell formula for multicomponent diffusion

The Teorell formula with combination of Onsager's definition of the diffusion force gives :$\backslash mathbf\_i\; =\; \backslash mathfrak\; a\_i\; \backslash sum\_j\; L\_\; X\_j,$ where $\backslash mathfrak$ is the mobility of the ''i''th component, $a\_i$ is its activity, $L\_$ is the matrix of the coefficients, $X\_j$ is the thermodynamic diffusion force, $X\_j=\; -\backslash nabla\; \backslash frac$. For the isothermal perfect systems, $X\_j\; =\; -\; R\; \backslash frac$. Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion: :$\backslash frac\; =\; \backslash sum\_j\; \backslash nabla\; \backslash cdot\; \backslash left(D\_\backslash frac\; \backslash nabla\; n\_j\backslash right),$ where $D\_$ is the matrix of coefficients. The Diffusion#The theory of diffusion in gases based on Boltzmann's equation, Chapman–Enskog formulas for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.Jumps on the surface and in solids

Surface diffusion, Diffusion of reagents on the surface of a catalyst 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 $A\_1,A\_2,\backslash ldots,\; A\_m$ on the surface. Their surface concentrations are $c\_1,c\_2,\backslash ldots,\; c\_m.$ 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 $z=c\_0$. The sum of all $c\_i$ (including free places) is constant, the density of adsorption places ''b''. The jump model gives for the diffusion flux of $A\_i$ (''i'' = 1, ..., ''n''): :$\backslash mathbf\_i=-D\_i[z\; \backslash ,\; \backslash nabla\; c\_i\; -\; c\_i\; \backslash nabla\; z]\backslash ,\; .$ The corresponding diffusion equation is: :$\backslash frac=-\; \backslash operatorname\backslash mathbf\_i=D\_i[z\; \backslash ,\; \backslash Delta\; c\_i\; -\; c\_i\; \backslash ,\; \backslash Delta\; z]\; \backslash ,\; .$ Due to the conservation law, $z=b-\backslash sum\_^n\; c\_i\; \backslash ,\; ,$ and we have the system of ''m'' diffusion equations. For one component we get Fick's law and linear equations because $(b-c)\; \backslash ,\backslash nabla\; c-\; c\backslash ,\backslash nabla(b-c)\; =\; b\backslash ,\backslash nabla\; c$. 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 :$\backslash mathbf\_i=-\backslash sum\_j\; D\_[c\_j\; \backslash ,\backslash nabla\; c\_i\; -\; c\_i\; \backslash ,\backslash nabla\; c\_j]$ :$\backslash frac=\backslash sum\_j\; D\_[c\_j\; \backslash ,\; \backslash Delta\; c\_i\; -\; c\_i\; \backslash ,\backslash Delta\; c\_j]$ where $D\_\; =\; D\_\; \backslash geq\; 0$ is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration $c\_0$. Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.Diffusion in porous media

For diffusion in porous media the basic equations are (if Φ is constant): :$\backslash mathbf=-\; \backslash phi\; D\; \backslash ,\backslash nabla\; n^m$ :$\backslash frac\; =\; D\; \backslash ,\; \backslash Delta\; n^m\; \backslash ,\; ,$ 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. 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 Darcy's law: the volumetric flux of a gas in the porous media is :$q=-\backslash frac\backslash ,\backslash nabla\; p$ where ''k'' is the Permeation, permeability of the medium, ''μ'' is the viscosity and ''p'' is the pressure. The advective molar flux is given as ''J'' = ''nq'' and for $p\; \backslash sim\; n^\backslash gamma$ 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: $\backslash upsilon=\; q/\backslash phi$ Combining the advective molar flux with the diffusive flux gives the advection dispersion equation $\backslash frac\; =\; D\; \backslash ,\; \backslash Delta\; n^m\; \backslash \; -\; \backslash nu\backslash cdot\; \backslash nabla\; n^m,$ For underground water infiltration, the Boussinesq approximation (buoyancy), Boussinesq approximation gives the same equation with ''m'' = 2. For plasma with the high level of radiation, the Yakov Borisovich Zel'dovich, Zeldovich–Raizer equation gives ''m'' > 4 for the heat transfer.Diffusion in physics

Diffusion coefficient in kinetic theory of gases

The diffusion coefficient $D$ is the coefficient in the Fick's laws of diffusion, Fick's first law $J=-\; D\; \backslash ,\; \backslash partial\; n/\backslash partial\; x$, where ''J'' is the diffusion flux (amount of substance) 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'' (self-diffusionAccording to IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations that represents chemists in individual countries. It is a member of the International Science Cou ...

). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient
:$D=\backslash frac\; \backslash ell\; v\_T\; =\; \backslash frac\backslash sqrt\; \backslash frac\backslash ,\; ,$
where ''k''temperature
Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy
Thermal radiation in visible light can be seen on this hot metalwork.
Thermal energy refers to several distinct physical concept ...

, ''P'' is the pressure
Pressure (symbol: ''p'' or ''P'') is the force
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

, $\backslash ell$ is the mean free path
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

, and ''vThe theory of diffusion in gases based on Boltzmann's equation

In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, $f\_i(x,c,t)$, where ''t'' is the time moment, ''x'' is position and ''c'' is velocity of molecule of the ''i''th component of the mixture. Each component has its mean velocity $C\_i(x,t)\; =\; \backslash frac\; \backslash int\_c\; c\; f(x,c,t)\; \backslash ,\; dc$. If the velocities $C\_i(x,t)$ do not coincide then there exists ''diffusion''. In the Chapman–Enskog theory, Chapman–Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities: * individual concentrations of particles, $n\_i(x,t)=\backslash int\_c\; f\_i(x,c,t)\backslash ,\; dc$ (particles per volume), * density of momentum $\backslash sum\_i\; m\_i\; n\_i\; C\_i(x,t)$ (''mDiffusion of electrons in solids

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 diffusion current. Diffusion current can also be described by Fick's laws of diffusion, Fick's first law :$J=-\; D\; \backslash ,\; \backslash partial\; n/\backslash partial\; x\backslash ,\; ,$ where ''J'' is the diffusion current density (amount of substance) per unit area per unit time, ''n'' (for ideal mixtures) is the electron density, ''x'' is the position [length].Diffusion in geophysics

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. 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.Dialysis

Dialysis works on the principles of the diffusion of solutes and ultrafiltration of fluid across a semi-permeable membrane. 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.'' Mosby’s Dictionary of Medicine, Nursing, & Health Professions''. 7th ed. St. Louis, MO; Mosby: 2006 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 glomerulus.Random walk (random motion)

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.)Separation of diffusion from convection in gases

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 transport phenomena, transport phenomenon,convection
Convection is single or multiphase fluid flow that occurs Spontaneous process, spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When t ...

. To separate diffusion in these cases, special efforts are needed.
Therefore, some often cited examples of diffusion are ''wrong'': If cologne is sprayed in one place, it can soon be smelled in the entire room, but a simple calculation shows that this can't be due to diffusion. Convective motion persists in the room because of the temperature [inhomogeneity]. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).
In contrast, heat conduction 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.
Other types of diffusion

* Anisotropic diffusion, also known as the Perona–Malik equation, enhances high gradients * Anomalous diffusion, in porous medium * Atomic diffusion, in solids * Bohm diffusion, spread of plasma across magnetic fields * Eddy diffusion, in coarse-grained description of turbulent flow * Effusion of a gas through small holes * Electronics, Electronic diffusion, resulting in an current (electricity), electric current called the diffusion current * Facilitated diffusion, present in some organisms * Gaseous diffusion, used for isotope separation * Heat equation, diffusion of thermal energy * Itō diffusion, mathematisation of Brownian motion, continuous stochastic process. *Knudsen diffusion
In physics, Knudsen diffusion, named after Martin Knudsen, is a means of diffusion
File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solution, solute molecules on the left sid ...

of gas in long pores with frequent wall collisions
* Lévy flight
* Molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperature
Temperature is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy, present ...

, diffusion of molecules from more dense to less dense areas
* Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
* Photon diffusion
* Plasma diffusion
* Random walk, model for diffusion
* Reverse diffusion, against the concentration gradient, in phase separation
* Rotational diffusion, random reorientation of molecules
* Surface diffusion, diffusion of adparticles on a surface
* Taxis is an animal's directional movement activity in response to a stimulus
** Kinesis (biology), Kinesis is an animal's non-directional movement activity in response to a stimulus
* Trans-cultural diffusion, diffusion of cultural traits across geographical area
* Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid
See also

* * * * * * * *References

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