# Online machine learning

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File:Diffusion v2 20101120.ogv

**Diffusion** is one of several transport phenomena that occur in nature. A distinguishing feature of diffusion is that it results in mixing or mass transport, without requiring bulk motion. Thus, diffusion should not be confused with convection, or advection, which are other transport mechanisms that utilize bulk motion to move particles from one place to another. In Latin, "diffundere" means "to spread out".

There are two ways to introduce 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*.^{[1]}

In the phenomenological approach, according to Fick's laws, the diffusion flux is proportional to the negative gradient of concentrations. It goes from regions of higher concentration to regions of lower concentration. Some time later, various generalizations of Fick's laws were developed in the frame of thermodynamics and non-equilibrium thermodynamics.^{[2]}

From the *atomistic point of view*, diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion, the moving molecules are self-propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein.^{[3]}

The concept of diffusion is widely used in: physics (particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas and of price values). The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals.

## History of diffusion in physics

In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, Pliny the Elder had previously described the cementation process which produces steel from the element iron (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colours 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:^{[4]}

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

The measurements of Graham contributed to James Clerk Maxwell deriving, in 1867, the coefficient of diffusion for CO_{2} in 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 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 that is similar to Fourier's law for heat conduction (1822) and Ohm's law for electrical current (1827).

Robert Boyle demonstrated diffusion in solids in the 17th century^{[5]} by penetration of Zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied till 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. :^{[6]}

"... 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 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 the Boltzmann equation, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.^{[7]}

In 1920–1921 George de Hevesy measured self-diffusion using radioisotopes. He studied self-diffusion of radioactive isotopes of lead in liquid and solid lead.

Yakov Frenkel (sometimes, Jakov/Jacov Frenkel) proposed, and elaborated in 1926, 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.

Some time later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.^{[6]}

Henry Eyring, with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion.^{[8]} The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law.^{[9]}

## Basic models of diffusion

### Diffusion flux

Each model of diffusion expresses the **diffusion flux** through concentrations, densities and their derivatives. Flux is a vector $\mathbf{J}$. The transfer of a physical quantity $N$ through a small area $\mathrm{\Delta}S$ with normal $\nu$ per time $\mathrm{\Delta}t$ is

where $(\mathbf{J},\nu )$ is the inner product and $o(...)$ is the little-o notation. If we use the notation of vector area $\mathrm{\Delta}\mathbf{S}=\nu \mathrm{\Delta}S$ then

The dimension of the diffusion flux is [flux]=[quantity]/([time]·[area]). The diffusing physical quantity $N$ may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, $n$, the diffusion equation has the form

where $W$ is intensity of any local source of this quantity (the rate of a chemical reaction, for example).
For the diffusion equation, the **no-flux boundary conditions** can be formulated as $(\mathbf{J}(x),\nu (x))=0$ on the boundary, where $\nu$ is the normal to the boundary at point $x$.

### Fick's law and equations

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Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient:

The corresponding diffusion equation (Fick's second law) is

where $\mathrm{\Delta}$ is the Laplace operator,

### Onsager's equations for multicomponent diffusion and thermodiffusion

Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, $-\mathrm{\nabla}n$.

In 1931, Lars Onsager^{[10]} included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For
multi-component transport,

where $\mathbf{J}}_{i$ is the flux of the *i*th physical quantity (component) and $X}_{j$ is the *j*th thermodynamic force.

The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density *s* (he used the term "force" in quotation marks or "driving force"):

where $n}_{i$ are the "thermodynamic coordinates".
For the heat and mass transfer one can take ${n}_{0}=u$ (the density of internal energy) and $n}_{i$ is the concentration of the *i*th component. The corresponding driving forces are the space vectors

- ${X}_{0}=\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\frac{1}{T}\phantom{\rule{0.5em}{0ex}},\phantom{\rule{0.278em}{0ex}}\phantom{\rule{0.278em}{0ex}}\phantom{\rule{0.278em}{0ex}}{X}_{i}=-\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\frac{{\mu}_{i}}{T}\phantom{\rule{0.278em}{0ex}}(i>0),$ because $\mathrm{d}}s=\frac{1}{T}\mathrm{d}u-{\sum}_{i\ge 1}\frac{{\mu}_{i}}{T}\mathrm{d}{n}_{i$

where *T* is the absolute temperature and $\mu}_{i$ is the chemical potential of the *i*th component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.

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

where the derivatives of *s* are calculated at equilibrium *n*^{*}.
The matrix of the *kinetic coefficients* $L}_{ij$ 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 $D}_{ik$of 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*=const. 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, $-(1/T)\mathrm{\nabla}{\mu}_{j}$, and 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 $\sum}_{j}{L}_{ij}{X}_{j$ can be measured. For example, in the original work of Onsager^{[10]} the thermodynamic forces include additional multiplier *T*, whereas in the Course of Theoretical Physics^{[11]} 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 effect the measurable quantities.

### Nondiagonal diffusion must be nonlinear

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

If the matrix of diffusion coefficients is diagonal then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is non-diagonal, for example, ${D}_{12}\ne 0$, and consider the state with ${c}_{2}=\dots ={c}_{n}=0$. At this state, $\partial {n}_{2}/\partial t={D}_{12}\mathrm{\Delta}{n}_{1}$. If ${D}_{12}\mathrm{\Delta}{n}_{1}(x)<0$ at some points then ${n}_{2}(x)$ becomes negative at these points in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. Non-diagonal equations of multicomponent diffusion must be non-linear.^{[9]}

### Einstein's mobility and Teorell formula

The Einstein relation (kinetic theory) connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force)^{[12]}

where
*D* is the diffusion constant; *μ* is the "mobility"; *k _{B}* is Boltzmann's constant;

*T*is the absolute temperature.

Below, to combine in the same formula the chemical potential *μ* and the mobility, we use for mobility the notation $\mathfrak{m}$.

The mobility—based approach was further applied by T. Teorell.^{[13]} In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula:

**the flux is equal to mobility×concentration×force per gram ion**.

This is the so-called *Teorell formula*.

The force under isothermal conditions consists of two parts:

- Diffusion force caused by concentration gradient: $-RT\frac{1}{n}\mathrm{\nabla}n=-RT\mathrm{\nabla}(\mathrm{ln}(n/{n}^{\mathrm{e}\mathrm{q}})).$
- Electrostatic force caused by electric potential gradient: $q\mathrm{\nabla}\phi .$

Here *R* is the gas constant, *T* is the absolute temperature, *n* is the concentration, the equilibrium concentration is marked by a superscript "eq", *q* is the charge and *φ* is the electric potential.

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

The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is^{[9]}

where *μ* is the chemical potential, *μ*_{0} is the standard value of the chemical potential.
The expression $a=\mathrm{exp}\left(\frac{\mu -{\mu}_{0}}{RT}\right)$ is the so-called activity. 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^{[9]}

The standard derivation of the activity includes a normalization factor and for small concentrations $a=n/{n}^{\ominus}+o(n/{n}^{\ominus})$, where $n}^{\ominus$ is the standard concentration. Therefore this formula for the flux describes the flux of the normalized dimensionless quantity, $n/{n}^{\ominus}$,

#### Teorell formula for multicomponent diffusion

The Teorell formula with combination of Onsager's definition of the diffusion force gives

where $\mathfrak{m}}_{\mathfrak{i}$ is the mobility of the *i*th component, $a}_{i$ is its activity, $L}_{ij$ is the matrix of the coefficients, $X}_{j$ is the themodynamic diffusion force, $X}_{j}=-\mathrm{\nabla}\frac{{\mu}_{j}}{T$. For the isothermal perfect systems, $X}_{j}=-R\frac{\mathrm{\nabla}{n}_{j}}{{n}_{j}$. Therefore, the Einstein-Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion:

where $D}_{ij$ is the matrix of coefficients. The Chapman-Enskog formulas for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.

### Jumps on the surface and in solids

Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.

The system includes several reagents $A}_{1},{A}_{2},\dots {A}_{m$ on the surface. Their surface concentrations are $c}_{1},{c}_{2},\dots {c}_{m$. The surface is a lattice of the adsorption places. Each
reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free paces is $z={c}_{0}$. The sum of all $c}_{i$ (including free places) is constant, the density of adsorption places *b*.

The jump model gives for the diffusion flux of $A}_{i$ (*i*=1,...,*n*):

The corresponding diffusion equation is:^{[9]}

Due to the conservation law, $z=b-{\displaystyle \sum _{i=1}^{n}}{c}_{i}\phantom{\rule{0.167em}{0ex}},$ and we
have the system of *m* diffusion equations. For one component we get Fick's law and linear equations because $(b-c)\mathrm{\nabla}c-c\mathrm{\nabla}(b-c)=b\mathrm{\nabla}c$. For two and more components the equations are nonlinear.

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

- ${\mathbf{J}}_{i}=-{\sum}_{j}{D}_{ij}[{c}_{j}\mathrm{\nabla}{c}_{i}-{c}_{i}\mathrm{\nabla}{c}_{j}]$
- $\frac{\partial {c}_{i}}{\partial t}={\sum}_{j}{D}_{ij}[{c}_{j}\mathrm{\Delta}{c}_{i}-{c}_{i}\mathrm{\Delta}{c}_{j}]$

where ${D}_{ij}={D}_{ji}\ge 0$ is a symmetric matrix of coefficients which characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration $c}_{0$.

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

### Diffusion in porous media

For diffusion in porous media the basic equations are:^{[14]}

- $\mathbf{J}=-D\mathrm{\nabla}{n}^{m}$
- $\frac{\partial n}{\partial t}=D\mathrm{\Delta}{n}^{m}\phantom{\rule{0.167em}{0ex}},$

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

For diffusion of gases in porous media this equation is the formalisation of Darcy's law: the velocity of a gas in the porous media is

where *k* is the permeability of the medium, *μ* is the viscosity and *p* is the pressure. The flux *J=nv* and for $p\sim {n}^{\gamma}$ Darcy's law gives the equation of diffusion in porous media with *m=γ+1*.

For underground water infiltration the Boussinesq approximation gives the same equation with *m*=2.

For plasma with the high level of radiation the Zeldovich-Raizer equation gives *m>4* for the heat transfer.

## Diffusion in physics

### Elementary theory of diffusion coefficient in gases

The diffusion coefficient $D$ is the coefficient in the Fick's first law $J=-D\partial n/\partial x$, where * J* is the diffusion flux (amount of substance) per unit area per unit time, *n* (for ideal mixtures) is the concentration, * x* is the position [length].

Let us consider two gases with molecules of the same diameter *d* and mass *m* (self-diffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient

where *k*_{B} is the Boltzmann constant, *T* is the temperature, *P* is the pressure, $\ell$ is the mean free path, and *v _{T}* is the mean thermal speed:

We can see that the diffusion coefficient in the mean free path approximation grows with *T* as *T*^{3/2} and decreases with *P* as 1/*P*. If we use for *P* the ideal gas law *P=RnT* with the total concentration *n*, then we can see that for given concentration *n* the diffusion coefficient grows with *T* as *T*^{1/2} and for given temperature it decreases with the total concentration as *1/n*.

For two different gases, A and B, with molecular masses *m*_{A}, *m*_{B} and molecular diameters *d*_{A}, *d*_{B}, the mean free path estimate of the diffusion coefficient of A in B and B in A is:

### The theory of diffusion in gases based on Boltzmann's equation

In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, ${f}_{i}(x,c,t)$, where *t* is the time moment, *x* is position and *c* is velocity of molecule of the *i*th component of the mixture. Each component has its mean velocity ${C}_{i}(x,t)=\frac{1}{{n}_{i}}{\int}_{c}cf(x,c,t)\phantom{\rule{0.167em}{0ex}}dc$. If the velocities ${C}_{i}(x,t)$ do not concide then there exists *diffusion*.

In the Chapman-Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:^{[7]}

- individual concentrations of particles, ${n}_{i}(x,t)={\int}_{c}{f}_{i}(x,c,t)\phantom{\rule{0.167em}{0ex}}dc$ (particles per volume),
- density of moment ${\sum}_{i}{m}_{i}{n}_{i}{C}_{i}(x,t)$ (
*m*is the_{i}*i*th particle mass), - density of kinetic energy ${\sum}_{i}({n}_{i}\frac{{m}_{i}{C}_{i}^{2}(x,t)}{2}+{\int}_{c}\frac{{m}_{i}({c}_{i}-{C}_{i}(x,t){)}^{2}}{2}{f}_{i}(x,c,t)\phantom{\rule{0.167em}{0ex}}dc)$.

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

- $\frac{3}{2}{k}_{\mathrm{B}}T=\frac{1}{n}{\int}_{c}\frac{{m}_{i}({c}_{i}-{C}_{i}(x,t){)}^{2}}{2}{f}_{i}(x,c,t)\phantom{\rule{0.167em}{0ex}}dc$; $P={k}_{\mathrm{B}}nT$,

where $n={\sum}_{i}{n}_{i}$ is the total density.

For two gases, the difference between velocities, $C}_{1}-{C}_{2$ is given by the expression:^{[7]}

where $F}_{i$ is the force applied to the molecules of the *i*th component and $k}_{T$ is the thermodiffusion ratio.

The coefficient *D*_{12} is positive. This is the diffusion coefficient. Four terms in the formula for *C*_{1}-*C*_{2} describe four main effects in the diffusion of gases:

- $\mathrm{\nabla}\left(\frac{{n}_{1}}{n}\right)$ describes the flux of the first component from the areas with the high ratio
*n*_{1}/*n*to the areas with lower values of this ratio (and, analogously the flux of the second component from high*n*_{2}/*n*to low*n*_{2}/*n*because*n*_{2}/*n*=1-*n*_{1}/*n*); - $\frac{{n}_{1}{n}_{2}({m}_{2}-{m}_{1})}{n({m}_{1}{n}_{1}+{m}_{2}{n}_{2})}\mathrm{\nabla}P$ describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is barodiffusion;
- $\frac{{m}_{1}{n}_{1}{m}_{2}{n}_{2}}{P({m}_{1}{n}_{1}+{m}_{2}{n}_{2})}({F}_{1}-{F}_{2})$ describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient.
- ${k}_{T}\frac{1}{T}\mathrm{\nabla}T$ describes thermodiffusion, the diffusion flux caused by the temperature gradient.

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

In the first approximation,^{[7]}

- $D}_{12}=\frac{3}{2n({d}_{1}+{d}_{2}{)}^{2}}{\left[\frac{kT({m}_{1}+{m}_{2})}{2\pi {m}_{1}{m}_{2}}\right]}^{1/2$ for rigid spheres;
- $D}_{12}=\frac{3}{8n{A}_{1}(\nu )\mathrm{\Gamma}(3-\frac{2}{\nu -1})}{\left[\frac{kT({m}_{1}+{m}_{2})}{2\pi {m}_{1}{m}_{2}}\right]}^{1/2}{\left(\frac{2kT}{{\kappa}_{12}}\right)}^{\frac{2}{\nu -1}$ for repulsing force $\kappa}_{12}{r}^{-\nu$.

The number ${A}_{1}(\nu )$ is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book^{[7]})

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

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

where $\rho}_{i}={m}_{i}{n}_{i$ is the mass concentration of the *i*th species, $\rho ={\sum}_{i}{\rho}_{i}$ is the mass density.

By definition, the diffusion velocity of the *i*th component is ${v}_{i}={C}_{i}-V$, ${\sum}_{i}{\rho}_{i}{v}_{i}=0$.
The mass transfer of the *i*th component is described by the continuity equation

where $W}_{i$ is the net mass production rate in chemical reactions, ${\sum}_{i}{W}_{i}=0$.

In these equations, the term $\mathrm{\nabla}({\rho}_{i}V)$ describes advection of the *i*th component and the term $\mathrm{\nabla}({\rho}_{i}{v}_{i})$ represents diffusion of this component.

In 1948, Wendell H. Furry proposed to use the *form* of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.^{[15]} For the diffusion velocities in multicomponent gases (*N* components) they used

- ${v}_{i}=-({\displaystyle \sum _{j=1}^{N}}{D}_{ij}{\mathbf{d}}_{j}+{D}_{i}^{(T)}\mathrm{\nabla}(\mathrm{ln}T))\phantom{\rule{0.167em}{0ex}};$
- ${\mathbf{d}}_{j}=\mathrm{\nabla}{X}_{j}+({X}_{j}-{Y}_{j})\mathrm{\nabla}(\mathrm{ln}P)+{\mathbf{g}}_{j}\phantom{\rule{0.167em}{0ex}};$
- ${\mathbf{g}}_{j}=\frac{\rho}{P}({Y}_{j}{\displaystyle \sum _{k=1}^{N}}{Y}_{k}({f}_{k}-{f}_{j}))\phantom{\rule{0.167em}{0ex}}.$

Here, $D}_{ij$ is the diffusion coefficient matrix, $D}_{i}^{(T)$ is the thermal diffusion coefficient, $f}_{i$ is the body force per unite mass acting on the *i*th species, ${X}_{i}={P}_{i}/P$ is the partial pressure fraction of the *i*th species (and $P}_{i$ is the partial pressure), ${Y}_{i}={\rho}_{i}/\rho$ is the mass fraction of the *i*th species, and ${\sum}_{i}{X}_{i}={\sum}_{i}{Y}_{i}=1$.

### Separation of diffusion from convection in gases

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

Under normal conditions, molecular diffusion dominates only on length scales between nanometer and millimeter. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection, and to study diffusion on the larger scale, special efforts are needed.

Therefore, some often cited examples of diffusion are *wrong*: If cologne is sprayed in one place, it will soon be smelled in the entire room, but a simple calculation shows that this can't be due to diffusion. Convective motion persists in the room because the temperature inhomogeneity. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.

### Other types of diffusion

- Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
- Anomalous diffusion,
^{[16]}in porous medium - Atomic diffusion, in solids
- Eddy diffusion, in coarse-grained description of turbulent flow
- Effusion of a gas through small holes
- Electronic diffusion, resulting in an electric current called the diffusion current
- Facilitated diffusion, present in some organisms
- Gaseous diffusion, used for isotope separation
- Heat equation, diffusion of thermal energy
- Itō diffusion, mathematisation of Brownian motion, continuous stochastic process.
- Knudsen diffusion of gas in long pores with frequent wall collisions
- Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
- Photon diffusion
- Plasma diffusion
- Random walk,
^{[17]}model for diffusion - Reverse diffusion, against the concentration gradient, in phase separation
- Rotational diffusion, random reorientations of molecules
- Surface diffusion, diffusion of adparticles on a surface
- Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid

## See also

## References

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## External links

- Diffusion in a Bipolar Junction Transistor Demo
- Diffusion Furnace for doping of semiconductor wafers. POCl3 doping of Silicon.
- A Java applet implementing Diffusion

- ↑ J. Philibert (2005). One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2, 1.1–1.10.
- ↑ S.R. De Groot, P. Mazur (1962).
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑
*Diffusion Processes*, Thomas Graham Symposium, ed. J.N. Sherwood, A.V. Chadwick, W.M.Muir, F.L. Swinton, Gordon and Breach, London, 1971. - ↑ L.W. Barr (1997), In:
*Diffusion in Materials, DIMAT 96*, ed. H.Mehrer, Chr. Herzig, N.A. Stolwijk, H. Bracht, Scitec Publications, Vol.1, pp. 1–9. - ↑
^{6.0}^{6.1}One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑
^{7.0}^{7.1}^{7.2}^{7.3}^{7.4}S. Chapman, T. G. Cowling (1970)*The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases*, Cambridge University Press (3rd edition), ISBN 052140844X. - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑
^{9.0}^{9.1}^{9.2}^{9.3}^{9.4}One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑
^{10.0}^{10.1}One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ S. Bromberg, K.A. Dill (2002), Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, ISBN 0815320515.
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, ISBN 0198569033.
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534