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64-529: Onsager is a surname. Notable people with the surname include: Lars Onsager (1903–1976), Norwegian–American physical chemist and theoretical physicist Søren Onsager (1878–1946), Norwegian painter See also [ edit ] Onsager Medal , an award in the fields of chemistry, physics and mathematics Onsager reciprocal relations , certain relations between flows and forces in thermodynamic systems [REDACTED] Surname list This page lists people with

128-518: A concentration gradient , a change in pressure over a distance is called a pressure gradient , and a change in temperature over a distance is called a temperature gradient . The word diffusion derives from the Latin word, diffundere , which means "to spread out". A distinguishing feature of diffusion is that it depends on particle random walk , and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow,

192-500: A daughter. After the trip to Europe , he was hired by Yale University , where he remained for most of the rest of his life, retiring in 1972. At Yale, he had been hired as a postdoctoral fellow, but it was discovered that he had never received a Ph.D. While he had submitted an outline of his work in reciprocal relations to the Norwegian Institute of Technology , they had decided it was too incomplete to qualify as

256-569: A doctoral dissertation. He was told that he could submit one of his published papers to the Yale faculty as a dissertation, but insisted on doing a new research project instead. His dissertation laid the mathematical background for his interpretation of deviations from Ohm's law in weak electrolytes. It dealt with the solutions of the Mathieu equation of period 4 π {\displaystyle 4\pi } and certain related functions and

320-432: 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 Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827). Robert Boyle demonstrated diffusion in solids in the 17th century by penetration of zinc into a copper coin. Nevertheless, diffusion in solids

384-477: A theoretical explanation of the superfluid properties of liquid helium in 1949; two years later the physicist Richard Feynman independently proposed the same theory. He also worked on the theories of liquid crystals and the electrical properties of ice . While on a Fulbright scholarship to the University of Cambridge , he worked on the magnetic properties of metals . He developed important ideas on

448-562: Is In case the diffusion coefficient is independent of x {\displaystyle x} , Fick's second law can be simplified to where Δ {\displaystyle \Delta } is the Laplace operator , 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

512-479: Is where ( J , ν ) {\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})} is the inner product and o ( ⋯ ) {\displaystyle o(\cdots )} is the little-o notation . If we use the notation of vector area Δ S = ν Δ S {\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S} then The dimension of

576-648: 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 statistics , probability theory , information theory , neural networks , finance , and marketing . The concept of diffusion is widely used in many fields, including physics ( particle diffusion ), chemistry , biology , sociology , economics , statistics , data science , and finance (diffusion of people, ideas, data and price values). The central idea of diffusion, however,

640-536: Is a vector J {\displaystyle \mathbf {J} } representing the quantity and direction of transfer. Given a small area Δ S {\displaystyle \Delta S} with normal ν {\displaystyle {\boldsymbol {\nu }}} , the transfer of a physical quantity N {\displaystyle N} through the area Δ S {\displaystyle \Delta S} per time Δ t {\displaystyle \Delta t}

704-407: Is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection. A gradient is the change in the value of a quantity; for example, concentration, pressure , or temperature with the change in another variable, usually distance . A change in concentration over a distance is called

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768-567: Is different from Wikidata All set index articles Lars Onsager Lars Onsager was born in Kristiania (now Oslo ), Norway . His father was a lawyer . After completing secondary school in Oslo, he attended the Norwegian Institute of Technology (NTH) in Trondheim , graduating as a chemical engineer in 1925. While there he worked through A Course of Modern Analysis , which

832-436: Is intensity of any local source of this quantity (for example, the rate of a chemical reaction). For the diffusion equation, the no-flux boundary conditions can be formulated as ( J ( x ) , ν ( x ) ) = 0 {\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0} on the boundary, where ν {\displaystyle {\boldsymbol {\nu }}}

896-412: Is the j {\displaystyle j} th thermodynamic force and L i j {\displaystyle L_{ij}} is Onsager's matrix of kinetic transport coefficients . The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s {\displaystyle s} (he used

960-528: Is the antigradient of concentration, − ∇ n {\displaystyle -\nabla n} . In 1931, Lars Onsager included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport, where J i {\displaystyle \mathbf {J} _{i}} is the flux of the i {\displaystyle i} th physical quantity (component), X j {\displaystyle X_{j}}

1024-435: Is the characteristic of advection . The term convection is used to describe the combination of both transport phenomena . If a diffusion process can be described by Fick's laws , it is called a normal diffusion (or Fickian diffusion); Otherwise, it is called an anomalous diffusion (or non-Fickian diffusion). When talking about the extent of diffusion, two length scales are used in two different scenarios: "Bulk flow"

1088-415: 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 bulk motion and diffusion occur is human breathing. First, there is a "bulk flow" process. The lungs are located in the thoracic cavity , which expands as

1152-434: Is the normal to the boundary at point x {\displaystyle x} . Fick's first law: The diffusion flux, J {\displaystyle \mathbf {J} } , is proportional to the negative gradient of spatial concentration, n ( x , t ) {\displaystyle n(x,t)} : where D is the diffusion coefficient . The corresponding diffusion equation (Fick's second law)

1216-749: Is then 4 × 4. He repeated this for up to six 1D Ising models, resulting in transfer matrices of up to 64 × 64. He diagonalized all of them and found that all the eigenvalues were of a special form, so he guessed that the algebra of the problem was an associative algebra (later called the Onsager algebra ). The solution involved generalized quaternion algebra and the theory of elliptic functions, which he learned from A Course of Modern Analysis . He remained in Florida until his death from an aneurysm in Coral Gables, Florida in 1976. Onsager

1280-458: Is universally recognized that atomic defects are necessary to mediate diffusion in crystals. Henry Eyring , with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion. The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law. Each model of diffusion expresses the diffusion flux with the use of concentrations, densities and their derivatives. Flux

1344-484: Is used to model the stellar atmospheres of chemically peculiar stars . Diffusion of the elements is critical in understanding the surface composition of degenerate white dwarf stars and their evolution over time. In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, Pliny the Elder had previously described the cementation process , which produces steel from

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1408-457: The i {\displaystyle i} th component. The corresponding driving forces are the space vectors where T is the absolute temperature and μ i {\displaystyle \mu _{i}} is the chemical potential of the i {\displaystyle i} th component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore,

1472-459: The Boltzmann equation , which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years. In 1920–1921, George de Hevesy measured self-diffusion using radioisotopes . He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead. Yakov Frenkel (sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926,

1536-496: The Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein . The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals. In chemistry and materials science , diffusion also refers to the movement of fluid molecules in porous solids. Different types of diffusion are distinguished in porous solids. Molecular diffusion occurs when

1600-692: The Eidgenössische Technische Hochschule (ETH) , where he remained until 1928. In 1928 he went to the United States to take a faculty position at the Johns Hopkins University in Baltimore, Maryland . At JHU he had to teach freshman classes in chemistry , and it quickly became apparent that, while he was a genius at developing theories in physical chemistry, he had little talent for teaching. He

1664-642: The Gunnerus Library in Trondheim. Diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential . It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition . Diffusion

1728-450: The blood in the body. Third, there is another "bulk flow" process. The pumping action of the heart then transports the blood around the body. As the left ventricle of the heart contracts, the volume decreases, which increases the pressure in the ventricle. This creates a pressure gradient between the heart and the capillaries, and blood moves through blood vessels by bulk flow down the pressure gradient. There are two ways to introduce

1792-472: The kinetic coefficients L i j {\displaystyle L_{ij}} should be symmetric ( Onsager reciprocal relations ) and positive definite ( for the entropy growth ). The transport equations are Here, all the indexes i , j , k = 0, 1, 2, ... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix D i k {\displaystyle D_{ik}} of

1856-411: The surname Onsager . If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name (s) to the link. Retrieved from " https://en.wikipedia.org/w/index.php?title=Onsager&oldid=1164069227 " Category : Surnames Hidden categories: Articles with short description Short description

1920-413: The 2D Ising model, Onsager began by diagonalizing increasingly large transfer matrices. He said that it's because he had a lot of time during WWII. He began by computing the 2 × 2 transfer matrix of the 1D Ising model, which is already solved by Ising himself. He then computed the transfer matrix of the "Ising ladder", meaning two 1D Ising models side-by-side, connected by links. The transfer matrix

1984-517: The Yale chemistry faculty. His statistical mechanics course was nicknamed "Sadistical Mechanics" by the students. His research at Brown was concerned mainly with the effects on diffusion of temperature gradients , and produced the Onsager reciprocal relations , a set of equations published in 1929 and, in an expanded form, in 1931, in statistical mechanics whose importance went unrecognized for many years. However, their value became apparent during

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2048-452: The alveoli and the blood in the capillaries that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that the concentration of carbon dioxide 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

2112-550: The application of mathematics to problems in physics and chemistry and, in a sense, could be considered to be continuing in the same areas Gibbs had pioneered. In 1947, he was elected to the National Academy of Sciences , the American Academy of Arts and Sciences in 1949, and in 1950 he joined the ranks of Alpha Chi Sigma . After World War II , Onsager researched new topics of interest. He proposed

2176-426: The cell, the probability that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) is into the cell. In other words, there is a net movement of oxygen molecules down the concentration gradient. In astronomy , atomic diffusion

2240-402: The collision with another molecule is more likely than the collision with the pore walls. Under such conditions, the diffusivity is similar to that in a non-confined space and is proportional to the mean free path. Knudsen diffusion 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

2304-416: The concept of the mean free path . In the same year, James Clerk Maxwell developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion was developed by Albert Einstein , Marian Smoluchowski and Jean-Baptiste Perrin . Ludwig Boltzmann , in the development of the atomistic backgrounds of the macroscopic transport processes , introduced

2368-682: The decades following World War II , and by 1968 they were considered important enough to gain Onsager that year's Nobel Prize in Chemistry . In 1933, when the Great Depression limited Brown's ability to support a faculty member who was only useful as a researcher and not a teacher, he was let go by Brown. He traveled to Austria to visit electrochemist Hans Falkenhagen . He met Falkenhagen's sister-in-law, Margrethe Arledter. They were married on September 7, 1933, and had three sons and

2432-462: The diffusing particles. In molecular diffusion , 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 Robert Brown , who found that minute particle suspended in a liquid medium and just large enough to be visible under an optical microscope exhibit a rapid and continually irregular motion of particles known as Brownian movement. The theory of

2496-400: The diffusion flux is proportional to the negative gradient of concentrations. It goes from regions of higher concentration to regions of lower concentration. Sometime later, various generalizations of Fick's laws were developed in the frame of thermodynamics and non-equilibrium thermodynamics . From the atomistic point of view , diffusion is considered as a result of the random walk of

2560-595: The diffusion ( i , k  > 0), thermodiffusion ( i  > 0, k  = 0 or k  > 0, i  = 0) and thermal conductivity ( i = k = 0 ) coefficients. Under isothermal conditions T  = constant. The relevant thermodynamic potential is the free energy (or the free entropy ). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, − ( 1 / T ) ∇ μ j {\displaystyle -(1/T)\,\nabla \mu _{j}} , and

2624-397: The diffusion flux is [flux] = [quantity]/([time]·[area]). The diffusing physical quantity N {\displaystyle N} may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity . For its density, n {\displaystyle n} , the diffusion equation has the form where W {\displaystyle W}

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2688-496: The element iron (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of stained glass or earthenware and Chinese ceramics . In modern science, the first systematic experimental study of diffusion was performed by Thomas Graham . He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833: "...gases of different nature, when brought into contact, do not arrange themselves according to their density,

2752-409: The first step in external respiration. This expansion leads to an increase in volume of the alveoli in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air 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

2816-541: The heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time." The measurements of Graham contributed to James Clerk Maxwell deriving, in 1867, the coefficient of diffusion for CO 2 in the air. The error rate is less than 5%. In 1855, Adolf Fick , the 26-year-old anatomy demonstrator from Zürich, proposed his law of diffusion . He used Graham's research, stating his goal as "the development of

2880-472: The idea of diffusion in crystals through local defects (vacancies and interstitial atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data. Sometime later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it

2944-657: The lower right corner of the stone. The Norwegian Institute of Technology established the Lars Onsager Lecture and The Lars Onsager Professorship in 1993 to award outstanding scientists in the scientific fields of Lars Onsager; Chemistry, Physics and Mathematics. The American Physical Society established Lars Onsager Prize in statistical physics in 1993. In 1997 his sons and daughter donated his scientific works and professional belongings to NTNU (before 1996 NTH ) in Trondheim , Norway as his alma mater. These are now organized as The Lars Onsager Archive at

3008-401: The matrix of diffusion coefficients is ( i,k  > 0). There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations ∑ j L i j X j {\textstyle \sum _{j}L_{ij}X_{j}} can be measured. For example, in

3072-433: The notion of diffusion : either a phenomenological approach starting with Fick's laws of diffusion and their mathematical consequences, or a physical and atomistic one, by considering the random walk of the diffusing particles . In the phenomenological approach, diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion . According to Fick's laws,

3136-406: The occasional outstanding one. His two courses on statistical mechanics were nicknamed "Advanced Norwegian I" and "Advanced Norwegian II" for being incomprehensible. During the late 1930s, Onsager researched the dipole theory of dielectrics , making improvements for another topic that had been studied by Peter Debye. However, when he submitted his paper to a journal that Debye edited in 1936, it

3200-404: The pore walls becomes gradually more likely and the diffusivity is lower. Finally there is configurational diffusion, which happens if the molecules have comparable size to that of the pore. Under this condition, the diffusivity is much lower compared to molecular diffusion and small differences in the kinetic diameter of the molecule cause large differences in diffusivity . Biologists often use

3264-399: The pressure of the air and that in the alveoli are equal, that is, the movement of air by bulk flow stops once there is no longer a pressure gradient. Second, there is a "diffusion" process. The air arriving in the alveoli has a higher concentration of oxygen than the "stale" air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in

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3328-769: The quantization of magnetic flux in metals. He was awarded the Lorentz Medal in 1958, Willard Gibbs Award in 1962, and the Nobel Prize in Chemistry in 1968. He was elected a member of the American Philosophical Society in 1959 and a Foreign Member of the Royal Society (ForMemRS) in 1975 . In 1972 Onsager retired from Yale and became emeritus. He then became a member of the Center for Theoretical Studies, University of Miami , and

3392-407: The term "force" in quotation marks or "driving force"): where n i {\displaystyle n_{i}} are the "thermodynamic coordinates". For the heat and mass transfer one can take n 0 = u {\displaystyle n_{0}=u} (the density of internal energy) and n i {\displaystyle n_{i}} is the concentration of

3456-421: The terms "net movement" or "net diffusion" to describe the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes so long as there is a higher concentration of oxygen outside the cell. However, because the movement of molecules is random, occasionally oxygen molecules move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside

3520-444: The terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients. For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium: where the derivatives of s {\displaystyle s} are calculated at equilibrium n ∗ {\displaystyle n^{*}} . The matrix of

3584-465: Was appointed Distinguished University Professor of Physics. At the University of Miami he remained active in guiding and inspiring postdoctoral students as his teaching skills, although not his lecturing skills, had improved during the course of his career. He developed interests in semiconductor physics, biophysics and radiation chemistry. However, his death came before he could produce any breakthroughs comparable to those of his earlier years. To solve

3648-437: Was awarded an honorary degree , doctor techn. honoris causa, at the Norwegian Institute of Technology , later part of Norwegian University of Science and Technology . In 1945, Onsager was naturalized as an American citizen, and the same year he was awarded the title of J. Willard Gibbs Professor of Theoretical Chemistry . This was particularly appropriate because Onsager, like Willard Gibbs , had been involved primarily in

3712-403: Was beyond the comprehension of the chemistry and physics faculty. Only when some members of the mathematics department, including the chairman Einar Hille (who also liked A Course of Modern Analysis ), insisted that the work was good enough that they would grant the doctorate if the chemistry department would not, was he granted a Ph.D. in chemistry in 1935. Even before the dissertation

3776-735: Was buried next to John Gamble Kirkwood at New Haven's Grove Street Cemetery . While Kirkwood's tombstone has a long list of awards and positions, including the American Chemical Society Award in Pure Chemistry, the Richards Medal, and the Lewis Award, Onsager's tombstone, in its original form, simply said "Nobel Laureate". When Onsager's wife Gretel died in 1991 and was buried there, his children added an asterisk after "Nobel Laureate" and "*etc." in

3840-536: Was dismissed by JHU after one semester. On leaving JHU, he accepted a position (involving the teaching of statistical mechanics to graduate students in chemistry) at Brown University in Providence, Rhode Island , where it became clear that he was no better at teaching advanced students than freshmen, but he made significant contributions to statistical mechanics and thermodynamics . His graduate student Raymond Fuoss worked under him and eventually joined him on

3904-399: Was finished, he was appointed assistant professor in 1934, and promoted to associate professor in 1940. He quickly showed at Yale the same traits he had at JHU and Brown: he produced brilliant theoretical research, but was incapable of giving a lecture at a level that a student (even a graduate student) could comprehend. He was also unable to direct the research of graduate students, except for

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3968-469: Was instrumental in his later work. In 1925 he arrived at a correction to the Debye-Hückel theory of electrolytic solutions , to specify Brownian movement of ions in solution, and during 1926 published it. He traveled to Zürich , where Peter Debye was teaching, and confronted Debye, telling him his theory was wrong. He impressed Debye so much that he was invited to become Debye's assistant at

4032-450: Was not systematically studied until the second part of the 19th century. William Chandler Roberts-Austen , the well-known British metallurgist and former assistant of Thomas Graham studied systematically solid state diffusion on the example of gold in lead in 1896. : "... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals." In 1858, Rudolf Clausius introduced

4096-442: Was rejected. Debye would not accept Onsager's ideas until after World War II . During the 1940s, Onsager studied the statistical-mechanical theory of phase transitions in solids , deriving a mathematically elegant theory which was enthusiastically received. In what is widely considered a tour de force of mathematical physics, he obtained the exact solution for the two dimensional Ising model in zero field in 1944. In 1960 he

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