11108:. 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. 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
11140:
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.)
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11080:
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.
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618:"...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."
69:
11148:
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
6227:
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,
416:
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
11139:
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
10987:
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
9107:
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
554:
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
45:
in a glass of water. At first, the particles are all near one top corner of the glass. If the particles randomly move around ("diffuse") in the water, they eventually become distributed randomly and uniformly from an area of high concentration to an area of low, and organized (diffusion continues,
11079:
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
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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
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4276:
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
565:
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,
8123:
7905:
8625:{\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\},}
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2677:
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.
420:
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
197:(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.
2831:
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As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to
6780:
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134:. Diffusion is a stochastic process 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
6971:
641:. 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
393:"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
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58:
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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,
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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
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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.
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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
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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.
1974:
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522:, 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
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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
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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.
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Stauffer, Philip H.; Vrugt, Jasper A.; Turin, H. Jake; Gable, Carl W.; Soll, Wendy E. (2009). "Untangling
Diffusion from Advection in Unsaturated Porous Media: Experimental Data, Modeling, and Parameter Uncertainty".
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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.
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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,
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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.
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88:
With an enormous number of solute molecules, all randomness is gone: The solute appears to move smoothly and deterministically from high-concentration areas to low-concentration areas. There is no microscopic
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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.
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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.
429:
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
5744:
1417:
7263:{\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}}}\,,}
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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
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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
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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
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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:
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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
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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".
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3211:{\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\,\Delta n_{k}\ .}
56:
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There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations
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is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration
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745:
developed
Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.
322:
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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
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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,
4649:
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1495:
796:
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9685:
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
8212:
518:, 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
7488:
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2015:
660:, 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. :
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The apparent random motion of atoms, ions or molecules explained. Substances appear to move randomly due to collisions with other substances. From the iBook
8964:
4480:. 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
76:
molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.
11681:
J.F. Kincaid; H. Eyring; A.E. Stearn (1941). "The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State".
8826:
3604:
9587:{\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}
9178:
11491:
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566:
occasionally oxygen molecules move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the
1271:
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because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a
8955:
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
8956:
8118:{\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,}
2024:
included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport,
1752:
7900:{\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).}
545:, diffusion also refers to the movement of fluid molecules in porous solids. Different types of diffusion are distinguished in porous solids.
351:
2232:
4384:{\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}
6527:
5871:
729:(sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and
6143:
531:
1088:
11661:
The
Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases
2826:{\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}\ ,}
250:, and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of
7345:
4412:
664:"... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals."
3780:
9706:
11523:"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen"
1355:
is intensity of any local source of this quantity (for example, the rate of a chemical reaction). For the diffusion equation, the
6775:{\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}}}\,,}
5672:
3514:{\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^{*}}}
2202:
The thermodynamic forces for the transport processes were introduced by
Onsager as the space gradients of the derivatives of the
1362:
489:
diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion
4651:
is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity
4969:
8723:
3598:
The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form
2030:
12106:
11835:
11581:
6966:{\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}
2196:
708:, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.
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For the linear
Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium:
11599:, Thomas Graham Symposium, ed. J.N. Sherwood, A.V. Chadwick, W.M.Muir, F.L. Swinton, Gordon and Breach, London, 1971.
11358:
3581:
can be measured. For example, in the original work of
Onsager the thermodynamic forces include additional multiplier
130:. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in
11004:
6245:
5132:
Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.
3236:
are related to the internal energy (0) and various components. The expression in the square brackets is the matrix
680:
developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and
10902:
3848:
9113:
6566:
5604:
5117:
657:
583:
In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example,
5376:
5209:
2616:{\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}
408:, which expands as the first step in external respiration. This expansion leads to an increase in volume of the
118:
is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher
11997:
11859:
11668:
7604:
7501:
approximation, all the distribution functions are expressed through the densities of the conserved quantities:
6506:
5749:
If all particles can exchange their positions with their closest neighbours then a simple generalization gives
5150:
504:
412:
in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the
84:
With more molecules, there is a statistical trend that the solute fills the container more and more uniformly.
11364:
10225:
10096:{\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}
3586:
3322:
9597:
595:(Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of
11717:
11626:
10850:
9925:
3728:
3319:). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials,
2925:
2917:
1985:
9829:
9775:
8777:
to the areas with lower values of this ratio (and, analogously the flux of the second component from high
1422:
827:
10996:
10659:{\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}
10182:
6561:
6001:
1470:
1207:{\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}
999:{\displaystyle \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}
638:
474:
266:
10416:{\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}
4225:
6641:). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient
6372:
6326:
4021:
31:
9700:
is the mass average velocity. It is defined through the momentum density and the mass concentrations:
5116:
are also explained. Moreover, unlike the
Einstein approach, a velocity can be defined, leading to the
4280:
The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is
3534:
2102:
11442:
11374:
7498:
4553:
The standard derivation of the activity includes a normalization factor and for small concentrations
4065:
394:
288:
9678:
is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book)
4654:
294:
11332:
10771:
9879:
9643:
6405:
Combining the advective molar flux with the diffusive flux gives the advection dispersion equation
6080:
Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.
5591:{\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}\,.}
2921:
357:
276:
When talking about the extent of diffusion, two length scales are used in two different scenarios:
10136:
8131:
7281:
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3695:
2515:{\displaystyle X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}
1478:
779:
11105:
10702:
8177:
4109:. The term "gram-ion" ("gram-particle") is used for a quantity of a substance that contains the
752:
611:
42:
17:
7451:
1052:
530:. The concept of diffusion is typically applied to any subject matter involving random walks in
10532:{\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}
4543:{\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}
1994:
748:
131:
11851:
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10988:
ends), forming a gradient of electron density. This process generates current, referred to as
5120:, connecting the competition between friction and random forces in defining the temperature.
3904:
2633:
895:
872:
804:
454:
11827:
11369:
11220:
11211:
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terms. This effect should not be confused with barodiffusion caused by the pressure gradient.
7278:
In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function,
5271:
2335:
1841:
30:
This article is about the generic concept of the time-dependent process. For other uses, see
11874:"Studies on the "Diffusion Effect" upon Ionic Distribution. Some Theoretical Considerations"
10672:
3239:
2889:
1500:
614:. He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833:
12115:
12078:
12020:
11992:
J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press,
11949:
11885:
11787:
11612:, ed. H.Mehrer, Chr. Herzig, N.A. Stolwijk, H. Bracht, Scitec Publications, Vol.1, pp. 1–9.
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1969:{\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}
1716:{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D(x)\,\nabla n(x,t))\,.}
1218:
693:
492:
225:
6485:{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}
3950:
connects the diffusion coefficient and the mobility (the ratio of the particle's terminal
723:. He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead.
8:
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We can see that the diffusion coefficient in the mean free path approximation grows with
6615:
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701:
689:
677:
623:
610:
In modern science, the first systematic experimental study of diffusion was performed by
588:
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and their mathematical consequences, or a physical and atomistic one, by considering the
270:
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In porous media, the average linear velocity (ν), is related to the volumetric flux as:
514:, diffusion is considered as a result of the random walk of the diffusing particles. In
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11753:
11733:
11392:
11383: – Physical or chemical process by which one substance becomes attached to another
11264:
11230:
6793:
6543:
5092:
is the mobility of the particle in the fluid or gas, which can be calculated using the
5073:
4937:
is the mobility of the particle in the fluid or gas, which can be calculated using the
4399:
4036:
3315: = constant. The relevant thermodynamic potential is the free energy (or the
3309:
2839:
2660:
2395:
2209:
2178:
2131:
1729:
1612:
1444:
1338:
1248:
1242:
1224:
852:
742:
705:
642:
409:
327:
143:
139:
127:
11458:
5669:
diffusion equations. For one component we get Fick's law and linear equations because
12202:
12143:
12131:
12036:
11993:
11975:
11913:
11855:
11831:
11664:
11577:
11470:
11462:
11428:
11324:
11312:
11290:
11247:
11226:
11215:
10989:
10982:
9100:{\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}
5136:
4810:
4265:
is the concentration, the equilibrium concentration is marked by a superscript "eq",
4110:
4052:
1079:
847:
756:
551:
542:
459:
269:, it is called a normal diffusion (or Fickian diffusion); Otherwise, it is called an
221:
123:
12048:
11757:
7058:, the mean free path estimate of the diffusion coefficient of A in B and B in A is:
72:
Diffusion from a microscopic and b macroscopic point of view. Initially, there are
12123:
12086:
12028:
11965:
11957:
11903:
11893:
11795:
11743:
11690:
11569:
11542:
11454:
11279:
11274:
11183:
10272:
8948:{\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}
3681:{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}
1859:
1608:
1241:
may be the number of particles, mass, energy, electric charge, or any other scalar
712:
697:
604:
556:
284:
11241:
5128:
4809:
The Einstein model neglects the inertia of the diffusing partial. The alternative
776:
with the use of concentrations, densities and their derivatives. Flux is a vector
27:
Transport of dissolved species from the highest to the lowest concentration region
11819:
11495:
11398:
11164:
11101:
9154:
7273:
6236:
4914:{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}
2332:
are the "thermodynamic coordinates". For the heat and mass transfer one can take
685:
681:
669:
584:
527:
523:
405:
280:
147:
5112:
At long time scales, Einstein's result is recovered, but short time scales, the
1325:{\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}
470:
11878:
Proceedings of the National Academy of Sciences of the United States of America
11395: – Mathematical theory on behavior of connected clusters in a random graph
11302:
11195:
11189:
11088:
6833:
6638:
6232:
3951:
716:
673:
500:
426:
287:
point source (for example, one single spray of perfume)—the square root of the
99:
12063:
11253:
8703:
is positive. This is the diffusion coefficient. Four terms in the formula for
7023:
and for given temperature it decreases with the total concentration as 1/
4794:{\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot .}
4212:{\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}
12221:
12135:
12040:
11852:
Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology
11547:
11522:
11466:
11235:
8766:
describes the flux of the first component from the areas with the high ratio
6997:
6231:
For diffusion of gases in porous media this equation is the formalization of
1046:
726:
596:
119:
12159:"Watershed Hydrology, Geomorphology, and Ecohydraulics :: TFD Modeling"
11748:
11721:
1221:
of the diffusion flux is = /(·). The diffusing physical quantity
11979:
11935:
11917:
11800:
11775:
11268:
11093:
3316:
2021:
720:
652:
646:
443:
190:
11898:
11474:
11429:
Flow equations and frames of reference for isothermal diffusion in liquids
11327:
is an animal's non-directional movement activity in response to a stimulus
11096:, where blood is red, dialysing fluid is blue, and the membrane is yellow.
6088:
For diffusion in porous media the basic equations are (if Φ is constant):
12032:
11716:
11361: – Combination of the diffusion and convection (advection) equations
11284:
11207:
6801:
4114:
1828:{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=D\,\Delta n(x,t)\ ,}
1083:
738:
600:
567:
479:
422:
247:
209:
122:
to a region of lower concentration. Diffusion is driven by a gradient in
107:
11694:
11389: – Migration of molecules to a region of lower solute concentration
11377: – Gaseous diffusion through body tissue at constant total pressure
11961:
11815:
11489:
One and a half century of diffusion: Fick, Einstein, before and beyond.
11321:
is an animal's directional movement activity in response to a stimulus
11157:
6532:
6528:
Kinetic theory of gases § Diffusion coefficient and diffusion flux
6293:
6228:
the porosity terms cancel out and the second equation above is formed.
2295:{\displaystyle X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{i}}}\,,}
634:
437:
Third, there is another "bulk flow" process. The pumping action of the
255:
186:
135:
12090:
5328:(including free places) is constant, the density of adsorption places
204:
is the change in the value of a quantity; for example, concentration,
68:
6301:
2412:
th component. The corresponding driving forces are the space vectors
1497:, is proportional to the negative gradient of spatial concentration,
700:, in the development of the atomistic backgrounds of the macroscopic
562:
538:
354:
in one dimension—the diffusion length. In Fickian diffusion, this is
345:
251:
246:
A distinguishing feature of diffusion is that it depends on particle
182:
178:
170:
155:
12127:
11244:, mathematisation of Brownian motion, continuous stochastic process.
11160:. To separate diffusion in these cases, special efforts are needed.
5988:{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}}
449:
11380:
11201:
11143:
10967:
6809:
6521:
6194:{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}
5140:
798:
representing the quantity and direction of transfer. Given a small
755:
to Frenkel's quasichemical model of diffusion. The analogy between
496:
213:
205:
201:
11738:
11573:
11180:, also known as the Perona–Malik equation, enhances high gradients
9341:{\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left^{1/2}}
7342:
th component of the mixture. Each component has its mean velocity
4099:
the flux is equal to mobility × concentration × force per gram-ion
2226:(he used the term "force" in quotation marks or "driving force"):
1118:{\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}
161:
The concept of diffusion is widely used in many fields, including
64:
Time lapse video of diffusion a dye dissolved in water into a gel.
12185:
Mosby's Dictionary of Medicine, Nursing, & Health Professions
11386:
11100:
Dialysis works on the principles of the diffusion of solutes and
6356:
Darcy's law gives the equation of diffusion in porous media with
2203:
1464:
759:
and diffusion leads to various nonlinear versions of Fick's law.
194:
174:
162:
151:
11680:
11355: – Diffusion process with a non-linear relationship to time
7441:{\textstyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}
4409:
is the standard value of the chemical potential. The expression
111:
Three-dimensional rendering of diffusion of purple dye in water.
97:
to be one in the bottom panel. This apparent force is called an
11341:, transport of mass, heat, or momentum within a turbulent fluid
4469:{\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}
1979:
73:
11936:
Bian, Xin; Kim, Changho; Karniadakis, George Em (2016-08-14).
11568:. Vol. 4. Singapore: WORLD SCIENTIFIC. pp. 150–151.
7274:
The theory of diffusion in gases based on Boltzmann's equation
3838:{\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}
11318:
9762:{\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}
4120:
The force under isothermal conditions consists of two parts:
4058:
Below, to combine in the same formula the chemical potential
3955:
637:, the 26-year-old anatomy demonstrator from Zürich, proposed
526:
and the atomistic backgrounds of diffusion were developed by
438:
431:
401:
236:
90:
11722:"Quasichemical Models of Multicomponent Nonlinear Diffusion"
11427:, R.L. Baldwin, P.J. Dunlop, L.J. Gosting, G. Kegeles (1960)
11261:, diffusion of molecules from more dense to less dense areas
37:
11562:
Pescarmona, P.P. (2020). Gitis, V.; Rothenberg, G. (eds.).
5746:. For two and more components the equations are nonlinear.
5739:{\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}
4277:
equations violate this simple and physically obvious rule.
4222:
Electrostatic force caused by electric potential gradient:
3437:
3101:
2722:
1412:{\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}
799:
592:
47:
11814:
3593:
224:, and a change in temperature over a distance is called a
12103:
11293:, against the concentration gradient, in phase separation
9689:
for a given temperature has always the same character, 1/
5052:{\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}
413:
9157:, the diffusion flux caused by the temperature gradient.
8759:{\displaystyle \nabla \,\left({\frac {n_{1}}{n}}\right)}
7030:
For two different gases, A and B, with molecular masses
2089:{\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}
216:. A change in concentration over a distance is called a
12009:
11335:, diffusion of cultural traits across geographical area
10133:
is the net mass production rate in chemical reactions,
6629:
Consider two gases with molecules of the same diameter
11624:
10905:
10139:
9928:
9832:
8717:
describe four main effects in the diffusion of gases:
8134:
7607:
7511:
7348:
3537:
626:
deriving, in 1867, the coefficient of diffusion for CO
11007:
10853:
10826:
10774:
10743:
10705:
10675:
10546:
10430:
10293:
10228:
10185:
10112:
9991:
9882:
9778:
9709:
9646:
9600:
9356:
9181:
9116:
8967:
8829:
8726:
8672:
8641:
8223:
8180:
7927:
7693:
7454:
7284:
7067:
6852:
6818:
6650:
6569:
6546:
6413:
6375:
6329:
6248:
6146:
6097:
6056:
6004:
5874:
5758:
5675:
5607:
5476:
5379:
5341:
5307:
5274:
5212:
5153:
5123:
4972:
4822:
4695:
4657:
4630:
4559:
4489:
4415:
4289:
4228:
4130:
4068:
3967:
3907:
3851:
3783:
3731:
3698:
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3379:
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3242:
2940:
2892:
2862:
2842:
2690:
2663:
2636:
2528:
2421:
2398:
2371:
2338:
2311:
2235:
2212:
2181:
2154:
2134:
2105:
2033:
1997:
1871:
1844:
1755:
1732:
1624:
1541:
1503:
1481:
1447:
1425:
1365:
1341:
1274:
1251:
1227:
1134:
1091:
1055:
1015:
924:
898:
875:
855:
830:
807:
782:
575:
of oxygen molecules down the concentration gradient.
360:
330:
297:
11443:"Bulk flow and diffusion in the airways of the lung"
11403:
Pages displaying wikidata descriptions as a fallback
1726:
In case the diffusion coefficient is independent of
11776:"Reciprocal Relations in Irreversible Processes. I"
11627:"Heroes and Highlights in the History of Diffusion"
10976:
9169:transport and differ from advection or convection.
4062:and the mobility, we use for mobility the notation
4006:{\displaystyle D={\frac {\mu \,k_{\text{B}}T}{q}},}
1038:{\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}
220:, a change in pressure over a distance is called a
11406:Pages displaying short descriptions with no spaces
11250:of gas in long pores with frequent wall collisions
11044:
10957:
10887:
10839:
10808:
10756:
10729:
10691:
10658:
10531:
10415:
10260:
10210:
10168:
10125:
10095:
9967:
9914:
9861:
9814:
9761:
9670:
9629:
9586:
9340:
9145:
9099:
8947:
8758:
8685:
8654:
8624:
8206:
8174:For two gases, the difference between velocities,
8163:
8117:
7899:
7665:
7590:
7482:
7440:
7318:
7262:
6965:
6824:
6774:
6602:
6552:
6484:
6395:
6348:
6277:
6193:
6132:{\displaystyle \mathbf {J} =-\phi D\,\nabla n^{m}}
6131:
6069:
6042:
5987:
5859:
5738:
5657:
5590:
5456:
5354:
5320:
5293:
5260:
5198:
5051:
4913:
4793:
4678:
4643:
4617:{\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}
4616:
4542:
4468:
4383:
4241:
4211:
4124:Diffusion force caused by concentration gradient:
4078:
4005:
3929:
3893:
3837:
3769:
3717:
3680:
3573:
3513:
3362:
3258:
3210:
2908:
2875:
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2649:
2615:
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2404:
2384:
2357:
2324:
2294:
2218:
2187:
2167:
2140:
2120:
2088:
2009:
1968:
1850:
1827:
1738:
1715:
1593:{\displaystyle \mathbf {J} =-D(x)\,\nabla n(x,t),}
1592:
1524:
1489:
1453:
1433:
1411:
1347:
1324:
1257:
1233:
1206:
1117:
1070:
1037:
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907:
884:
861:
838:
816:
790:
578:
376:
336:
316:
12156:
11712:
11710:
11708:
11706:
11704:
11655:
11653:
11651:
11649:
11647:
11198:, in coarse-grained description of turbulent flow
11149:is necessary and this may be a non-trivial task.
7591:{\textstyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}
6505:For plasma with the high level of radiation, the
5860:{\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}}
5206:on the surface. Their surface concentrations are
450:Diffusion in the context of different disciplines
12219:
11931:
11929:
11927:
11431:. The Journal of Chemical Physics 33(5):1505–13.
11144:Separation of diffusion from convection in gases
6522:Diffusion coefficient in kinetic theory of gases
7011:, then we can see that for given concentration
5335:The jump model gives for the diffusion flux of
4113:of ions (particles). The common modern term is
767:
291:from this point. In Fickian diffusion, this is
11769:
11767:
11701:
11644:
11115:
11067:(for ideal mixtures) is the electron density,
11045:{\displaystyle J=-D\,\partial n/\partial x\,,}
10764:is the body force per unit mass acting on the
6278:{\displaystyle q=-{\frac {k}{\mu }}\,\nabla p}
4089:
3370:, and the matrix of diffusion coefficients is
1465:Normal single component concentration gradient
465:There are two ways to introduce the notion of
11924:
10958:{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}
9872:By definition, the diffusion velocity of the
8662:is the force applied to the molecules of the
6208:is the diffusion coefficient, Φ is porosity,
212:with the change in another variable, usually
11663:, Cambridge University Press (3rd edition),
11620:
11618:
4953:is the random force applied to the particle.
3894:{\displaystyle D_{12}\,\Delta c_{1}(x)<0}
1980:Multicomponent diffusion and thermodiffusion
762:
630:in the air. The error rate is less than 5%.
258:is used to describe the combination of both
12199:Aspects and Applications of the Random Walk
11773:
11764:
11726:Mathematical Modelling of Natural Phenomena
11520:
11171:
11092:Schematic of semipermeable membrane during
10995:Diffusion current can also be described by
9146:{\displaystyle k_{T}{\frac {1}{T}}\nabla T}
6622:(for ideal mixtures) is the concentration,
6603:{\displaystyle J=-D\,\partial n/\partial x}
6224: = 1 corresponds to Fick's law).
5658:{\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}
4813:starts with Newton's second law of motion:
400:First, there is a "bulk flow" process. The
265:If a diffusion process can be described by
11871:
11561:
11135:, free license granted by IS3D, LLC, 2014.
11074:
7666:{\textstyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}
6911:
6910:
6909:
5457:{\displaystyle \mathbf {J} _{i}=-D_{i}\,.}
5261:{\displaystyle c_{1},c_{2},\ldots ,c_{m}.}
2493:
2456:
2455:
2454:
622:The measurements of Graham contributed to
491:. According to Fick's laws, the diffusion
388:
12061:
11969:
11907:
11897:
11799:
11747:
11737:
11615:
11546:
11192:, spread of plasma across magnetic fields
11038:
11020:
10652:
10525:
10491:
10409:
10385:
10089:
9755:
8730:
8077:
7882:
7581:
7431:
7256:
6959:
6905:
6768:
6582:
6513: > 4 for the heat transfer.
6440:
6268:
6187:
6173:
6115:
5968:
5941:
5840:
5813:
5729:
5704:
5691:
5651:
5584:
5574:
5547:
5467:The corresponding diffusion equation is:
5450:
5414:
5199:{\displaystyle A_{1},A_{2},\ldots ,A_{m}}
4991:
4232:
4169:
4150:
3980:
3862:
3821:
3661:
3346:
3188:
3143:
2764:
2595:
2550:
2288:
2082:
1797:
1746:, Fick's second law can be simplified to
1709:
1684:
1565:
1318:
1200:
1190:
1166:
1108:
992:
982:
960:
953:
751:, with co-authors, applied his theory of
446:by bulk flow down the pressure gradient.
80:A single molecule moves around randomly.
12180:
12178:
11447:British Journal of Diseases of the Chest
11119:
11087:
10966:
10816:is the partial pressure fraction of the
10268:represents diffusion of this component.
7505:individual concentrations of particles,
6536:Random collisions of particles in a gas.
6531:
6494:For underground water infiltration, the
5127:
591:, which produces steel from the element
453:
417:there is no longer a pressure gradient.
397:and diffusion occur is human breathing.
106:
67:
53:
36:
11720:, H.P. Sargsyan and H.A. Wahab (2011).
11315:, diffusion of adparticles on a surface
10261:{\displaystyle \nabla (\rho _{i}v_{i})}
6516:
4804:
4055:, that is, the charge of one electron.
3594:Nondiagonal diffusion must be nonlinear
3363:{\displaystyle -(1/T)\,\nabla \mu _{j}}
1441:is the normal to the boundary at point
1427:
1387:
1104:
1028:
946:
832:
93:pushing molecules rightward, but there
14:
12220:
10737:is the thermal diffusion coefficient,
9968:{\textstyle \sum _{i}\rho _{i}v_{i}=0}
9630:{\displaystyle \kappa _{12}r^{-\nu }.}
1475:Fick's first law: The diffusion flux,
1265:, the diffusion equation has the form
772:Each model of diffusion expresses the
12196:
12175:
12107:Earth Surface Processes and Landforms
10888:{\displaystyle Y_{i}=\rho _{i}/\rho }
10699:is the diffusion coefficient matrix,
9862:{\textstyle \rho =\sum _{i}\rho _{i}}
7015:the diffusion coefficient grows with
6311:The advective molar flux is given as
5147:The system includes several reagents
3770:{\displaystyle c_{2}=\cdots =c_{n}=0}
3274: > 0), thermodiffusion (
2365:(the density of internal energy) and
2148:th physical quantity (component) and
12187:. 7th ed. St. Louis, MO; Mosby: 2006
12064:"Multicomponent diffusion revisited"
11498:Diffusion Fundamentals, 2, 1.1–1.10.
11440:
9815:{\displaystyle \rho _{i}=m_{i}n_{i}}
5137:Diffusion of reagents on the surface
1434:{\displaystyle {\boldsymbol {\nu }}}
839:{\displaystyle {\boldsymbol {\nu }}}
11299:, random reorientation of molecules
10211:{\displaystyle \nabla (\rho _{i}V)}
6043:{\displaystyle D_{ij}=D_{ji}\geq 0}
4751:
4500:
4300:
4071:
24:
11059:is the diffusion current density (
11032:
11021:
10492:
10446:
10386:
10229:
10186:
10047:
10022:
10010:
9995:
9681:We can see that the dependence on
9403:
9137:
8939:
8727:
8608:
8447:
8304:
8100:
7944:
7490:do not coincide then there exists
7237:
7222:
7173:
7146:
7105:
7077:
7074:
6938:
6868:
6702:
6594:
6583:
6498:gives the same equation with
6466:
6441:
6425:
6417:
6269:
6174:
6158:
6150:
6116:
5969:
5942:
5893:
5878:
5841:
5814:
5730:
5705:
5692:
5575:
5548:
5495:
5480:
5441:
5415:
5124:Jumps on the surface and in solids
5094:Einstein relation (kinetic theory)
4998:
4939:Einstein relation (kinetic theory)
4762:
4740:
4728:
4699:
4514:
4355:
4242:{\displaystyle q\,\nabla \varphi }
4233:
4170:
4151:
3948:Einstein relation (kinetic theory)
3863:
3822:
3802:
3784:
3662:
3626:
3611:
3472:
3442:
3347:
3189:
3144:
3130:
3107:
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2728:
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2435:
2272:
2255:
2249:
2001:
1939:
1910:
1872:
1845:
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1759:
1685:
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1628:
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1301:
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1278:
1191:
1182:
1167:
1155:
1135:
1109:
1092:
983:
976:
961:
954:
925:
899:
876:
808:
487:In the phenomenological approach,
25:
12239:
11659:S. Chapman, T. G. Cowling (1970)
11625:H. Mehrer; N.A. Stolwijk (2009).
9979:th component is described by the
9822:is the mass concentration of the
6396:{\displaystyle \upsilon =q/\phi }
6349:{\displaystyle p\sim n^{\gamma }}
3941:
3574:{\textstyle \sum _{j}L_{ij}X_{j}}
2657:is the chemical potential of the
643:Fourier's law for heat conduction
11610:Diffusion in Materials, DIMAT 96
11507:S.R. De Groot, P. Mazur (1962).
10977:Diffusion of electrons in solids
10549:
10515:
10433:
10351:
6239:of a gas in the porous media is
6099:
5761:
5518:
5382:
4491:
4291:
2982:
2630:is the absolute temperature and
2121:{\displaystyle \mathbf {J} _{i}}
2108:
2036:
1543:
1483:
1370:
1308:
1186:
1159:
1148:
1096:
1020:
938:
784:
495:is proportional to the negative
12190:
12150:
12097:
12055:
12003:
11986:
11865:
11850:S. Bromberg, K.A. Dill (2002),
11844:
11808:
11674:
11063:) per unit area per unit time,
8087:
7338:is velocity of molecule of the
6618:) per unit area per unit time,
6083:
5366: = 1, ...,
5118:Fluctuation-dissipation theorem
4079:{\displaystyle {\mathfrak {m}}}
658:William Chandler Roberts-Austen
579:History of diffusion in physics
243:, which means "to spread out".
11938:"111 years of Brownian motion"
11826:. Vol. 5 (3rd ed.).
11602:
11590:
11555:
11514:
11509:Non-equilibrium Thermodynamics
11501:
11481:
11434:
11418:
10722:
10716:
10644:
10618:
10507:
10495:
10488:
10462:
10401:
10389:
10380:
10374:
10255:
10232:
10205:
10189:
10073:
10050:
10041:
10025:
9665:
9657:
9479:
9453:
9433:
9406:
9400:
9392:
9286:
9260:
9234:
9207:
9094:
9068:
9062:
9016:
8933:
8887:
8879:
8853:
8693:is the thermodiffusion ratio.
8582:
8556:
8550:
8504:
8441:
8395:
8384:
8358:
8074:
8056:
8031:
8027:
8015:
7989:
7879:
7861:
7836:
7832:
7820:
7794:
7759:
7747:
7660:
7648:
7578:
7560:
7534:
7522:
7477:
7465:
7428:
7410:
7371:
7359:
7313:
7295:
7244:
7213:
5982:
5928:
5854:
5800:
5720:
5708:
5688:
5676:
5581:
5541:
5447:
5408:
5046:
5041:
5032:
5007:
4982:
4976:
4908:
4902:
4785:
4782:
4779:
4771:
4759:
4746:
4723:
4702:
4679:{\displaystyle n/n^{\ominus }}
4611:
4590:
4534:
4531:
4523:
4508:
4375:
4372:
4364:
4349:
4206:
4203:
4182:
4173:
3924:
3918:
3882:
3876:
3725:, and consider the state with
3467:
3455:
3343:
3329:
3125:
3119:
2856:are calculated at equilibrium
2746:
2740:
2506:
2494:
2267:
2261:
1934:
1922:
1890:
1878:
1816:
1804:
1777:
1765:
1706:
1703:
1691:
1681:
1675:
1669:
1646:
1634:
1584:
1572:
1562:
1556:
1519:
1507:
1400:
1397:
1391:
1380:
1374:
1366:
1197:
1179:
1163:
1144:
1065:
1059:
1032:
1016:
989:
973:
950:
934:
672:introduced the concept of the
505:non-equilibrium thermodynamics
317:{\displaystyle {\sqrt {2nDt}}}
13:
1:
11459:10.1016/S0007-0971(66)80044-X
11441:Muir, D. C. F. (1966-10-01).
11412:
11365:Diffusion-limited aggregation
11359:Convection–diffusion equation
11238:, diffusion of thermal energy
10809:{\displaystyle X_{i}=P_{i}/P}
10179:In these equations, the term
10169:{\textstyle \sum _{i}W_{i}=0}
9915:{\displaystyle v_{i}=C_{i}-V}
9671:{\displaystyle A_{1}({\nu })}
9161:All these effects are called
8164:{\textstyle n=\sum _{i}n_{i}}
7007:with the total concentration
5601:Due to the conservation law,
4261:is the absolute temperature,
3587:Course of Theoretical Physics
649:for electric current (1827).
587:had previously described the
377:{\displaystyle 2{\sqrt {Dt}}}
352:Constant concentration source
11565:Handbook of Porous Materials
11204:of a gas through small holes
10895:is the mass fraction of the
9172:In the first approximation,
8214:is given by the expression:
7319:{\displaystyle f_{i}(x,c,t)}
6220: > 1, the case
5102:is the mass of the particle.
4947:is the mass of the particle.
4644:{\displaystyle n^{\ominus }}
3718:{\displaystyle D_{12}\neq 0}
2931:The transport equations are
2918:Onsager reciprocal relations
2392:is the concentration of the
1986:Onsager reciprocal relations
1490:{\displaystyle \mathbf {J} }
1082:. If we use the notation of
791:{\displaystyle \mathbf {J} }
768:Definition of diffusion flux
458:Diffusion furnaces used for
273:(or non-Fickian diffusion).
7:
11511:. North-Holland, Amsterdam.
11345:
11223:, present in some organisms
11210:diffusion, resulting in an
11116:Random walk (random motion)
11083:
10730:{\displaystyle D_{i}^{(T)}}
10218:describes advection of the
9975:. The mass transfer of the
8810: = 1 –
8207:{\displaystyle C_{1}-C_{2}}
7918:are defined in 3D space as
6843:is the mean thermal speed:
6216: > 0 (usually
4273:is the electric potential.
4090:Diffusion across a membrane
1357:no-flux boundary conditions
10:
12244:
10980:
10847:is the partial pressure),
10222:th component and the term
7687:density of kinetic energy
7483:{\displaystyle C_{i}(x,t)}
6560:is the coefficient in the
6540:The diffusion coefficient
6525:
5665:and we have the system of
3958:). For charged particles:
1983:
1468:
1071:{\displaystyle o(\cdots )}
482:of the diffusing particles
32:Diffusion (disambiguation)
29:
11375:Isobaric counterdiffusion
11267:ex. the diffusion of the
2836:where the derivatives of
2010:{\displaystyle -\nabla n}
763:Basic models of diffusion
471:phenomenological approach
289:mean squared displacement
11548:10.1002/andp.19053220806
11333:Trans-cultural diffusion
11172:Other types of diffusion
7910:The kinetic temperature
7044:and molecular diameters
6496:Boussinesq approximation
3930:{\displaystyle c_{2}(x)}
2650:{\displaystyle \mu _{i}}
1471:Fick's laws of diffusion
908:{\displaystyle \Delta t}
885:{\displaystyle \Delta S}
817:{\displaystyle \Delta S}
475:Fick's laws of diffusion
348:of this Brownian motion;
12157:Gregory B. Pasternack.
11133:Cell Membrane Transport
11106:semi-permeable membrane
11075:Diffusion in geophysics
7598:(particles per volume),
6614:is the diffusion flux (
6509:–Raizer equation gives
5294:{\displaystyle z=c_{0}}
4776:external force per mole
4528:external force per mole
4369:external force per mole
2358:{\displaystyle n_{0}=u}
1851:{\displaystyle \Delta }
1615:(Fick's second law) is
1419:on the boundary, where
753:absolute reaction rates
512:atomistic point of view
389:Diffusion vs. bulk flow
12163:pasternack.ucdavis.edu
12077:(7): 073101–073101–8.
11801:10.1103/PhysRev.37.405
11634:Diffusion Fundamentals
11608:L.W. Barr (1997), In:
11136:
11097:
11046:
10973:
10959:
10889:
10841:
10810:
10758:
10731:
10693:
10692:{\displaystyle D_{ij}}
10660:
10607:
10533:
10417:
10335:
10284:components) they used
10262:
10212:
10170:
10127:
10097:
9969:
9916:
9863:
9816:
9763:
9672:
9631:
9588:
9342:
9147:
9101:
8949:
8760:
8687:
8656:
8626:
8208:
8171:is the total density.
8165:
8119:
7901:
7667:
7592:
7484:
7442:
7320:
7264:
6967:
6826:
6776:
6604:
6554:
6537:
6486:
6397:
6350:
6279:
6212:is the concentration,
6195:
6133:
6071:
6044:
5989:
5861:
5740:
5659:
5640:
5592:
5458:
5356:
5322:
5295:
5262:
5200:
5133:
5053:
4915:
4795:
4680:
4645:
4618:
4544:
4470:
4385:
4243:
4213:
4105:This is the so-called
4080:
4007:
3931:
3895:
3839:
3771:
3719:
3682:
3575:
3515:
3364:
3260:
3259:{\displaystyle D_{ik}}
3221:Here, all the indexes
3212:
2926:for the entropy growth
2910:
2909:{\displaystyle L_{ij}}
2877:
2850:
2827:
2671:
2651:
2617:
2516:
2406:
2386:
2359:
2326:
2296:
2220:
2189:
2169:
2142:
2122:
2090:
2011:
1970:
1852:
1829:
1740:
1717:
1594:
1526:
1525:{\displaystyle n(x,t)}
1491:
1455:
1435:
1413:
1349:
1326:
1259:
1235:
1208:
1119:
1072:
1039:
1000:
909:
886:
863:
840:
818:
792:
666:
620:
462:
378:
338:
318:
218:concentration gradient
132:spinodal decomposition
112:
104:
65:
51:
11899:10.1073/pnas.21.3.152
11828:Butterworth-Heinemann
11749:10.1051/mmnp/20116509
11487:J. Philibert (2005).
11307:spin magnetic moments
11287:, model for diffusion
11221:Facilitated diffusion
11178:Anisotropic diffusion
11130:
11091:
11047:
10970:
10960:
10890:
10842:
10840:{\displaystyle P_{i}}
10811:
10759:
10757:{\displaystyle f_{i}}
10732:
10694:
10661:
10587:
10534:
10418:
10315:
10263:
10213:
10171:
10128:
10126:{\displaystyle W_{i}}
10098:
9970:
9917:
9869:is the mass density.
9864:
9817:
9764:
9673:
9632:
9589:
9343:
9148:
9102:
8950:
8761:
8688:
8686:{\displaystyle k_{T}}
8657:
8655:{\displaystyle F_{i}}
8627:
8209:
8166:
8120:
7902:
7668:
7593:
7485:
7443:
7321:
7265:
6968:
6827:
6825:{\displaystyle \ell }
6777:
6605:
6555:
6535:
6487:
6398:
6351:
6280:
6196:
6134:
6072:
6070:{\displaystyle c_{0}}
6045:
5990:
5862:
5741:
5660:
5620:
5593:
5459:
5357:
5355:{\displaystyle A_{i}}
5323:
5321:{\displaystyle c_{i}}
5296:
5263:
5201:
5131:
5054:
4916:
4796:
4681:
4646:
4619:
4545:
4471:
4386:
4257:is the gas constant,
4244:
4214:
4081:
4008:
3932:
3901:at some points, then
3896:
3840:
3772:
3720:
3683:
3576:
3516:
3365:
3310:isothermal conditions
3261:
3213:
2916:should be symmetric (
2911:
2878:
2876:{\displaystyle n^{*}}
2851:
2828:
2672:
2652:
2618:
2517:
2407:
2387:
2385:{\displaystyle n_{i}}
2360:
2327:
2325:{\displaystyle n_{i}}
2297:
2221:
2190:
2170:
2168:{\displaystyle X_{j}}
2143:
2123:
2091:
2012:
1971:
1853:
1830:
1741:
1718:
1609:diffusion coefficient
1595:
1527:
1492:
1456:
1436:
1414:
1359:can be formulated as
1350:
1327:
1260:
1236:
1209:
1120:
1073:
1040:
1001:
910:
887:
864:
841:
819:
793:
662:
616:
457:
379:
339:
319:
110:
71:
63:
40:
12033:10.2136/vzj2008.0055
11774:Onsager, L. (1931).
11521:A. Einstein (1905).
11297:Rotational diffusion
11154:transport phenomenon
11005:
10903:
10851:
10824:
10772:
10741:
10703:
10673:
10544:
10428:
10291:
10275:proposed to use the
10226:
10183:
10137:
10110:
9989:
9926:
9880:
9830:
9776:
9707:
9644:
9598:
9594:for repulsing force
9354:
9179:
9114:
8965:
8827:
8724:
8670:
8639:
8221:
8178:
8132:
7925:
7691:
7605:
7601:density of momentum
7509:
7452:
7448:. If the velocities
7346:
7330:is the time moment,
7282:
7065:
6850:
6816:
6648:
6567:
6544:
6517:Diffusion in physics
6411:
6373:
6327:
6246:
6144:
6095:
6054:
6002:
5872:
5756:
5673:
5605:
5474:
5377:
5339:
5305:
5272:
5210:
5151:
5084:absolute temperature
4970:
4820:
4805:Ballistic time scale
4693:
4655:
4628:
4557:
4487:
4413:
4287:
4226:
4128:
4066:
4045:absolute temperature
3965:
3905:
3849:
3781:
3729:
3696:
3605:
3535:
3528: > 0).
3377:
3323:
3292:thermal conductivity
3290: = 0) and
3240:
2938:
2890:
2885:kinetic coefficients
2883:. The matrix of the
2860:
2840:
2688:
2661:
2634:
2526:
2419:
2396:
2369:
2336:
2309:
2233:
2210:
2179:
2152:
2132:
2103:
2031:
1995:
1869:
1842:
1753:
1730:
1622:
1611:. The corresponding
1539:
1501:
1479:
1445:
1423:
1363:
1339:
1272:
1249:
1225:
1132:
1089:
1053:
1013:
922:
896:
873:
853:
846:, the transfer of a
828:
805:
780:
694:Jean-Baptiste Perrin
676:. In the same year,
639:his law of diffusion
358:
328:
295:
226:temperature gradient
12120:2001ESPL...26..409P
12083:2006PhFl...18g3101L
12025:2009VZJ.....8..510S
12013:Vadose Zone Journal
11954:2016SMat...12.6331B
11890:1935PNAS...21..152T
11872:T. Teorell (1935).
11854:, Garland Science,
11824:Statistical Physics
11792:1931PhRv...37..405O
11695:10.1021/cr60090a005
11597:Diffusion Processes
11539:1905AnP...322..549E
11353:Anomalous diffusion
11339:Turbulent diffusion
11259:Molecular diffusion
11061:amount of substance
10726:
10384:
9981:continuity equation
7746:
7115:
6984:and decreases with
6712:
6616:amount of substance
4028:is the "mobility",
3286: > 0,
3278: > 0,
2197:thermodynamic force
2128:is the flux of the
1956:
1245:. For its density,
702:transport processes
690:Marian Smoluchowski
678:James Clerk Maxwell
624:James Clerk Maxwell
589:cementation process
547:Molecular diffusion
516:molecular diffusion
404:are located in the
271:anomalous diffusion
260:transport phenomena
41:Some particles are
12197:Weiss, G. (1994).
12062:S. H. Lam (2006).
11962:10.1039/c6sm01153e
11494:2013-12-13 at the
11393:Percolation theory
11370:Darken's equations
11265:Momentum diffusion
11231:isotope separation
11137:
11104:of fluid across a
11098:
11071:is the position .
11042:
10974:
10955:
10938:
10915:
10885:
10837:
10806:
10754:
10727:
10706:
10689:
10656:
10529:
10413:
10364:
10258:
10208:
10166:
10149:
10123:
10093:
9965:
9938:
9912:
9859:
9848:
9812:
9759:
9728:
9668:
9627:
9584:
9348:for rigid spheres;
9338:
9143:
9097:
8945:
8756:
8683:
8652:
8622:
8204:
8161:
8150:
8115:
7897:
7732:
7703:
7684:th particle mass),
7663:
7617:
7588:
7480:
7438:
7316:
7260:
7099:
6963:
6822:
6794:Boltzmann constant
6772:
6696:
6626:is the position .
6600:
6550:
6538:
6482:
6393:
6346:
6275:
6191:
6129:
6067:
6040:
5985:
5914:
5857:
5786:
5736:
5655:
5588:
5454:
5352:
5318:
5291:
5258:
5196:
5134:
5074:Boltzmann constant
5049:
4911:
4791:
4676:
4641:
4614:
4540:
4466:
4400:chemical potential
4381:
4269:is the charge and
4239:
4209:
4076:
4037:Boltzmann constant
4022:diffusion constant
4003:
3927:
3891:
3835:
3767:
3715:
3678:
3647:
3571:
3547:
3511:
3421:
3360:
3282: = 0 or
3266:of the diffusion (
3256:
3208:
3085:
3061:
3013:
2906:
2873:
2846:
2823:
2719:
2667:
2647:
2613:
2577:
2512:
2402:
2382:
2355:
2322:
2292:
2216:
2185:
2165:
2138:
2118:
2086:
2058:
2007:
1966:
1942:
1905:
1848:
1825:
1736:
1713:
1613:diffusion equation
1590:
1522:
1487:
1451:
1431:
1409:
1345:
1322:
1255:
1243:extensive quantity
1231:
1204:
1115:
1068:
1035:
996:
905:
882:
859:
836:
814:
788:
743:Walter H. Schottky
706:Boltzmann equation
463:
374:
334:
314:
167:particle diffusion
144:information theory
140:probability theory
128:chemical potential
113:
105:
66:
52:
12201:. North-Holland.
12091:10.1063/1.2221312
12071:Physics of Fluids
11948:(30): 6331–6346.
11837:978-0-7506-3372-7
11583:978-981-12-2328-0
11313:Surface diffusion
11291:Reverse diffusion
11248:Knudsen diffusion
11227:Gaseous diffusion
11216:diffusion current
11128:
10990:diffusion current
10983:Diffusion current
10929:
10906:
10570:
10140:
10017:
9929:
9839:
9753:
9719:
9581:
9559:
9511:
9437:
9431:
9318:
9244:
9135:
9066:
8937:
8750:
8666:th component and
8606:
8554:
8445:
8326:
8287:
8141:
8044:
7964:
7936:
7849:
7766:
7694:
7608:
7392:
7254:
7182:
7180:
7153:
7127:
7126:
7094:
6957:
6956:
6903:
6884:
6766:
6729:
6728:
6691:
6665:
6553:{\displaystyle D}
6456:
6432:
6308:is the pressure.
6266:
6165:
5905:
5900:
5777:
5502:
5301:. The sum of all
4894:
4874:
4858:
4811:Langevin equation
4777:
4735:
4529:
4476:is the so-called
4460:
4370:
4343:
4200:
4148:
4053:elementary charge
3998:
3988:
3777:. At this state,
3638:
3633:
3585:, whereas in the
3538:
3486:
3406:
3404:
3204:
3158:
3070:
3046:
2998:
2966:
2922:positive definite
2849:{\displaystyle s}
2819:
2779:
2704:
2670:{\displaystyle i}
2593:
2562:
2548:
2491:
2450:
2446:
2405:{\displaystyle i}
2286:
2219:{\displaystyle s}
2188:{\displaystyle j}
2141:{\displaystyle i}
2049:
1962:
1958:
1896:
1821:
1789:
1739:{\displaystyle x}
1658:
1454:{\displaystyle x}
1348:{\displaystyle W}
1293:
1258:{\displaystyle n}
1234:{\displaystyle N}
1080:little-o notation
869:through the area
862:{\displaystyle N}
848:physical quantity
757:reaction kinetics
704:, introduced the
684:was developed by
552:Knudsen diffusion
543:materials science
460:thermal oxidation
372:
337:{\displaystyle n}
312:
235:derives from the
222:pressure gradient
124:Gibbs free energy
61:
16:(Redirected from
12235:
12213:
12212:
12194:
12188:
12182:
12173:
12172:
12170:
12169:
12154:
12148:
12147:
12101:
12095:
12094:
12068:
12059:
12053:
12052:
12007:
12001:
11990:
11984:
11983:
11973:
11933:
11922:
11921:
11911:
11901:
11869:
11863:
11848:
11842:
11841:
11812:
11806:
11805:
11803:
11771:
11762:
11761:
11751:
11741:
11714:
11699:
11698:
11678:
11672:
11657:
11642:
11641:
11631:
11622:
11613:
11606:
11600:
11594:
11588:
11587:
11559:
11553:
11552:
11550:
11518:
11512:
11505:
11499:
11485:
11479:
11478:
11438:
11432:
11422:
11407:
11404:
11280:Plasma diffusion
11275:Photon diffusion
11212:electric current
11184:Atomic diffusion
11129:
11051:
11049:
11048:
11043:
11031:
10997:Fick's first law
10964:
10962:
10961:
10956:
10948:
10947:
10937:
10925:
10924:
10914:
10899:th species, and
10894:
10892:
10891:
10886:
10881:
10876:
10875:
10863:
10862:
10846:
10844:
10843:
10838:
10836:
10835:
10820:th species (and
10815:
10813:
10812:
10807:
10802:
10797:
10796:
10784:
10783:
10763:
10761:
10760:
10755:
10753:
10752:
10736:
10734:
10733:
10728:
10725:
10714:
10698:
10696:
10695:
10690:
10688:
10687:
10665:
10663:
10662:
10657:
10651:
10647:
10643:
10642:
10630:
10629:
10617:
10616:
10606:
10601:
10586:
10585:
10571:
10563:
10558:
10557:
10552:
10538:
10536:
10535:
10530:
10524:
10523:
10518:
10487:
10486:
10474:
10473:
10458:
10457:
10442:
10441:
10436:
10422:
10420:
10419:
10414:
10408:
10404:
10383:
10372:
10360:
10359:
10354:
10348:
10347:
10334:
10329:
10303:
10302:
10273:Wendell H. Furry
10267:
10265:
10264:
10259:
10254:
10253:
10244:
10243:
10217:
10215:
10214:
10209:
10201:
10200:
10175:
10173:
10172:
10167:
10159:
10158:
10148:
10132:
10130:
10129:
10124:
10122:
10121:
10102:
10100:
10099:
10094:
10088:
10087:
10072:
10071:
10062:
10061:
10037:
10036:
10018:
10016:
10008:
10007:
10006:
9993:
9974:
9972:
9971:
9966:
9958:
9957:
9948:
9947:
9937:
9921:
9919:
9918:
9913:
9905:
9904:
9892:
9891:
9876:th component is
9868:
9866:
9865:
9860:
9858:
9857:
9847:
9821:
9819:
9818:
9813:
9811:
9810:
9801:
9800:
9788:
9787:
9768:
9766:
9765:
9760:
9754:
9749:
9748:
9747:
9738:
9737:
9727:
9717:
9677:
9675:
9674:
9669:
9664:
9656:
9655:
9636:
9634:
9633:
9628:
9623:
9622:
9610:
9609:
9593:
9591:
9590:
9585:
9583:
9582:
9580:
9566:
9564:
9560:
9558:
9557:
9548:
9537:
9530:
9529:
9525:
9516:
9512:
9510:
9509:
9508:
9499:
9498:
9482:
9478:
9477:
9465:
9464:
9445:
9438:
9436:
9432:
9430:
9416:
9399:
9391:
9390:
9371:
9366:
9365:
9347:
9345:
9344:
9339:
9337:
9336:
9332:
9323:
9319:
9317:
9316:
9315:
9306:
9305:
9289:
9285:
9284:
9272:
9271:
9252:
9245:
9243:
9242:
9241:
9232:
9231:
9219:
9218:
9196:
9191:
9190:
9152:
9150:
9149:
9144:
9136:
9128:
9126:
9125:
9106:
9104:
9103:
9098:
9093:
9092:
9080:
9079:
9067:
9065:
9061:
9060:
9051:
9050:
9038:
9037:
9028:
9027:
9011:
9010:
9009:
9000:
8999:
8990:
8989:
8980:
8979:
8969:
8954:
8952:
8951:
8946:
8938:
8936:
8932:
8931:
8922:
8921:
8909:
8908:
8899:
8898:
8882:
8878:
8877:
8865:
8864:
8852:
8851:
8842:
8841:
8831:
8765:
8763:
8762:
8757:
8755:
8751:
8746:
8745:
8736:
8696:The coefficient
8692:
8690:
8689:
8684:
8682:
8681:
8661:
8659:
8658:
8653:
8651:
8650:
8631:
8629:
8628:
8623:
8618:
8614:
8607:
8599:
8597:
8596:
8581:
8580:
8568:
8567:
8555:
8553:
8549:
8548:
8539:
8538:
8526:
8525:
8516:
8515:
8499:
8498:
8497:
8488:
8487:
8478:
8477:
8468:
8467:
8457:
8446:
8444:
8440:
8439:
8430:
8429:
8417:
8416:
8407:
8406:
8387:
8383:
8382:
8370:
8369:
8357:
8356:
8347:
8346:
8336:
8331:
8327:
8322:
8321:
8312:
8298:
8297:
8288:
8286:
8285:
8284:
8275:
8274:
8264:
8263:
8254:
8246:
8245:
8233:
8232:
8213:
8211:
8210:
8205:
8203:
8202:
8190:
8189:
8170:
8168:
8167:
8162:
8160:
8159:
8149:
8124:
8122:
8121:
8116:
8105:
8104:
8103:
8055:
8054:
8045:
8040:
8039:
8038:
8014:
8013:
8001:
8000:
7988:
7987:
7977:
7975:
7974:
7965:
7957:
7949:
7948:
7947:
7937:
7929:
7906:
7904:
7903:
7898:
7893:
7889:
7860:
7859:
7850:
7845:
7844:
7843:
7819:
7818:
7806:
7805:
7793:
7792:
7782:
7780:
7779:
7767:
7762:
7745:
7740:
7731:
7730:
7720:
7718:
7717:
7702:
7672:
7670:
7669:
7664:
7647:
7646:
7637:
7636:
7627:
7626:
7616:
7597:
7595:
7594:
7589:
7559:
7558:
7549:
7548:
7521:
7520:
7489:
7487:
7486:
7481:
7464:
7463:
7447:
7445:
7444:
7439:
7403:
7402:
7393:
7391:
7390:
7378:
7358:
7357:
7334:is position and
7325:
7323:
7322:
7317:
7294:
7293:
7269:
7267:
7266:
7261:
7255:
7253:
7252:
7251:
7242:
7241:
7240:
7227:
7226:
7225:
7208:
7207:
7206:
7202:
7185:
7183:
7181:
7179:
7178:
7177:
7176:
7159:
7154:
7152:
7151:
7150:
7149:
7132:
7130:
7128:
7125:
7124:
7114:
7109:
7108:
7098:
7097:
7095:
7087:
7082:
7081:
7080:
6992:. If we use for
6972:
6970:
6969:
6964:
6958:
6955:
6947:
6943:
6942:
6941:
6927:
6926:
6921:
6920:
6904:
6902:
6898:
6897:
6885:
6880:
6877:
6873:
6872:
6871:
6860:
6831:
6829:
6828:
6823:
6781:
6779:
6778:
6773:
6767:
6765:
6764:
6763:
6750:
6749:
6745:
6732:
6730:
6727:
6723:
6722:
6711:
6706:
6705:
6695:
6694:
6692:
6684:
6679:
6678:
6666:
6658:
6609:
6607:
6606:
6601:
6593:
6562:Fick's first law
6559:
6557:
6556:
6551:
6502: = 2.
6491:
6489:
6488:
6483:
6478:
6477:
6454:
6453:
6452:
6433:
6431:
6423:
6415:
6402:
6400:
6399:
6394:
6389:
6364: + 1.
6355:
6353:
6352:
6347:
6345:
6344:
6284:
6282:
6281:
6276:
6267:
6259:
6200:
6198:
6197:
6192:
6186:
6185:
6166:
6164:
6156:
6148:
6138:
6136:
6135:
6130:
6128:
6127:
6102:
6076:
6074:
6073:
6068:
6066:
6065:
6049:
6047:
6046:
6041:
6033:
6032:
6017:
6016:
5994:
5992:
5991:
5986:
5981:
5980:
5967:
5966:
5954:
5953:
5940:
5939:
5927:
5926:
5913:
5901:
5899:
5891:
5890:
5889:
5876:
5866:
5864:
5863:
5858:
5853:
5852:
5839:
5838:
5826:
5825:
5812:
5811:
5799:
5798:
5785:
5770:
5769:
5764:
5745:
5743:
5742:
5737:
5664:
5662:
5661:
5656:
5650:
5649:
5639:
5634:
5597:
5595:
5594:
5589:
5573:
5572:
5560:
5559:
5540:
5539:
5527:
5526:
5521:
5503:
5501:
5493:
5492:
5491:
5478:
5463:
5461:
5460:
5455:
5440:
5439:
5427:
5426:
5407:
5406:
5391:
5390:
5385:
5361:
5359:
5358:
5353:
5351:
5350:
5327:
5325:
5324:
5319:
5317:
5316:
5300:
5298:
5297:
5292:
5290:
5289:
5267:
5265:
5264:
5259:
5254:
5253:
5235:
5234:
5222:
5221:
5205:
5203:
5202:
5197:
5195:
5194:
5176:
5175:
5163:
5162:
5114:ballistic regime
5058:
5056:
5055:
5050:
5045:
5044:
5031:
5003:
5002:
5001:
4931:is the position.
4920:
4918:
4917:
4912:
4895:
4893:
4885:
4877:
4875:
4867:
4859:
4857:
4856:
4855:
4842:
4838:
4837:
4827:
4800:
4798:
4797:
4792:
4778:
4775:
4755:
4754:
4736:
4734:
4726:
4722:
4721:
4712:
4697:
4685:
4683:
4682:
4677:
4675:
4674:
4665:
4650:
4648:
4647:
4642:
4640:
4639:
4623:
4621:
4620:
4615:
4610:
4609:
4600:
4583:
4582:
4573:
4549:
4547:
4546:
4541:
4530:
4527:
4504:
4503:
4494:
4475:
4473:
4472:
4467:
4465:
4461:
4459:
4451:
4450:
4449:
4433:
4390:
4388:
4387:
4382:
4371:
4368:
4348:
4344:
4342:
4334:
4333:
4332:
4316:
4304:
4303:
4294:
4248:
4246:
4245:
4240:
4218:
4216:
4215:
4210:
4202:
4201:
4198:
4192:
4149:
4141:
4085:
4083:
4082:
4077:
4075:
4074:
4012:
4010:
4009:
4004:
3999:
3994:
3990:
3989:
3986:
3975:
3936:
3934:
3933:
3928:
3917:
3916:
3900:
3898:
3897:
3892:
3875:
3874:
3861:
3860:
3844:
3842:
3841:
3836:
3834:
3833:
3820:
3819:
3801:
3796:
3795:
3776:
3774:
3773:
3768:
3760:
3759:
3741:
3740:
3724:
3722:
3721:
3716:
3708:
3707:
3687:
3685:
3684:
3679:
3674:
3673:
3660:
3659:
3646:
3634:
3632:
3624:
3623:
3622:
3609:
3580:
3578:
3577:
3572:
3570:
3569:
3560:
3559:
3546:
3520:
3518:
3517:
3512:
3510:
3509:
3508:
3507:
3491:
3487:
3485:
3484:
3483:
3470:
3454:
3453:
3440:
3434:
3433:
3420:
3405:
3397:
3392:
3391:
3369:
3367:
3366:
3361:
3359:
3358:
3339:
3305:) coefficients.
3304:
3265:
3263:
3262:
3257:
3255:
3254:
3235:
3217:
3215:
3214:
3209:
3202:
3201:
3200:
3187:
3183:
3182:
3181:
3180:
3179:
3163:
3159:
3157:
3156:
3155:
3142:
3141:
3128:
3115:
3114:
3104:
3098:
3097:
3084:
3060:
3042:
3041:
3026:
3025:
3012:
2991:
2990:
2985:
2967:
2965:
2957:
2956:
2955:
2942:
2915:
2913:
2912:
2907:
2905:
2904:
2882:
2880:
2879:
2874:
2872:
2871:
2855:
2853:
2852:
2847:
2832:
2830:
2829:
2824:
2817:
2816:
2815:
2803:
2802:
2801:
2800:
2784:
2780:
2778:
2777:
2776:
2763:
2762:
2749:
2736:
2735:
2725:
2718:
2700:
2699:
2676:
2674:
2673:
2668:
2656:
2654:
2653:
2648:
2646:
2645:
2622:
2620:
2619:
2614:
2612:
2611:
2602:
2601:
2594:
2589:
2588:
2579:
2576:
2555:
2549:
2541:
2533:
2521:
2519:
2518:
2513:
2492:
2487:
2486:
2477:
2466:
2465:
2448:
2447:
2439:
2431:
2430:
2411:
2409:
2408:
2403:
2391:
2389:
2388:
2383:
2381:
2380:
2364:
2362:
2361:
2356:
2348:
2347:
2331:
2329:
2328:
2323:
2321:
2320:
2301:
2299:
2298:
2293:
2287:
2285:
2284:
2283:
2270:
2253:
2245:
2244:
2225:
2223:
2222:
2217:
2194:
2192:
2191:
2186:
2174:
2172:
2171:
2166:
2164:
2163:
2147:
2145:
2144:
2139:
2127:
2125:
2124:
2119:
2117:
2116:
2111:
2095:
2093:
2092:
2087:
2081:
2080:
2071:
2070:
2057:
2045:
2044:
2039:
2016:
2014:
2013:
2008:
1975:
1973:
1972:
1967:
1960:
1959:
1957:
1955:
1950:
1937:
1918:
1917:
1907:
1904:
1860:Laplace operator
1857:
1855:
1854:
1849:
1834:
1832:
1831:
1826:
1819:
1790:
1788:
1780:
1757:
1745:
1743:
1742:
1737:
1722:
1720:
1719:
1714:
1659:
1657:
1649:
1626:
1599:
1597:
1596:
1591:
1546:
1531:
1529:
1528:
1523:
1496:
1494:
1493:
1488:
1486:
1460:
1458:
1457:
1452:
1440:
1438:
1437:
1432:
1430:
1418:
1416:
1415:
1410:
1390:
1373:
1354:
1352:
1351:
1346:
1331:
1329:
1328:
1323:
1311:
1294:
1292:
1284:
1276:
1264:
1262:
1261:
1256:
1240:
1238:
1237:
1232:
1213:
1211:
1210:
1205:
1189:
1162:
1151:
1124:
1122:
1121:
1116:
1107:
1099:
1077:
1075:
1074:
1069:
1044:
1042:
1041:
1036:
1031:
1023:
1005:
1003:
1002:
997:
949:
941:
914:
912:
911:
906:
891:
889:
888:
883:
868:
866:
865:
860:
845:
843:
842:
837:
835:
823:
821:
820:
815:
797:
795:
794:
789:
787:
737:Sometime later,
713:George de Hevesy
698:Ludwig Boltzmann
605:Chinese ceramics
534:of individuals.
383:
381:
380:
375:
373:
365:
343:
341:
340:
335:
323:
321:
320:
315:
313:
299:
62:
46:but with no net
21:
12243:
12242:
12238:
12237:
12236:
12234:
12233:
12232:
12218:
12217:
12216:
12209:
12195:
12191:
12183:
12176:
12167:
12165:
12155:
12151:
12128:10.1002/esp.189
12102:
12098:
12066:
12060:
12056:
12008:
12004:
11991:
11987:
11934:
11925:
11870:
11866:
11849:
11845:
11838:
11813:
11809:
11780:Physical Review
11772:
11765:
11715:
11702:
11679:
11675:
11658:
11645:
11629:
11623:
11616:
11607:
11603:
11595:
11591:
11584:
11560:
11556:
11519:
11515:
11506:
11502:
11496:Wayback Machine
11486:
11482:
11439:
11435:
11423:
11419:
11415:
11410:
11405:
11402:
11401: – journal
11399:Social Networks
11348:
11305:, diffusion of
11174:
11165:heat conduction
11146:
11120:
11118:
11102:ultrafiltration
11086:
11077:
11027:
11006:
11003:
11002:
10985:
10979:
10943:
10939:
10933:
10920:
10916:
10910:
10904:
10901:
10900:
10877:
10871:
10867:
10858:
10854:
10852:
10849:
10848:
10831:
10827:
10825:
10822:
10821:
10798:
10792:
10788:
10779:
10775:
10773:
10770:
10769:
10748:
10744:
10742:
10739:
10738:
10715:
10710:
10704:
10701:
10700:
10680:
10676:
10674:
10671:
10670:
10638:
10634:
10625:
10621:
10612:
10608:
10602:
10591:
10581:
10577:
10576:
10572:
10562:
10553:
10548:
10547:
10545:
10542:
10541:
10519:
10514:
10513:
10482:
10478:
10469:
10465:
10453:
10449:
10437:
10432:
10431:
10429:
10426:
10425:
10373:
10368:
10355:
10350:
10349:
10340:
10336:
10330:
10319:
10314:
10310:
10298:
10294:
10292:
10289:
10288:
10249:
10245:
10239:
10235:
10227:
10224:
10223:
10196:
10192:
10184:
10181:
10180:
10154:
10150:
10144:
10138:
10135:
10134:
10117:
10113:
10111:
10108:
10107:
10083:
10079:
10067:
10063:
10057:
10053:
10032:
10028:
10009:
10002:
9998:
9994:
9992:
9990:
9987:
9986:
9953:
9949:
9943:
9939:
9933:
9927:
9924:
9923:
9900:
9896:
9887:
9883:
9881:
9878:
9877:
9853:
9849:
9843:
9831:
9828:
9827:
9806:
9802:
9796:
9792:
9783:
9779:
9777:
9774:
9773:
9743:
9739:
9733:
9729:
9723:
9718:
9716:
9708:
9705:
9704:
9660:
9651:
9647:
9645:
9642:
9641:
9615:
9611:
9605:
9601:
9599:
9596:
9595:
9570:
9565:
9553:
9549:
9538:
9536:
9532:
9531:
9521:
9517:
9504:
9500:
9494:
9490:
9483:
9473:
9469:
9460:
9456:
9446:
9444:
9440:
9439:
9420:
9415:
9395:
9386:
9382:
9375:
9370:
9361:
9357:
9355:
9352:
9351:
9328:
9324:
9311:
9307:
9301:
9297:
9290:
9280:
9276:
9267:
9263:
9253:
9251:
9247:
9246:
9237:
9233:
9227:
9223:
9214:
9210:
9200:
9195:
9186:
9182:
9180:
9177:
9176:
9155:thermodiffusion
9127:
9121:
9117:
9115:
9112:
9111:
9088:
9084:
9075:
9071:
9056:
9052:
9046:
9042:
9033:
9029:
9023:
9019:
9012:
9005:
9001:
8995:
8991:
8985:
8981:
8975:
8971:
8970:
8968:
8966:
8963:
8962:
8927:
8923:
8917:
8913:
8904:
8900:
8894:
8890:
8883:
8873:
8869:
8860:
8856:
8847:
8843:
8837:
8833:
8832:
8830:
8828:
8825:
8824:
8816:
8805:
8794:
8783:
8772:
8741:
8737:
8735:
8731:
8725:
8722:
8721:
8716:
8709:
8702:
8677:
8673:
8671:
8668:
8667:
8646:
8642:
8640:
8637:
8636:
8598:
8592:
8588:
8576:
8572:
8563:
8559:
8544:
8540:
8534:
8530:
8521:
8517:
8511:
8507:
8500:
8493:
8489:
8483:
8479:
8473:
8469:
8463:
8459:
8458:
8456:
8435:
8431:
8425:
8421:
8412:
8408:
8402:
8398:
8388:
8378:
8374:
8365:
8361:
8352:
8348:
8342:
8338:
8337:
8335:
8317:
8313:
8311:
8307:
8303:
8299:
8293:
8289:
8280:
8276:
8270:
8266:
8265:
8259:
8255:
8253:
8241:
8237:
8228:
8224:
8222:
8219:
8218:
8198:
8194:
8185:
8181:
8179:
8176:
8175:
8155:
8151:
8145:
8133:
8130:
8129:
8099:
8098:
8094:
8050:
8046:
8034:
8030:
8009:
8005:
7996:
7992:
7983:
7979:
7978:
7976:
7970:
7966:
7956:
7943:
7942:
7938:
7928:
7926:
7923:
7922:
7855:
7851:
7839:
7835:
7814:
7810:
7801:
7797:
7788:
7784:
7783:
7781:
7775:
7771:
7741:
7736:
7726:
7722:
7721:
7719:
7713:
7709:
7708:
7704:
7698:
7692:
7689:
7688:
7678:
7642:
7638:
7632:
7628:
7622:
7618:
7612:
7606:
7603:
7602:
7554:
7550:
7544:
7540:
7516:
7512:
7510:
7507:
7506:
7459:
7455:
7453:
7450:
7449:
7398:
7394:
7386:
7382:
7377:
7353:
7349:
7347:
7344:
7343:
7289:
7285:
7283:
7280:
7279:
7276:
7247:
7243:
7236:
7235:
7231:
7221:
7220:
7216:
7209:
7198:
7194:
7190:
7186:
7184:
7172:
7171:
7167:
7163:
7158:
7145:
7144:
7140:
7136:
7131:
7129:
7120:
7116:
7110:
7104:
7103:
7096:
7086:
7073:
7072:
7068:
7066:
7063:
7062:
7057:
7050:
7043:
7036:
6948:
6937:
6936:
6932:
6928:
6925:
6916:
6912:
6893:
6889:
6879:
6878:
6867:
6866:
6862:
6861:
6859:
6851:
6848:
6847:
6841:
6817:
6814:
6813:
6791:
6759:
6755:
6751:
6741:
6737:
6733:
6731:
6718:
6714:
6713:
6707:
6701:
6700:
6693:
6683:
6674:
6670:
6657:
6649:
6646:
6645:
6589:
6568:
6565:
6564:
6545:
6542:
6541:
6530:
6524:
6519:
6473:
6469:
6448:
6444:
6424:
6416:
6414:
6412:
6409:
6408:
6385:
6374:
6371:
6370:
6340:
6336:
6328:
6325:
6324:
6296:of the medium,
6258:
6247:
6244:
6243:
6237:volumetric flux
6181:
6177:
6157:
6149:
6147:
6145:
6142:
6141:
6123:
6119:
6098:
6096:
6093:
6092:
6086:
6061:
6057:
6055:
6052:
6051:
6025:
6021:
6009:
6005:
6003:
6000:
5999:
5976:
5972:
5962:
5958:
5949:
5945:
5935:
5931:
5919:
5915:
5909:
5892:
5885:
5881:
5877:
5875:
5873:
5870:
5869:
5848:
5844:
5834:
5830:
5821:
5817:
5807:
5803:
5791:
5787:
5781:
5765:
5760:
5759:
5757:
5754:
5753:
5674:
5671:
5670:
5645:
5641:
5635:
5624:
5606:
5603:
5602:
5568:
5564:
5555:
5551:
5535:
5531:
5522:
5517:
5516:
5494:
5487:
5483:
5479:
5477:
5475:
5472:
5471:
5435:
5431:
5422:
5418:
5402:
5398:
5386:
5381:
5380:
5378:
5375:
5374:
5346:
5342:
5340:
5337:
5336:
5312:
5308:
5306:
5303:
5302:
5285:
5281:
5273:
5270:
5269:
5249:
5245:
5230:
5226:
5217:
5213:
5211:
5208:
5207:
5190:
5186:
5171:
5167:
5158:
5154:
5152:
5149:
5148:
5126:
5071:
5027:
5020:
5016:
4997:
4996:
4992:
4971:
4968:
4967:
4886:
4878:
4876:
4866:
4851:
4847:
4843:
4833:
4829:
4828:
4826:
4821:
4818:
4817:
4807:
4774:
4750:
4749:
4727:
4717:
4713:
4708:
4698:
4696:
4694:
4691:
4690:
4670:
4666:
4661:
4656:
4653:
4652:
4635:
4631:
4629:
4626:
4625:
4605:
4601:
4596:
4578:
4574:
4569:
4558:
4555:
4554:
4526:
4499:
4498:
4490:
4488:
4485:
4484:
4452:
4445:
4441:
4434:
4432:
4428:
4414:
4411:
4410:
4408:
4367:
4335:
4328:
4324:
4317:
4315:
4311:
4299:
4298:
4290:
4288:
4285:
4284:
4227:
4224:
4223:
4197:
4193:
4188:
4140:
4129:
4126:
4125:
4111:Avogadro number
4107:Teorell formula
4092:
4070:
4069:
4067:
4064:
4063:
4034:
3985:
3981:
3976:
3974:
3966:
3963:
3962:
3944:
3912:
3908:
3906:
3903:
3902:
3870:
3866:
3856:
3852:
3850:
3847:
3846:
3829:
3825:
3815:
3811:
3797:
3791:
3787:
3782:
3779:
3778:
3755:
3751:
3736:
3732:
3730:
3727:
3726:
3703:
3699:
3697:
3694:
3693:
3669:
3665:
3652:
3648:
3642:
3625:
3618:
3614:
3610:
3608:
3606:
3603:
3602:
3596:
3565:
3561:
3552:
3548:
3542:
3536:
3533:
3532:
3503:
3499:
3492:
3479:
3475:
3471:
3449:
3445:
3441:
3439:
3436:
3435:
3426:
3422:
3410:
3396:
3384:
3380:
3378:
3375:
3374:
3354:
3350:
3335:
3324:
3321:
3320:
3295:
3247:
3243:
3241:
3238:
3237:
3222:
3196:
3192:
3175:
3171:
3164:
3151:
3147:
3137:
3133:
3129:
3110:
3106:
3105:
3103:
3100:
3099:
3090:
3086:
3074:
3066:
3062:
3050:
3037:
3033:
3018:
3014:
3002:
2986:
2981:
2980:
2958:
2951:
2947:
2943:
2941:
2939:
2936:
2935:
2897:
2893:
2891:
2888:
2887:
2867:
2863:
2861:
2858:
2857:
2841:
2838:
2837:
2811:
2807:
2796:
2792:
2785:
2772:
2768:
2758:
2754:
2750:
2731:
2727:
2726:
2724:
2721:
2720:
2708:
2695:
2691:
2689:
2686:
2685:
2662:
2659:
2658:
2641:
2637:
2635:
2632:
2631:
2607:
2603:
2597:
2596:
2584:
2580:
2578:
2566:
2551:
2540:
2529:
2527:
2524:
2523:
2482:
2478:
2476:
2461:
2457:
2438:
2426:
2422:
2420:
2417:
2416:
2397:
2394:
2393:
2376:
2372:
2370:
2367:
2366:
2343:
2339:
2337:
2334:
2333:
2316:
2312:
2310:
2307:
2306:
2279:
2275:
2271:
2254:
2252:
2240:
2236:
2234:
2231:
2230:
2211:
2208:
2207:
2180:
2177:
2176:
2159:
2155:
2153:
2150:
2149:
2133:
2130:
2129:
2112:
2107:
2106:
2104:
2101:
2100:
2076:
2072:
2063:
2059:
2053:
2040:
2035:
2034:
2032:
2029:
2028:
1996:
1993:
1992:
1988:
1982:
1951:
1946:
1938:
1913:
1909:
1908:
1906:
1900:
1870:
1867:
1866:
1843:
1840:
1839:
1781:
1758:
1756:
1754:
1751:
1750:
1731:
1728:
1727:
1650:
1627:
1625:
1623:
1620:
1619:
1542:
1540:
1537:
1536:
1502:
1499:
1498:
1482:
1480:
1477:
1476:
1473:
1467:
1446:
1443:
1442:
1426:
1424:
1421:
1420:
1386:
1369:
1364:
1361:
1360:
1340:
1337:
1336:
1307:
1285:
1277:
1275:
1273:
1270:
1269:
1250:
1247:
1246:
1226:
1223:
1222:
1185:
1158:
1147:
1133:
1130:
1129:
1103:
1095:
1090:
1087:
1086:
1054:
1051:
1050:
1027:
1019:
1014:
1011:
1010:
945:
937:
923:
920:
919:
897:
894:
893:
874:
871:
870:
854:
851:
850:
831:
829:
826:
825:
806:
803:
802:
783:
781:
778:
777:
770:
765:
686:Albert Einstein
682:Brownian motion
670:Rudolf Clausius
629:
585:Pliny the Elder
581:
528:Albert Einstein
524:Brownian motion
452:
406:thoracic cavity
391:
364:
359:
356:
355:
329:
326:
325:
298:
296:
293:
292:
281:Brownian motion
148:neural networks
54:
35:
28:
23:
22:
15:
12:
11:
5:
12241:
12231:
12230:
12215:
12214:
12208:978-0444816061
12207:
12189:
12174:
12149:
12096:
12054:
12002:
11985:
11923:
11864:
11843:
11836:
11807:
11763:
11732:(5): 184–262.
11700:
11673:
11643:
11614:
11601:
11589:
11582:
11554:
11513:
11500:
11480:
11453:(4): 169–176.
11433:
11416:
11414:
11411:
11409:
11408:
11396:
11390:
11384:
11378:
11372:
11367:
11362:
11356:
11349:
11347:
11344:
11343:
11342:
11336:
11330:
11329:
11328:
11316:
11310:
11303:Spin diffusion
11300:
11294:
11288:
11282:
11277:
11272:
11271:velocity field
11262:
11256:
11251:
11245:
11239:
11233:
11224:
11218:
11205:
11199:
11196:Eddy diffusion
11193:
11190:Bohm diffusion
11187:
11181:
11173:
11170:
11145:
11142:
11117:
11114:
11085:
11082:
11076:
11073:
11053:
11052:
11041:
11037:
11034:
11030:
11026:
11023:
11019:
11016:
11013:
11010:
10981:Main article:
10978:
10975:
10954:
10951:
10946:
10942:
10936:
10932:
10928:
10923:
10919:
10913:
10909:
10884:
10880:
10874:
10870:
10866:
10861:
10857:
10834:
10830:
10805:
10801:
10795:
10791:
10787:
10782:
10778:
10751:
10747:
10724:
10721:
10718:
10713:
10709:
10686:
10683:
10679:
10667:
10666:
10655:
10650:
10646:
10641:
10637:
10633:
10628:
10624:
10620:
10615:
10611:
10605:
10600:
10597:
10594:
10590:
10584:
10580:
10575:
10569:
10566:
10561:
10556:
10551:
10539:
10528:
10522:
10517:
10512:
10509:
10506:
10503:
10500:
10497:
10494:
10490:
10485:
10481:
10477:
10472:
10468:
10464:
10461:
10456:
10452:
10448:
10445:
10440:
10435:
10423:
10412:
10407:
10403:
10400:
10397:
10394:
10391:
10388:
10382:
10379:
10376:
10371:
10367:
10363:
10358:
10353:
10346:
10343:
10339:
10333:
10328:
10325:
10322:
10318:
10313:
10309:
10306:
10301:
10297:
10257:
10252:
10248:
10242:
10238:
10234:
10231:
10207:
10204:
10199:
10195:
10191:
10188:
10165:
10162:
10157:
10153:
10147:
10143:
10120:
10116:
10104:
10103:
10092:
10086:
10082:
10078:
10075:
10070:
10066:
10060:
10056:
10052:
10049:
10046:
10043:
10040:
10035:
10031:
10027:
10024:
10021:
10015:
10012:
10005:
10001:
9997:
9964:
9961:
9956:
9952:
9946:
9942:
9936:
9932:
9911:
9908:
9903:
9899:
9895:
9890:
9886:
9856:
9852:
9846:
9842:
9838:
9835:
9809:
9805:
9799:
9795:
9791:
9786:
9782:
9770:
9769:
9758:
9752:
9746:
9742:
9736:
9732:
9726:
9722:
9715:
9712:
9667:
9663:
9659:
9654:
9650:
9638:
9637:
9626:
9621:
9618:
9614:
9608:
9604:
9579:
9576:
9573:
9569:
9563:
9556:
9552:
9547:
9544:
9541:
9535:
9528:
9524:
9520:
9515:
9507:
9503:
9497:
9493:
9489:
9486:
9481:
9476:
9472:
9468:
9463:
9459:
9455:
9452:
9449:
9443:
9435:
9429:
9426:
9423:
9419:
9414:
9411:
9408:
9405:
9402:
9398:
9394:
9389:
9385:
9381:
9378:
9374:
9369:
9364:
9360:
9349:
9335:
9331:
9327:
9322:
9314:
9310:
9304:
9300:
9296:
9293:
9288:
9283:
9279:
9275:
9270:
9266:
9262:
9259:
9256:
9250:
9240:
9236:
9230:
9226:
9222:
9217:
9213:
9209:
9206:
9203:
9199:
9194:
9189:
9185:
9159:
9158:
9142:
9139:
9134:
9131:
9124:
9120:
9109:
9096:
9091:
9087:
9083:
9078:
9074:
9070:
9064:
9059:
9055:
9049:
9045:
9041:
9036:
9032:
9026:
9022:
9018:
9015:
9008:
9004:
8998:
8994:
8988:
8984:
8978:
8974:
8960:
8944:
8941:
8935:
8930:
8926:
8920:
8916:
8912:
8907:
8903:
8897:
8893:
8889:
8886:
8881:
8876:
8872:
8868:
8863:
8859:
8855:
8850:
8846:
8840:
8836:
8822:
8814:
8803:
8792:
8781:
8770:
8754:
8749:
8744:
8740:
8734:
8729:
8714:
8707:
8700:
8680:
8676:
8649:
8645:
8633:
8632:
8621:
8617:
8613:
8610:
8605:
8602:
8595:
8591:
8587:
8584:
8579:
8575:
8571:
8566:
8562:
8558:
8552:
8547:
8543:
8537:
8533:
8529:
8524:
8520:
8514:
8510:
8506:
8503:
8496:
8492:
8486:
8482:
8476:
8472:
8466:
8462:
8455:
8452:
8449:
8443:
8438:
8434:
8428:
8424:
8420:
8415:
8411:
8405:
8401:
8397:
8394:
8391:
8386:
8381:
8377:
8373:
8368:
8364:
8360:
8355:
8351:
8345:
8341:
8334:
8330:
8325:
8320:
8316:
8310:
8306:
8302:
8296:
8292:
8283:
8279:
8273:
8269:
8262:
8258:
8252:
8249:
8244:
8240:
8236:
8231:
8227:
8201:
8197:
8193:
8188:
8184:
8158:
8154:
8148:
8144:
8140:
8137:
8126:
8125:
8114:
8111:
8108:
8102:
8097:
8093:
8090:
8086:
8083:
8080:
8076:
8073:
8070:
8067:
8064:
8061:
8058:
8053:
8049:
8043:
8037:
8033:
8029:
8026:
8023:
8020:
8017:
8012:
8008:
8004:
7999:
7995:
7991:
7986:
7982:
7973:
7969:
7963:
7960:
7955:
7952:
7946:
7941:
7935:
7932:
7908:
7907:
7896:
7892:
7888:
7885:
7881:
7878:
7875:
7872:
7869:
7866:
7863:
7858:
7854:
7848:
7842:
7838:
7834:
7831:
7828:
7825:
7822:
7817:
7813:
7809:
7804:
7800:
7796:
7791:
7787:
7778:
7774:
7770:
7765:
7761:
7758:
7755:
7752:
7749:
7744:
7739:
7735:
7729:
7725:
7716:
7712:
7707:
7701:
7697:
7685:
7676:
7662:
7659:
7656:
7653:
7650:
7645:
7641:
7635:
7631:
7625:
7621:
7615:
7611:
7599:
7587:
7584:
7580:
7577:
7574:
7571:
7568:
7565:
7562:
7557:
7553:
7547:
7543:
7539:
7536:
7533:
7530:
7527:
7524:
7519:
7515:
7499:Chapman–Enskog
7479:
7476:
7473:
7470:
7467:
7462:
7458:
7437:
7434:
7430:
7427:
7424:
7421:
7418:
7415:
7412:
7409:
7406:
7401:
7397:
7389:
7385:
7381:
7376:
7373:
7370:
7367:
7364:
7361:
7356:
7352:
7315:
7312:
7309:
7306:
7303:
7300:
7297:
7292:
7288:
7275:
7272:
7271:
7270:
7259:
7250:
7246:
7239:
7234:
7230:
7224:
7219:
7215:
7212:
7205:
7201:
7197:
7193:
7189:
7175:
7170:
7166:
7162:
7157:
7148:
7143:
7139:
7135:
7123:
7119:
7113:
7107:
7102:
7093:
7090:
7085:
7079:
7076:
7071:
7055:
7048:
7041:
7034:
6974:
6973:
6962:
6954:
6951:
6946:
6940:
6935:
6931:
6924:
6919:
6915:
6908:
6901:
6896:
6892:
6888:
6883:
6876:
6870:
6865:
6858:
6855:
6839:
6834:mean free path
6821:
6789:
6783:
6782:
6771:
6762:
6758:
6754:
6748:
6744:
6740:
6736:
6726:
6721:
6717:
6710:
6704:
6699:
6690:
6687:
6682:
6677:
6673:
6669:
6664:
6661:
6656:
6653:
6639:self-diffusion
6599:
6596:
6592:
6588:
6585:
6581:
6578:
6575:
6572:
6549:
6523:
6520:
6518:
6515:
6481:
6476:
6472:
6468:
6465:
6462:
6459:
6451:
6447:
6443:
6439:
6436:
6430:
6427:
6422:
6419:
6392:
6388:
6384:
6381:
6378:
6343:
6339:
6335:
6332:
6286:
6285:
6274:
6271:
6265:
6262:
6257:
6254:
6251:
6202:
6201:
6190:
6184:
6180:
6176:
6172:
6169:
6163:
6160:
6155:
6152:
6139:
6126:
6122:
6118:
6114:
6111:
6108:
6105:
6101:
6085:
6082:
6064:
6060:
6039:
6036:
6031:
6028:
6024:
6020:
6015:
6012:
6008:
5996:
5995:
5984:
5979:
5975:
5971:
5965:
5961:
5957:
5952:
5948:
5944:
5938:
5934:
5930:
5925:
5922:
5918:
5912:
5908:
5904:
5898:
5895:
5888:
5884:
5880:
5867:
5856:
5851:
5847:
5843:
5837:
5833:
5829:
5824:
5820:
5816:
5810:
5806:
5802:
5797:
5794:
5790:
5784:
5780:
5776:
5773:
5768:
5763:
5735:
5732:
5728:
5725:
5722:
5719:
5716:
5713:
5710:
5707:
5703:
5700:
5697:
5694:
5690:
5687:
5684:
5681:
5678:
5654:
5648:
5644:
5638:
5633:
5630:
5627:
5623:
5619:
5616:
5613:
5610:
5599:
5598:
5587:
5583:
5580:
5577:
5571:
5567:
5563:
5558:
5554:
5550:
5546:
5543:
5538:
5534:
5530:
5525:
5520:
5515:
5512:
5509:
5506:
5500:
5497:
5490:
5486:
5482:
5465:
5464:
5453:
5449:
5446:
5443:
5438:
5434:
5430:
5425:
5421:
5417:
5413:
5410:
5405:
5401:
5397:
5394:
5389:
5384:
5349:
5345:
5315:
5311:
5288:
5284:
5280:
5277:
5257:
5252:
5248:
5244:
5241:
5238:
5233:
5229:
5225:
5220:
5216:
5193:
5189:
5185:
5182:
5179:
5174:
5170:
5166:
5161:
5157:
5125:
5122:
5110:
5109:
5103:
5097:
5087:
5077:
5069:
5060:
5059:
5048:
5043:
5040:
5037:
5034:
5030:
5026:
5023:
5019:
5015:
5012:
5009:
5006:
5000:
4995:
4990:
4987:
4984:
4981:
4978:
4975:
4961:
4960:
4954:
4948:
4942:
4932:
4922:
4921:
4910:
4907:
4904:
4901:
4898:
4892:
4889:
4884:
4881:
4873:
4870:
4865:
4862:
4854:
4850:
4846:
4841:
4836:
4832:
4825:
4806:
4803:
4802:
4801:
4790:
4787:
4784:
4781:
4773:
4770:
4767:
4764:
4761:
4758:
4753:
4748:
4745:
4742:
4739:
4733:
4730:
4725:
4720:
4716:
4711:
4707:
4704:
4701:
4673:
4669:
4664:
4660:
4638:
4634:
4613:
4608:
4604:
4599:
4595:
4592:
4589:
4586:
4581:
4577:
4572:
4568:
4565:
4562:
4551:
4550:
4539:
4536:
4533:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4502:
4497:
4493:
4464:
4458:
4455:
4448:
4444:
4440:
4437:
4431:
4427:
4424:
4421:
4418:
4406:
4392:
4391:
4380:
4377:
4374:
4366:
4363:
4360:
4357:
4354:
4351:
4347:
4341:
4338:
4331:
4327:
4323:
4320:
4314:
4310:
4307:
4302:
4297:
4293:
4251:
4250:
4238:
4235:
4231:
4220:
4208:
4205:
4196:
4191:
4187:
4184:
4181:
4178:
4175:
4172:
4168:
4165:
4162:
4159:
4156:
4153:
4147:
4144:
4139:
4136:
4133:
4103:
4102:
4091:
4088:
4073:
4032:
4014:
4013:
4002:
3997:
3993:
3984:
3979:
3973:
3970:
3954:to an applied
3952:drift velocity
3943:
3942:Applied forces
3940:
3926:
3923:
3920:
3915:
3911:
3890:
3887:
3884:
3881:
3878:
3873:
3869:
3865:
3859:
3855:
3832:
3828:
3824:
3818:
3814:
3810:
3807:
3804:
3800:
3794:
3790:
3786:
3766:
3763:
3758:
3754:
3750:
3747:
3744:
3739:
3735:
3714:
3711:
3706:
3702:
3689:
3688:
3677:
3672:
3668:
3664:
3658:
3655:
3651:
3645:
3641:
3637:
3631:
3628:
3621:
3617:
3613:
3595:
3592:
3568:
3564:
3558:
3555:
3551:
3545:
3541:
3522:
3521:
3506:
3502:
3498:
3495:
3490:
3482:
3478:
3474:
3469:
3466:
3463:
3460:
3457:
3452:
3448:
3444:
3438:
3432:
3429:
3425:
3419:
3416:
3413:
3409:
3403:
3400:
3395:
3390:
3387:
3383:
3357:
3353:
3349:
3345:
3342:
3338:
3334:
3331:
3328:
3253:
3250:
3246:
3234:= 0, 1, 2, ...
3219:
3218:
3207:
3199:
3195:
3191:
3186:
3178:
3174:
3170:
3167:
3162:
3154:
3150:
3146:
3140:
3136:
3132:
3127:
3124:
3121:
3118:
3113:
3109:
3102:
3096:
3093:
3089:
3083:
3080:
3077:
3073:
3069:
3065:
3059:
3056:
3053:
3049:
3045:
3040:
3036:
3032:
3029:
3024:
3021:
3017:
3011:
3008:
3005:
3001:
2997:
2994:
2989:
2984:
2979:
2976:
2973:
2970:
2964:
2961:
2954:
2950:
2946:
2903:
2900:
2896:
2870:
2866:
2845:
2834:
2833:
2822:
2814:
2810:
2806:
2799:
2795:
2791:
2788:
2783:
2775:
2771:
2767:
2761:
2757:
2753:
2748:
2745:
2742:
2739:
2734:
2730:
2723:
2717:
2714:
2711:
2707:
2703:
2698:
2694:
2666:
2644:
2640:
2624:
2623:
2610:
2606:
2600:
2592:
2587:
2583:
2575:
2572:
2569:
2565:
2561:
2558:
2554:
2547:
2544:
2539:
2536:
2532:
2511:
2508:
2505:
2502:
2499:
2496:
2490:
2485:
2481:
2475:
2472:
2469:
2464:
2460:
2453:
2445:
2442:
2437:
2434:
2429:
2425:
2401:
2379:
2375:
2354:
2351:
2346:
2342:
2319:
2315:
2303:
2302:
2291:
2282:
2278:
2274:
2269:
2266:
2263:
2260:
2257:
2251:
2248:
2243:
2239:
2215:
2184:
2162:
2158:
2137:
2115:
2110:
2097:
2096:
2085:
2079:
2075:
2069:
2066:
2062:
2056:
2052:
2048:
2043:
2038:
2006:
2003:
2000:
1984:Main article:
1981:
1978:
1977:
1976:
1965:
1954:
1949:
1945:
1941:
1936:
1933:
1930:
1927:
1924:
1921:
1916:
1912:
1903:
1899:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1847:
1836:
1835:
1824:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1796:
1793:
1787:
1784:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1735:
1724:
1723:
1712:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1656:
1653:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1601:
1600:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1564:
1561:
1558:
1555:
1552:
1549:
1545:
1521:
1518:
1515:
1512:
1509:
1506:
1485:
1469:Main article:
1466:
1463:
1450:
1429:
1408:
1405:
1402:
1399:
1396:
1393:
1389:
1385:
1382:
1379:
1376:
1372:
1368:
1344:
1333:
1332:
1321:
1317:
1314:
1310:
1306:
1303:
1300:
1297:
1291:
1288:
1283:
1280:
1254:
1230:
1215:
1214:
1203:
1199:
1196:
1193:
1188:
1184:
1181:
1178:
1175:
1172:
1169:
1165:
1161:
1157:
1154:
1150:
1146:
1143:
1140:
1137:
1114:
1111:
1106:
1102:
1098:
1094:
1067:
1064:
1061:
1058:
1034:
1030:
1026:
1022:
1018:
1007:
1006:
995:
991:
988:
985:
981:
978:
975:
972:
969:
966:
963:
959:
956:
952:
948:
944:
940:
936:
933:
930:
927:
904:
901:
881:
878:
858:
834:
813:
810:
786:
774:diffusion flux
769:
766:
764:
761:
717:self-diffusion
711:In 1920–1921,
674:mean free path
627:
580:
577:
501:thermodynamics
473:starting with
451:
448:
427:carbon dioxide
390:
387:
386:
385:
371:
368:
363:
349:
333:
311:
308:
305:
302:
100:entropic force
26:
9:
6:
4:
3:
2:
12240:
12229:
12226:
12225:
12223:
12210:
12204:
12200:
12193:
12186:
12181:
12179:
12164:
12160:
12153:
12145:
12141:
12137:
12133:
12129:
12125:
12121:
12117:
12114:(4): 409–27.
12113:
12109:
12108:
12100:
12092:
12088:
12084:
12080:
12076:
12072:
12065:
12058:
12050:
12046:
12042:
12038:
12034:
12030:
12026:
12022:
12018:
12014:
12006:
11999:
11995:
11989:
11981:
11977:
11972:
11967:
11963:
11959:
11955:
11951:
11947:
11943:
11939:
11932:
11930:
11928:
11919:
11915:
11910:
11905:
11900:
11895:
11891:
11887:
11884:(3): 152–61.
11883:
11879:
11875:
11868:
11861:
11857:
11853:
11847:
11839:
11833:
11829:
11825:
11821:
11820:E.M. Lifshitz
11817:
11811:
11802:
11797:
11793:
11789:
11786:(4): 405–26.
11785:
11781:
11777:
11770:
11768:
11759:
11755:
11750:
11745:
11740:
11735:
11731:
11727:
11723:
11719:
11713:
11711:
11709:
11707:
11705:
11696:
11692:
11689:(2): 301–65.
11688:
11684:
11677:
11670:
11666:
11662:
11656:
11654:
11652:
11650:
11648:
11639:
11635:
11628:
11621:
11619:
11611:
11605:
11598:
11593:
11585:
11579:
11575:
11574:10.1142/11909
11571:
11567:
11566:
11558:
11549:
11544:
11540:
11536:
11533:(8): 549–60.
11532:
11528:
11524:
11517:
11510:
11504:
11497:
11493:
11490:
11484:
11476:
11472:
11468:
11464:
11460:
11456:
11452:
11448:
11444:
11437:
11430:
11426:
11425:J.G. Kirkwood
11421:
11417:
11400:
11397:
11394:
11391:
11388:
11385:
11382:
11379:
11376:
11373:
11371:
11368:
11366:
11363:
11360:
11357:
11354:
11351:
11350:
11340:
11337:
11334:
11331:
11326:
11323:
11322:
11320:
11317:
11314:
11311:
11308:
11304:
11301:
11298:
11295:
11292:
11289:
11286:
11283:
11281:
11278:
11276:
11273:
11270:
11266:
11263:
11260:
11257:
11255:
11252:
11249:
11246:
11243:
11242:Itō diffusion
11240:
11237:
11236:Heat equation
11234:
11232:
11228:
11225:
11222:
11219:
11217:
11213:
11209:
11206:
11203:
11200:
11197:
11194:
11191:
11188:
11185:
11182:
11179:
11176:
11175:
11169:
11166:
11163:In contrast,
11161:
11159:
11155:
11150:
11141:
11134:
11113:
11111:
11107:
11103:
11095:
11090:
11081:
11072:
11070:
11066:
11062:
11058:
11039:
11035:
11028:
11024:
11017:
11014:
11011:
11008:
11001:
11000:
10999:
10998:
10993:
10991:
10984:
10969:
10965:
10952:
10949:
10944:
10940:
10934:
10930:
10926:
10921:
10917:
10911:
10907:
10898:
10882:
10878:
10872:
10868:
10864:
10859:
10855:
10832:
10828:
10819:
10803:
10799:
10793:
10789:
10785:
10780:
10776:
10767:
10749:
10745:
10719:
10711:
10707:
10684:
10681:
10677:
10653:
10648:
10639:
10635:
10631:
10626:
10622:
10613:
10609:
10603:
10598:
10595:
10592:
10588:
10582:
10578:
10573:
10567:
10564:
10559:
10554:
10540:
10526:
10520:
10510:
10504:
10501:
10498:
10483:
10479:
10475:
10470:
10466:
10459:
10454:
10450:
10443:
10438:
10424:
10410:
10405:
10398:
10395:
10392:
10377:
10369:
10365:
10361:
10356:
10344:
10341:
10337:
10331:
10326:
10323:
10320:
10316:
10311:
10307:
10304:
10299:
10295:
10287:
10286:
10285:
10283:
10278:
10274:
10269:
10250:
10246:
10240:
10236:
10221:
10202:
10197:
10193:
10177:
10163:
10160:
10155:
10151:
10145:
10141:
10118:
10114:
10090:
10084:
10080:
10076:
10068:
10064:
10058:
10054:
10044:
10038:
10033:
10029:
10019:
10013:
10003:
9999:
9985:
9984:
9983:
9982:
9978:
9962:
9959:
9954:
9950:
9944:
9940:
9934:
9930:
9909:
9906:
9901:
9897:
9893:
9888:
9884:
9875:
9870:
9854:
9850:
9844:
9840:
9836:
9833:
9825:
9807:
9803:
9797:
9793:
9789:
9784:
9780:
9756:
9750:
9744:
9740:
9734:
9730:
9724:
9720:
9713:
9710:
9703:
9702:
9701:
9699:
9694:
9692:
9688:
9684:
9679:
9661:
9652:
9648:
9624:
9619:
9616:
9612:
9606:
9602:
9577:
9574:
9571:
9567:
9561:
9554:
9550:
9545:
9542:
9539:
9533:
9526:
9522:
9518:
9513:
9505:
9501:
9495:
9491:
9487:
9484:
9474:
9470:
9466:
9461:
9457:
9450:
9447:
9441:
9427:
9424:
9421:
9417:
9412:
9409:
9396:
9387:
9383:
9379:
9376:
9372:
9367:
9362:
9358:
9350:
9333:
9329:
9325:
9320:
9312:
9308:
9302:
9298:
9294:
9291:
9281:
9277:
9273:
9268:
9264:
9257:
9254:
9248:
9238:
9228:
9224:
9220:
9215:
9211:
9204:
9201:
9197:
9192:
9187:
9183:
9175:
9174:
9173:
9170:
9168:
9164:
9156:
9140:
9132:
9129:
9122:
9118:
9110:
9089:
9085:
9081:
9076:
9072:
9057:
9053:
9047:
9043:
9039:
9034:
9030:
9024:
9020:
9013:
9006:
9002:
8996:
8992:
8986:
8982:
8976:
8972:
8961:
8958:
8957:barodiffusion
8942:
8928:
8924:
8918:
8914:
8910:
8905:
8901:
8895:
8891:
8884:
8874:
8870:
8866:
8861:
8857:
8848:
8844:
8838:
8834:
8823:
8820:
8813:
8809:
8802:
8798:
8791:
8787:
8780:
8776:
8769:
8752:
8747:
8742:
8738:
8732:
8720:
8719:
8718:
8713:
8706:
8699:
8694:
8678:
8674:
8665:
8647:
8643:
8619:
8615:
8611:
8603:
8600:
8593:
8589:
8585:
8577:
8573:
8569:
8564:
8560:
8545:
8541:
8535:
8531:
8527:
8522:
8518:
8512:
8508:
8501:
8494:
8490:
8484:
8480:
8474:
8470:
8464:
8460:
8453:
8450:
8436:
8432:
8426:
8422:
8418:
8413:
8409:
8403:
8399:
8392:
8389:
8379:
8375:
8371:
8366:
8362:
8353:
8349:
8343:
8339:
8332:
8328:
8323:
8318:
8314:
8308:
8300:
8294:
8290:
8281:
8277:
8271:
8267:
8260:
8256:
8250:
8247:
8242:
8238:
8234:
8229:
8225:
8217:
8216:
8215:
8199:
8195:
8191:
8186:
8182:
8172:
8156:
8152:
8146:
8142:
8138:
8135:
8112:
8109:
8106:
8095:
8091:
8088:
8084:
8081:
8078:
8071:
8068:
8065:
8062:
8059:
8051:
8047:
8041:
8035:
8024:
8021:
8018:
8010:
8006:
8002:
7997:
7993:
7984:
7980:
7971:
7967:
7961:
7958:
7953:
7950:
7939:
7933:
7930:
7921:
7920:
7919:
7917:
7914:and pressure
7913:
7894:
7890:
7886:
7883:
7876:
7873:
7870:
7867:
7864:
7856:
7852:
7846:
7840:
7829:
7826:
7823:
7815:
7811:
7807:
7802:
7798:
7789:
7785:
7776:
7772:
7768:
7763:
7756:
7753:
7750:
7742:
7737:
7733:
7727:
7723:
7714:
7710:
7705:
7699:
7695:
7686:
7683:
7679:
7657:
7654:
7651:
7643:
7639:
7633:
7629:
7623:
7619:
7613:
7609:
7600:
7585:
7582:
7575:
7572:
7569:
7566:
7563:
7555:
7551:
7545:
7541:
7537:
7531:
7528:
7525:
7517:
7513:
7504:
7503:
7502:
7500:
7495:
7493:
7474:
7471:
7468:
7460:
7456:
7435:
7432:
7425:
7422:
7419:
7416:
7413:
7407:
7404:
7399:
7395:
7387:
7383:
7379:
7374:
7368:
7365:
7362:
7354:
7350:
7341:
7337:
7333:
7329:
7310:
7307:
7304:
7301:
7298:
7290:
7286:
7257:
7248:
7232:
7228:
7217:
7210:
7203:
7199:
7195:
7191:
7187:
7168:
7164:
7160:
7155:
7141:
7137:
7133:
7121:
7117:
7111:
7100:
7091:
7088:
7083:
7069:
7061:
7060:
7059:
7054:
7047:
7040:
7033:
7028:
7026:
7022:
7018:
7014:
7010:
7006:
7003: =
7002:
6999:
6998:ideal gas law
6995:
6991:
6987:
6983:
6979:
6960:
6952:
6949:
6944:
6933:
6929:
6922:
6917:
6913:
6906:
6899:
6894:
6890:
6886:
6881:
6874:
6863:
6856:
6853:
6846:
6845:
6844:
6842:
6835:
6819:
6811:
6807:
6803:
6799:
6795:
6788:
6769:
6760:
6756:
6752:
6746:
6742:
6738:
6734:
6724:
6719:
6715:
6708:
6697:
6688:
6685:
6680:
6675:
6671:
6667:
6662:
6659:
6654:
6651:
6644:
6643:
6642:
6640:
6636:
6632:
6627:
6625:
6621:
6617:
6613:
6597:
6590:
6586:
6579:
6576:
6573:
6570:
6563:
6547:
6534:
6529:
6514:
6512:
6508:
6503:
6501:
6497:
6492:
6479:
6474:
6470:
6463:
6460:
6457:
6449:
6445:
6437:
6434:
6428:
6420:
6406:
6403:
6390:
6386:
6382:
6379:
6376:
6368:
6365:
6363:
6360: =
6359:
6341:
6337:
6333:
6330:
6321:
6320:
6317: =
6316:
6312:
6309:
6307:
6303:
6299:
6295:
6291:
6272:
6263:
6260:
6255:
6252:
6249:
6242:
6241:
6240:
6238:
6234:
6229:
6225:
6223:
6219:
6215:
6211:
6207:
6188:
6182:
6178:
6170:
6167:
6161:
6153:
6140:
6124:
6120:
6112:
6109:
6106:
6103:
6091:
6090:
6089:
6081:
6078:
6062:
6058:
6037:
6034:
6029:
6026:
6022:
6018:
6013:
6010:
6006:
5977:
5973:
5963:
5959:
5955:
5950:
5946:
5936:
5932:
5923:
5920:
5916:
5910:
5906:
5902:
5896:
5886:
5882:
5868:
5849:
5845:
5835:
5831:
5827:
5822:
5818:
5808:
5804:
5795:
5792:
5788:
5782:
5778:
5774:
5771:
5766:
5752:
5751:
5750:
5747:
5733:
5726:
5723:
5717:
5714:
5711:
5701:
5698:
5695:
5685:
5682:
5679:
5668:
5652:
5646:
5642:
5636:
5631:
5628:
5625:
5621:
5617:
5614:
5611:
5608:
5585:
5578:
5569:
5565:
5561:
5556:
5552:
5544:
5536:
5532:
5528:
5523:
5513:
5510:
5507:
5504:
5498:
5488:
5484:
5470:
5469:
5468:
5451:
5444:
5436:
5432:
5428:
5423:
5419:
5411:
5403:
5399:
5395:
5392:
5387:
5373:
5372:
5371:
5369:
5365:
5347:
5343:
5333:
5331:
5313:
5309:
5286:
5282:
5278:
5275:
5255:
5250:
5246:
5242:
5239:
5236:
5231:
5227:
5223:
5218:
5214:
5191:
5187:
5183:
5180:
5177:
5172:
5168:
5164:
5159:
5155:
5145:
5142:
5138:
5130:
5121:
5119:
5115:
5107:
5104:
5101:
5098:
5095:
5091:
5088:
5085:
5081:
5078:
5075:
5068:
5065:
5064:
5063:
5038:
5035:
5028:
5024:
5021:
5017:
5013:
5010:
5004:
4993:
4988:
4985:
4979:
4973:
4966:
4965:
4964:
4958:
4955:
4952:
4949:
4946:
4943:
4940:
4936:
4933:
4930:
4927:
4926:
4925:
4905:
4899:
4896:
4890:
4887:
4882:
4879:
4871:
4868:
4863:
4860:
4852:
4848:
4844:
4839:
4834:
4830:
4823:
4816:
4815:
4814:
4812:
4788:
4768:
4765:
4756:
4743:
4737:
4731:
4718:
4714:
4709:
4705:
4689:
4688:
4687:
4671:
4667:
4662:
4658:
4636:
4632:
4606:
4602:
4597:
4593:
4587:
4584:
4579:
4575:
4570:
4566:
4563:
4560:
4537:
4520:
4517:
4511:
4505:
4495:
4483:
4482:
4481:
4479:
4462:
4456:
4453:
4446:
4442:
4438:
4435:
4429:
4425:
4422:
4419:
4416:
4405:
4401:
4397:
4378:
4361:
4358:
4352:
4345:
4339:
4336:
4329:
4325:
4321:
4318:
4312:
4308:
4305:
4295:
4283:
4282:
4281:
4278:
4274:
4272:
4268:
4264:
4260:
4256:
4236:
4229:
4221:
4194:
4189:
4185:
4179:
4176:
4166:
4163:
4160:
4157:
4154:
4145:
4142:
4137:
4134:
4131:
4123:
4122:
4121:
4118:
4116:
4112:
4108:
4100:
4097:
4096:
4095:
4087:
4061:
4056:
4054:
4050:
4046:
4042:
4038:
4031:
4027:
4023:
4019:
4000:
3995:
3991:
3982:
3977:
3971:
3968:
3961:
3960:
3959:
3957:
3953:
3949:
3939:
3921:
3913:
3909:
3888:
3885:
3879:
3871:
3867:
3857:
3853:
3830:
3826:
3816:
3812:
3808:
3805:
3798:
3792:
3788:
3764:
3761:
3756:
3752:
3748:
3745:
3742:
3737:
3733:
3712:
3709:
3704:
3700:
3675:
3670:
3666:
3656:
3653:
3649:
3643:
3639:
3635:
3629:
3619:
3615:
3601:
3600:
3599:
3591:
3588:
3584:
3566:
3562:
3556:
3553:
3549:
3543:
3539:
3529:
3527:
3504:
3500:
3496:
3493:
3488:
3480:
3476:
3464:
3461:
3458:
3450:
3446:
3430:
3427:
3423:
3417:
3414:
3411:
3407:
3401:
3398:
3393:
3388:
3385:
3381:
3373:
3372:
3371:
3355:
3351:
3340:
3336:
3332:
3326:
3318:
3314:
3311:
3306:
3302:
3298:
3293:
3289:
3285:
3281:
3277:
3273:
3269:
3251:
3248:
3244:
3233:
3229:
3225:
3205:
3197:
3193:
3184:
3176:
3172:
3168:
3165:
3160:
3152:
3148:
3138:
3134:
3122:
3116:
3111:
3094:
3091:
3087:
3081:
3078:
3075:
3071:
3067:
3063:
3057:
3054:
3051:
3047:
3043:
3038:
3034:
3030:
3027:
3022:
3019:
3015:
3009:
3006:
3003:
2999:
2995:
2992:
2987:
2977:
2974:
2971:
2968:
2962:
2952:
2948:
2934:
2933:
2932:
2929:
2927:
2923:
2919:
2901:
2898:
2894:
2886:
2868:
2864:
2843:
2820:
2812:
2808:
2797:
2793:
2789:
2786:
2781:
2773:
2769:
2759:
2755:
2743:
2737:
2732:
2715:
2712:
2709:
2705:
2701:
2696:
2692:
2684:
2683:
2682:
2679:
2664:
2642:
2638:
2629:
2608:
2604:
2590:
2585:
2581:
2573:
2570:
2567:
2563:
2559:
2556:
2545:
2542:
2537:
2534:
2509:
2503:
2500:
2497:
2488:
2483:
2479:
2470:
2467:
2462:
2458:
2451:
2443:
2440:
2432:
2427:
2423:
2415:
2414:
2413:
2399:
2377:
2373:
2352:
2349:
2344:
2340:
2317:
2313:
2289:
2280:
2276:
2264:
2258:
2246:
2241:
2237:
2229:
2228:
2227:
2213:
2205:
2200:
2198:
2182:
2160:
2156:
2135:
2113:
2083:
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2067:
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2060:
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2026:
2025:
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1947:
1943:
1931:
1928:
1925:
1919:
1914:
1901:
1897:
1893:
1887:
1884:
1881:
1875:
1865:
1864:
1863:
1861:
1822:
1813:
1810:
1807:
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727:Yakov Frenkel
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721:radioisotopes
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612:Thomas Graham
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597:stained glass
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12162:
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11269:hydrodynamic
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11099:
11094:hemodialysis
11078:
11068:
11064:
11056:
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10768:th species,
10765:
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6313:
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6297:
6294:permeability
6289:
6287:
6230:
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6217:
6213:
6209:
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6087:
6084:Porous media
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3317:free entropy
3312:
3307:
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3296:
3287:
3283:
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3231:
3227:
3223:
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2884:
2835:
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2304:
2201:
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2022:Lars Onsager
2019:
1989:
1837:
1725:
1604:
1602:
1474:
1356:
1334:
1216:
1008:
824:with normal
773:
771:
749:Henry Eyring
747:
736:
731:interstitial
725:
710:
667:
663:
653:Robert Boyle
651:
632:
621:
617:
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573:net movement
572:
561:
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520:Robert Brown
511:
509:
488:
486:
478:
466:
464:
436:
419:
399:
392:
275:
264:
245:
240:
232:
230:
199:
191:data science
160:
115:
114:
98:
94:
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81:
77:
11942:Soft Matter
11816:L.D. Landau
11718:A.N. Gorban
11285:Random walk
11254:Lévy flight
11229:, used for
11214:called the
11186:, in solids
9640:The number
6802:temperature
6233:Darcy's law
1084:vector area
739:Carl Wagner
645:(1822) and
601:earthenware
568:probability
557:diffusivity
480:random walk
469:: either a
423:capillaries
395:bulk motion
267:Fick's laws
254:. The term
248:random walk
210:temperature
12168:2017-06-12
12019:(2): 510.
11998:0198569033
11860:0815320515
11669:052140844X
11640:(1): 1–32.
11413:References
11208:Electronic
11158:convection
11110:glomerulus
9153:describes
6526:See also:
635:Adolf Fick
563:Biologists
256:convection
241:diffundere
187:statistics
136:statistics
12228:Diffusion
12144:129080402
12136:1096-9837
12041:1539-1663
11739:1012.2908
11683:Chem. Rev
11527:Ann. Phys
11467:0007-0971
11309:in solids
11033:∂
11022:∂
11015:−
10931:∑
10908:∑
10883:ρ
10869:ρ
10632:−
10589:∑
10565:ρ
10502:
10493:∇
10476:−
10447:∇
10396:
10387:∇
10317:∑
10308:−
10271:In 1948,
10237:ρ
10230:∇
10194:ρ
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10142:∑
10055:ρ
10048:∇
10030:ρ
10023:∇
10011:∂
10000:ρ
9996:∂
9941:ρ
9931:∑
9907:−
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9834:ρ
9781:ρ
9751:ρ
9731:ρ
9721:∑
9662:ν
9620:ν
9617:−
9603:κ
9575:−
9572:ν
9551:κ
9488:π
9425:−
9422:ν
9413:−
9404:Γ
9397:ν
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9163:diffusion
9138:∇
9082:−
8940:∇
8867:−
8728:∇
8609:∇
8570:−
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8372:−
8305:∇
8251:−
8235:−
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8003:−
7968:∫
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7773:∫
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7610:∑
7542:∫
7492:diffusion
7396:∫
7118:π
6950:π
6887:π
6854:ℓ
6820:ℓ
6716:π
6668:ℓ
6633:and mass
6595:∂
6584:∂
6577:−
6507:Zeldovich
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6464:⋅
6461:ν
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6442:Δ
6426:∂
6418:∂
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6377:υ
6342:γ
6334:∼
6302:viscosity
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5828:−
5815:∇
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5775:−
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5699:−
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5562:−
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5429:−
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5181:…
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5022:−
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4872:μ
4864:−
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4763:∇
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4719:⊖
4700:∂
4672:⊖
4637:⊖
4607:⊖
4580:⊖
4518:μ
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4512:−
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4426:
4359:μ
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4322:−
4319:μ
4309:
4237:φ
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4180:
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3823:Δ
3803:∂
3785:∂
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3640:∑
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3612:∂
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3505:∗
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3190:Δ
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3108:∂
3079:≥
3072:∑
3068:−
3055:≥
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3031:
3007:≥
3000:∑
2996:−
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2972:−
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2945:∂
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2805:∇
2798:∗
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2639:μ
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2273:∂
2256:∂
2250:∇
2051:∑
2020:In 1931,
2002:∇
1999:−
1940:∂
1911:∂
1898:∑
1873:Δ
1846:Δ
1799:Δ
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1760:∂
1686:∇
1667:⋅
1664:∇
1652:∂
1629:∂
1567:∇
1551:−
1428:ν
1388:ν
1305:⋅
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1299:−
1287:∂
1279:∂
1219:dimension
1192:Δ
1183:Δ
1168:Δ
1156:Δ
1136:Δ
1110:Δ
1105:ν
1093:Δ
1063:⋯
1029:ν
984:Δ
977:Δ
962:Δ
955:Δ
947:ν
926:Δ
900:Δ
892:per time
877:Δ
833:ν
809:Δ
715:measured
668:In 1858,
647:Ohm's law
633:In 1855,
539:chemistry
532:ensembles
510:From the
467:diffusion
346:dimension
285:impulsive
252:advection
233:diffusion
231:The word
183:economics
179:sociology
171:chemistry
156:marketing
116:Diffusion
43:dissolved
12222:Category
12049:46200956
11980:27396746
11918:16587950
11822:(1980).
11758:18961678
11492:Archived
11381:Sorption
11346:See also
11202:Effusion
11084:Dialysis
8799:because
7326:, where
6810:pressure
6610:, where
6323:and for
5141:catalyst
5108:is time.
4959:is time.
4624:, where
4478:activity
2522:because
2206:density
497:gradient
324:, where
214:distance
206:pressure
202:gradient
12116:Bibcode
12079:Bibcode
12021:Bibcode
11971:5476231
11950:Bibcode
11909:1076553
11886:Bibcode
11788:Bibcode
11535:Bibcode
11475:5969933
11387:Osmosis
11325:Kinesis
8788:to low
7680:is the
7497:In the
6832:is the
6808:is the
6800:is the
6792:is the
6300:is the
6292:is the
5082:is the
5072:is the
4398:is the
4051:is the
4043:is the
4035:is the
4020:is the
2204:entropy
2175:is the
1858:is the
1607:is the
1078:is the
1045:is the
410:alveoli
344:is the
195:finance
175:biology
163:physics
152:finance
95:appears
86:Bottom:
82:Middle:
18:Diffuse
12205:
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12134:
12047:
12039:
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11968:
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11667:
11580:
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11465:
11055:where
10972:holes.
10669:Here,
10106:where
9772:where
8635:where
8128:where
6836:, and
6785:where
6455:
6288:where
6235:: the
6204:where
5998:where
5062:where
4924:where
4394:where
4047:, and
4016:where
3308:Under
3203:
2920:) and
2818:
2626:where
2449:
2305:where
2099:where
1961:
1838:where
1820:
1603:where
1335:where
1009:where
719:using
283:of an
239:word,
193:, and
154:, and
74:solute
12140:S2CID
12067:(PDF)
12045:S2CID
11754:S2CID
11734:arXiv
11630:(PDF)
11319:Taxis
6988:as 1/
5139:of a
4253:Here
3956:force
3845:. If
1125:then
439:heart
432:blood
402:lungs
237:Latin
208:, or
91:force
12203:ISBN
12132:ISSN
12037:ISSN
11994:ISBN
11976:PMID
11914:PMID
11856:ISBN
11832:ISBN
11665:ISBN
11578:ISBN
11471:PMID
11463:ISSN
10277:form
9167:bulk
6996:the
6304:and
4115:mole
3946:The
3886:<
2501:>
1217:The
1049:and
800:area
741:and
692:and
603:and
593:iron
541:and
503:and
493:flux
78:Top:
48:flux
12124:doi
12087:doi
12029:doi
11966:PMC
11958:doi
11904:PMC
11894:doi
11796:doi
11744:doi
11691:doi
11570:doi
11543:doi
11455:doi
7019:as
7005:RnT
6980:as
5511:div
5370:):
4423:exp
4306:exp
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2195:th
915:is
599:or
537:In
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10399:T
10390:(
10381:)
10378:T
10375:(
10370:i
10366:D
10362:+
10357:j
10352:d
10345:j
10342:i
10338:D
10332:N
10327:1
10324:=
10321:j
10312:(
10305:=
10300:i
10296:v
10282:N
10280:(
10256:)
10251:i
10247:v
10241:i
10233:(
10220:i
10206:)
10203:V
10198:i
10190:(
10164:0
10161:=
10156:i
10152:W
10146:i
10119:i
10115:W
10091:,
10085:i
10081:W
10077:=
10074:)
10069:i
10065:v
10059:i
10051:(
10045:+
10042:)
10039:V
10034:i
10026:(
10020:+
10014:t
10004:i
9977:i
9963:0
9960:=
9955:i
9951:v
9945:i
9935:i
9910:V
9902:i
9898:C
9894:=
9889:i
9885:v
9874:i
9855:i
9845:i
9837:=
9824:i
9808:i
9804:n
9798:i
9794:m
9790:=
9785:i
9757:.
9745:i
9741:C
9735:i
9725:i
9714:=
9711:V
9698:V
9691:n
9687:n
9683:T
9666:)
9658:(
9653:1
9649:A
9625:.
9613:r
9578:1
9568:2
9562:)
9546:T
9543:k
9540:2
9534:(
9527:2
9523:/
9519:1
9514:]
9506:2
9502:m
9496:1
9492:m
9485:2
9480:)
9475:2
9471:m
9467:+
9462:1
9458:m
9454:(
9451:T
9448:k
9442:[
9434:)
9428:1
9418:2
9410:3
9407:(
9401:)
9393:(
9388:1
9384:A
9380:n
9377:8
9373:3
9368:=
9359:D
9334:2
9330:/
9326:1
9321:]
9313:2
9309:m
9303:1
9299:m
9292:2
9287:)
9282:2
9278:m
9274:+
9269:1
9265:m
9261:(
9258:T
9255:k
9249:[
9239:2
9235:)
9229:2
9225:d
9221:+
9216:1
9212:d
9208:(
9205:n
9202:2
9198:3
9193:=
9184:D
9141:T
9133:T
9130:1
9123:T
9119:k
9095:)
9090:2
9086:F
9077:1
9073:F
9069:(
9063:)
9058:2
9054:n
9048:2
9044:m
9040:+
9035:1
9031:n
9025:1
9021:m
9017:(
9014:P
9007:2
9003:n
8997:2
8993:m
8987:1
8983:n
8977:1
8973:m
8959:;
8943:P
8934:)
8929:2
8925:n
8919:2
8915:m
8911:+
8906:1
8902:n
8896:1
8892:m
8888:(
8885:n
8880:)
8875:1
8871:m
8862:2
8858:m
8854:(
8849:2
8845:n
8839:1
8835:n
8819:n
8817:/
8815:1
8812:n
8808:n
8806:/
8804:2
8801:n
8797:n
8795:/
8793:2
8790:n
8786:n
8784:/
8782:2
8779:n
8775:n
8773:/
8771:1
8768:n
8753:)
8748:n
8743:1
8739:n
8733:(
8715:2
8712:C
8710:−
8708:1
8705:C
8698:D
8679:T
8675:k
8664:i
8648:i
8644:F
8620:,
8616:}
8612:T
8604:T
8601:1
8594:T
8590:k
8586:+
8583:)
8578:2
8574:F
8565:1
8561:F
8557:(
8551:)
8546:2
8542:n
8536:2
8532:m
8528:+
8523:1
8519:n
8513:1
8509:m
8505:(
8502:P
8495:2
8491:n
8485:2
8481:m
8475:1
8471:n
8465:1
8461:m
8451:P
8442:)
8437:2
8433:n
8427:2
8423:m
8419:+
8414:1
8410:n
8404:1
8400:m
8396:(
8393:n
8390:P
8385:)
8380:1
8376:m
8367:2
8363:m
8359:(
8354:2
8350:n
8344:1
8340:n
8333:+
8329:)
8324:n
8319:1
8315:n
8309:(
8301:{
8291:D
8282:2
8278:n
8272:1
8268:n
8261:2
8257:n
8248:=
8243:2
8239:C
8230:1
8226:C
8200:2
8196:C
8187:1
8183:C
8157:i
8153:n
8147:i
8139:=
8136:n
8113:,
8110:T
8107:n
8101:B
8096:k
8092:=
8089:P
8085:;
8082:c
8079:d
8075:)
8072:t
8069:,
8066:c
8063:,
8060:x
8057:(
8052:i
8048:f
8042:2
8036:2
8032:)
8028:)
8025:t
8022:,
8019:x
8016:(
8011:i
8007:C
7998:i
7994:c
7990:(
7985:i
7981:m
7972:c
7962:n
7959:1
7954:=
7951:T
7945:B
7940:k
7934:2
7931:3
7916:P
7912:T
7895:.
7891:)
7887:c
7884:d
7880:)
7877:t
7874:,
7871:c
7868:,
7865:x
7862:(
7857:i
7853:f
7847:2
7841:2
7837:)
7833:)
7830:t
7827:,
7824:x
7821:(
7816:i
7812:C
7803:i
7799:c
7795:(
7790:i
7786:m
7777:c
7769:+
7764:2
7760:)
7757:t
7754:,
7751:x
7748:(
7743:2
7738:i
7734:C
7728:i
7724:m
7715:i
7711:n
7706:(
7700:i
7682:i
7677:i
7675:m
7673:(
7661:)
7658:t
7655:,
7652:x
7649:(
7644:i
7640:C
7634:i
7630:n
7624:i
7620:m
7614:i
7586:c
7583:d
7579:)
7576:t
7573:,
7570:c
7567:,
7564:x
7561:(
7556:i
7552:f
7546:c
7538:=
7535:)
7532:t
7529:,
7526:x
7523:(
7518:i
7514:n
7478:)
7475:t
7472:,
7469:x
7466:(
7461:i
7457:C
7436:c
7433:d
7429:)
7426:t
7423:,
7420:c
7417:,
7414:x
7411:(
7408:f
7405:c
7400:c
7388:i
7384:n
7380:1
7375:=
7372:)
7369:t
7366:,
7363:x
7360:(
7355:i
7351:C
7340:i
7336:c
7332:x
7328:t
7314:)
7311:t
7308:,
7305:c
7302:,
7299:x
7296:(
7291:i
7287:f
7258:,
7249:2
7245:)
7238:B
7233:d
7229:+
7223:A
7218:d
7214:(
7211:P
7204:2
7200:/
7196:3
7192:T
7188:4
7174:B
7169:m
7165:2
7161:1
7156:+
7147:A
7142:m
7138:2
7134:1
7122:3
7112:3
7106:B
7101:k
7092:3
7089:2
7084:=
7078:B
7075:A
7070:D
7056:B
7053:d
7049:A
7046:d
7042:B
7039:m
7035:A
7032:m
7025:n
7021:T
7017:T
7013:n
7009:n
7001:P
6994:P
6990:P
6986:P
6982:T
6978:T
6961:.
6953:m
6945:T
6939:B
6934:k
6930:8
6923:=
6918:T
6914:v
6907:,
6900:P
6895:2
6891:d
6882:2
6875:T
6869:B
6864:k
6857:=
6840:T
6838:v
6806:P
6798:T
6790:B
6787:k
6770:,
6761:2
6757:d
6753:P
6747:2
6743:/
6739:3
6735:T
6725:m
6720:3
6709:3
6703:B
6698:k
6689:3
6686:2
6681:=
6676:T
6672:v
6663:3
6660:1
6655:=
6652:D
6637:(
6635:m
6631:d
6624:x
6620:n
6612:J
6598:x
6591:/
6587:n
6580:D
6574:=
6571:J
6548:D
6511:m
6500:m
6480:,
6475:m
6471:n
6450:m
6446:n
6438:D
6435:=
6429:t
6421:n
6387:/
6383:q
6380:=
6362:γ
6358:m
6338:n
6331:p
6315:J
6306:p
6298:μ
6290:k
6273:p
6261:k
6253:=
6250:q
6222:m
6218:m
6214:m
6210:n
6206:D
6189:,
6183:m
6179:n
6171:D
6168:=
6162:t
6154:n
6125:m
6121:n
6113:D
6104:=
6100:J
6063:0
6059:c
6038:0
6030:i
6027:j
6023:D
6019:=
6014:j
6011:i
6007:D
5983:]
5978:j
5974:c
5964:i
5960:c
5951:i
5947:c
5937:j
5933:c
5929:[
5924:j
5921:i
5917:D
5911:j
5903:=
5897:t
5887:i
5883:c
5855:]
5850:j
5846:c
5836:i
5832:c
5823:i
5819:c
5809:j
5805:c
5801:[
5796:j
5793:i
5789:D
5783:j
5772:=
5767:i
5762:J
5734:c
5727:b
5724:=
5721:)
5718:c
5712:b
5709:(
5702:c
5696:c
5689:)
5686:c
5680:b
5677:(
5667:m
5653:,
5647:i
5643:c
5637:n
5632:1
5629:=
5626:i
5615:b
5612:=
5609:z
5586:.
5582:]
5579:z
5570:i
5566:c
5557:i
5553:c
5545:z
5542:[
5537:i
5533:D
5529:=
5524:i
5519:J
5505:=
5499:t
5489:i
5485:c
5452:.
5448:]
5445:z
5437:i
5433:c
5424:i
5420:c
5412:z
5409:[
5404:i
5400:D
5393:=
5388:i
5383:J
5368:n
5364:i
5362:(
5348:i
5344:A
5330:b
5314:i
5310:c
5287:0
5283:c
5279:=
5276:z
5256:.
5251:m
5247:c
5243:,
5237:,
5232:2
5228:c
5224:,
5219:1
5215:c
5192:m
5188:A
5184:,
5178:,
5173:2
5169:A
5165:,
5160:1
5156:A
5106:t
5100:m
5096:.
5090:μ
5086:.
5080:T
5076:;
5070:B
5067:k
5047:)
5042:)
5036:m
5033:(
5029:/
5025:t
5018:e
5011:1
5008:(
5005:T
4999:B
4994:k
4986:=
4983:)
4980:t
4977:(
4974:D
4957:t
4951:F
4945:m
4941:.
4935:μ
4929:x
4909:)
4906:t
4903:(
4900:F
4897:+
4891:t
4888:d
4883:x
4880:d
4869:1
4861:=
4853:2
4849:t
4845:d
4840:x
4835:2
4831:d
4824:m
4789:.
4786:]
4783:)
4780:)
4772:(
4760:(
4757:a
4752:m
4747:[
4738:=
4732:t
4724:)
4715:n
4710:/
4706:n
4703:(
4668:n
4663:/
4659:n
4633:n
4612:)
4603:n
4598:/
4594:n
4591:(
4588:o
4585:+
4576:n
4571:/
4567:n
4564:=
4561:a
4538:.
4535:)
4532:)
4524:(
4521:+
4509:(
4506:a
4501:m
4496:=
4492:J
4463:)
4457:T
4454:R
4447:0
4430:(
4420:=
4417:a
4407:0
4404:μ
4396:μ
4379:,
4376:)
4373:)
4365:(
4362:+
4350:(
4346:)
4340:T
4337:R
4330:0
4313:(
4301:m
4296:=
4292:J
4271:φ
4267:q
4263:n
4259:T
4255:R
4249:.
4230:q
4219:.
4207:)
4204:)
4195:n
4190:/
4186:n
4183:(
4174:(
4167:T
4164:R
4158:=
4155:n
4146:n
4143:1
4138:T
4135:R
4101:.
4072:m
4060:μ
4049:q
4041:T
4033:B
4030:k
4026:μ
4018:D
4001:,
3996:q
3992:T
3987:B
3983:k
3972:=
3969:D
3925:)
3922:x
3919:(
3914:2
3910:c
3889:0
3883:)
3880:x
3877:(
3872:1
3868:c
3854:D
3831:1
3827:c
3813:D
3809:=
3806:t
3799:/
3793:2
3789:c
3765:0
3762:=
3757:n
3753:c
3749:=
3743:=
3738:2
3734:c
3713:0
3701:D
3676:.
3671:j
3667:c
3657:j
3654:i
3650:D
3644:j
3636:=
3630:t
3620:i
3616:c
3583:T
3567:j
3563:X
3557:j
3554:i
3550:L
3544:j
3524:(
3501:n
3497:=
3494:n
3489:|
3481:k
3477:n
3468:)
3465:T
3462:,
3459:n
3456:(
3451:j
3431:j
3428:i
3424:L
3418:1
3412:j
3402:T
3399:1
3394:=
3389:k
3386:i
3382:D
3356:j
3344:)
3341:T
3337:/
3333:1
3330:(
3313:T
3301:k
3297:i
3294:(
3288:i
3284:k
3280:k
3276:i
3272:k
3270:,
3268:i
3252:k
3249:i
3245:D
3232:k
3228:j
3224:i
3206:.
3198:k
3194:n
3185:]
3173:n
3169:=
3166:n
3161:|
3153:k
3149:n
3139:j
3135:n
3126:)
3123:n
3120:(
3117:s
3112:2
3095:j
3092:i
3088:L
3082:0
3076:j
3064:[
3058:0
3052:k
3044:=
3039:j
3035:X
3023:j
3020:i
3016:L
3010:0
3004:j
2993:=
2988:i
2983:J
2969:=
2963:t
2953:i
2949:n
2924:(
2902:j
2899:i
2895:L
2865:n
2844:s
2821:,
2813:k
2809:n
2794:n
2790:=
2787:n
2782:|
2774:k
2770:n
2760:i
2756:n
2747:)
2744:n
2741:(
2738:s
2733:2
2716:0
2710:k
2702:=
2697:i
2693:X
2665:i
2643:i
2628:T
2609:i
2605:n
2599:d
2591:T
2586:i
2574:1
2568:i
2557:u
2553:d
2546:T
2543:1
2538:=
2535:s
2531:d
2510:,
2507:)
2504:0
2498:i
2495:(
2489:T
2484:i
2468:=
2463:i
2459:X
2452:,
2444:T
2441:1
2433:=
2428:0
2424:X
2400:i
2378:i
2374:n
2353:u
2350:=
2345:0
2341:n
2318:i
2314:n
2290:,
2281:i
2277:n
2268:)
2265:n
2262:(
2259:s
2247:=
2242:i
2238:X
2214:s
2183:j
2161:j
2157:X
2136:i
2114:i
2109:J
2084:,
2078:j
2074:X
2068:j
2065:i
2061:L
2055:j
2047:=
2042:i
2037:J
2005:n
1964:.
1953:2
1948:i
1944:x
1935:)
1932:t
1929:,
1926:x
1923:(
1920:n
1915:2
1902:i
1894:=
1891:)
1888:t
1885:,
1882:x
1879:(
1876:n
1823:,
1817:)
1814:t
1811:,
1808:x
1805:(
1802:n
1795:D
1792:=
1786:t
1778:)
1775:t
1772:,
1769:x
1766:(
1763:n
1734:x
1711:.
1707:)
1704:)
1701:t
1698:,
1695:x
1692:(
1689:n
1682:)
1679:x
1676:(
1673:D
1670:(
1661:=
1655:t
1647:)
1644:t
1641:,
1638:x
1635:(
1632:n
1605:D
1588:,
1585:)
1582:t
1579:,
1576:x
1573:(
1570:n
1563:)
1560:x
1557:(
1554:D
1548:=
1544:J
1520:)
1517:t
1514:,
1511:x
1508:(
1505:n
1484:J
1449:x
1407:0
1404:=
1401:)
1398:)
1395:x
1392:(
1384:,
1381:)
1378:x
1375:(
1371:J
1367:(
1343:W
1320:,
1316:W
1313:+
1309:J
1296:=
1290:t
1282:n
1253:n
1229:N
1202:.
1198:)
1195:t
1187:S
1180:(
1177:o
1174:+
1171:t
1164:)
1160:S
1153:,
1149:J
1145:(
1142:=
1139:N
1113:S
1101:=
1097:S
1066:)
1060:(
1057:o
1033:)
1025:,
1021:J
1017:(
994:,
990:)
987:t
980:S
974:(
971:o
968:+
965:t
958:S
951:)
943:,
939:J
935:(
932:=
929:N
903:t
880:S
857:N
812:S
785:J
628:2
384:.
370:t
367:D
362:2
332:n
310:t
307:D
304:n
301:2
165:(
103:.
34:.
20:)
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