Einstein relation response

The nature of rotational motion responsible for orientational disorder in plastic crystals is not completely understood and a variety of experimental techniques have been employed to investigate this interesting problem. There can be coupling between rotation and translation motion, the simplest form of the latter being self-diffusion. The diffusion constant D is given by the Einstein relation... 

In order to evaluate this expression, we need to know the force v / that is responsible for producing the molecular flux. It could be an external force such as an electric field acting on ions. Then evaluation of Eq. 18-48 would lead to the relationship between electric conductivity, viscosity, and diflusivity known as the Nernst-Einstein relation. 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

The Einstein relation states that, for particles (or a particle) midergoing thermal motion, the quantity D which characterizes the diffusion process is related to the quantity f which specifies the response to an external force. Substituting Eq. (3.9) into Eq. (3.6), one obtains... 

The Einstein relation states that the quantity D which characterizes the thermal motion is related to the quantity which specifies the response to the external force. We shall show later that the Einstein relation is a special case of a more general theorem, called the fluctuation dissipation theorem, which states that the characteristics of the spontaneous thermal flucutation are related to the characteristics of the response of the system to an external field. 

Two diffusion coefficients are of interest in MIECs the component diffusion coefficient, Dk, and the chemical diffusion coefficient, D. The component diffusion coefficient reflects the random walk of a chemical component. It is therefore equal to the tracer diffusion coefficient, except for a correlation factor which is of the order of unity. It is also proportional to the component mobility as given by the Nemst-Einstein relations. The chemical diffusion coefficient, I), reflects the transport of neutral mass under chemical potential gradients. In MIECs mass is carried by ions, and transport of neutral mass occurs via ambipolar motion of ions and electrons or holes so that the total electric current vanishes. b can be determined from steady-state permeation measurements, as mentioned in Section IV.H. However, D is usually determined from the time dependence of a response to a step change in a parameter, e.g., the applied current. Alternatively, D is determined from the response to an ac signal applied to the MIEC. 

This rate expression, responsible for the right side of Figure 5.8, is important in a liquid with a high viscosity. Use of the Stokes-Einstein relation (5.18) converts (5.24) to... 

Brownian motion is a random thermal motion of a particle inside a fluid medium. The collision between the fluid molecules and suspended microparticles are responsible for the Brownian motion. The Brownian motion consists of high frequencies and is not possible to be resolved easily. Average particle displacement after many velocity fluctuations is used as a measure of Brownian motion. The mean square diffusion distance, is proportional to DAt, where D is diffusion coefficient of the particle given by Einstein relation as... 

When solvated ions migrate within the electrolyte, the drag force applied by the surrounding solvent molecules is measured by solvent viscosity rj. Thus, in a solvent of lower viscosity, the solvated ions would move more easily in response to an applied electric field, as expressed by the Einstein—Stokes relation (eq 3). Solvents of low viscosity have always been considered the ideal candidates for electrolyte application however, their actual use was restricted because most of these solvents have low dielectric constants (Tables 1 and 2) and cannot dissociate ions effectively enough to prevent ion pairing. 

In many applications, one response from an instrument is related to the concentration of a single chemical component. This is referred to as univariate calibration because only one instrument response is used per sample. Multivariate calibration is the process of relating multiple responses from an instrument to a property or properties of a sample. The samples could be, for example, a mixture of chemical components in a process stream, and the goal is to predict the concentration levels of the different chemical components in the stream from infrared measurements. The methods are quite powerful, but as Dr. Einstein noted, the application of mathematics to reality is not without its limitations. It is, therefore, the obligation of the analyst to use them in a responsible manner. 

The previous chapter dealt with chemical bonding and the forces present between the atoms in molecules. Forces between atoms within a molecule are termed intramolecular forces and are responsible for chemical bonding. The interaction of valence electrons between atoms creates intramolecular forces, and this interaction dictates the chemical behavior of substances. Forces also exist between the molecules themselves, and these are collectively referred to as intermolecular forces. Intermolecular forces are mainly responsible for the physical characteristics of substances. One of the most obvious physical characteristics related to intermolecular force is the phase or physical state of matter. Solid, liquid, and gas are the three common states of matter. In addition to these three, two other states of matter exist—plasma and Bose-Einstein condensate. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

Within the linear response approximation, the rate of transport (mass, momentum, or energy) through a system is proportional to the gradient (of concentration, velocity, and temperature), with the transport coefficient being the proportionality constant. This proportionality constant can be computed using equilibrium description of the system through the so-called fluctuation dissipation theorems. One such equation, relating equilibrium fluctuations to the diffusion constant, is Einstein s well-known equation ... 

The determination of As requires the use of Smoluchowski-Einstein theory of fluctuations the latter postulates that the free energy of a system increases spontaneously by a value (SG) in response to a fluctuation of any of its characteristics. For the variable z of a system undergoing a fluctuation hz, its relation with SG is established as... 

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