Formula Einstein

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for... 

The visualization of light as an assembly of photons moving with light velocity dates back to Isaac Newton and was formulated quantitatively by Max Planck and Albert Einstein. Formula [1] below connects basic physical values ... 

It is well known that each transport coefficients is given by a Green-Kubo formula or, equivalently, by an Einstein formula ... 

The Helfand moment is the center of mass, energy or momentum of the moving particles, depending on whether the transport property is diffusion, heat conductivity, or viscosity. The Helfand moments associated with the different transport properties are given in Table III. Einstein formula shows that the Helfand moment undergoes a diffusive random walk, which suggests to set up a... 

Onsager reciprocity relations as well as the Green-Kubo and Einstein formulas for these coefficients ... 

Generally, these behave as Newtonian fluids and, for the case of an extremely dilute suspension of spherical non-interacting particles having a density equal to that of the continuous medium, we can apply the Einstein formula for a suspension of spheres ... 

As fluctuations are an intrinsic part of a thermodynamic system, a discussion of nonequilibrium structures is not complete without the consideration of the consequences of fluctuations. Unlike equilibrium systems, nonequilibrium systems do not have a general prescription, like the Einstein formula, to describe the fluctuations. Nonequilibrium fluctuations are highly specific. The importance of fluctuations appears clearly in the way they alter the macroscopic behavior in the vicinity of the bifurcation point and also in the way the coherence of a structure depends on the dimensionality of the system in the face of the destructive influence of fluctuations. 

The viscosity of such an idealized plastisol may be evaluated 37 39> by using the well-known Einstein formula ... 

The segmental diffusion constant is given by Einsteins formula (Equation 2) ... 

With rubber as a matrix (v = 0.5), ky has the value of 2.5, and we meet the well-known Einstein formula for the increase in viscosity as a result of the presence of spherical particles. 

For non-spherical particles the relations are different this is easiest demonstrated by the Einstein formula the coefficient ky increases when the particles are flattened, and even stronger upon elongation. An example of the latter case When a sphere is extended up to a length/diameter ratio of 10, ky increases from 2.5 to 6. Sticks and platelets, therefore, have a stronger effect this accounts for the fact that with a volume fraction of 20 % increases in stiffness up to a factor of 3 are measured, whereas the Kemer formula would predict a much lower value. 

To describe the viscosity of binary alloys, let us use Einstein formula for nondissociated low concentration solutions [9] ... 

Because of the small mass of the proton, the decrease of the transition dipole moment as we move to higher overtone bands of H3 is not as drastic as in ordinary molecules. The band origins, transition moments, relative intensities and Einstein s spontaneous emission probabilities theoretically calculated by Dinelli, Miller and Tennyson are listed in Table 1. Note that the value of Aij is larger for the 2v2(2) overtone band than for the Vj fundamental band because the factor in the Einstein formula overrides the reduction of j n. This explains the strong 2 pm overtone emission observed in Jupiter. ... 

The intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is... 

The simplest case to consider is steady flow of a dilute suspension of Newtonian drops or bubbles in a Newtonian medium. If the capillary number y a / F is small, so that the drops or bubbles do not deform under flow, then at steady state the viscosity of the suspension is given by Taylor s (1932) extension of the Einstein formula for solid spheres ... 

For the x-ray emission process, the transition probability( is also calculated from the dipole matrix similar to the case of the x-ray absorption, but the molecular state f in eq.(lO) is of occupied in this case. The transition probability corresponds to the spontaneous emission rate, then is given by Einstein formula as... 

In this section, we consider the advection and diffusion of a univariafe NDF n(t, x, ), where R+ is a passive scalar (i.e. the velocity u t, x) does not depend on ). For example, could denote the particle mass for fine particles in a liquid with very small Stokes number. The diffusion coefficient, F(4, is assumed to be a function of For example, using the Stokes-Einstein formula for diffusivity reported in Eq. (5.116) on page 187, F( ) = Fq/, where is the particle mass. The PBE for this case reads... 

Moreover, employing the Stokes-Einstein formula, we can write f f r(f)ndf = Fom/i-i, which is essentially the form that will be used in the numerical examples in Section 8.3.4. In summary, we will consider two variations of the moment-transport equations in the numerical examples in Section 8.3.4. The first example will use a closed moment system wherein the diffusivity does not depend on f ... 

Since the Stokes-Einstein formula is valid for finite-size particles, the NDF should be nonzero only for fo < f For convenience, we will allow 0 < f < oo and add fo to the denominator of the Stokes-Einstein formula in order to ensure that the diffusivity remains finite with f = 0. 

The friction constant, 7/3, may be related to an angular diffusion constant by use of the Einstein formula, D = kBT/I. For the Tyr-21 ring torsional motion in BPTI, one obtains D = 2.3 X 1011 s 1 at 308 K, the temperature of the simulation. This value is somewhat larger than experimental diffusion constants for the corresponding rotational motion of small aromatic molecules in organic solvents (e.g., the value for benzene in isopentane is 8 X 10l° s-1). 

Given in terms of viscosity, experimental results confirm the theoretical Einstein formula relating the diffusion coefficient to viscosity ... 

Einstein formula Ds = knTI6Tn)RU .to obtain at least apparent hydrodynamic... 

The value p depends primarily on the volume concentration of the disperse phase . The well-known Einstein formula [179]... 

The general form of this relationship agrees with the well-known Nernst-Planck-Einstein formula, according to which the diffusion coefficient is given by the relation... 

This constitutes the generalization of the Stokes-Einstein formulas (316). It applies for any choice of origin. The positive-definite character of the resistance matrix assures the existence of its inverse, as required by the preceding relation. 

The flux of oxygen through an ITM device is given by the Nernst-Einstein formula. 

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