Stationary equilibrium distribution

Figure 3. (Stationary) equilibrium distribution for p(s). ( ) refer to the result of our numerical simulation for f/a=, = 0.1, y 5. The accuracy of our data is evaluated to...

Both equations, the master equation and the Fokker-Planck equation, have one important property An arbitrary initial distribution - for instance, concentrated in a small interval dr (x) - finally develops into a stationary equilibrium distribution Fst( )-... 

Every chromatographic process is controlled by the equilibrium distribution of the solute between the mobile and stationary phases. The retention volume V describing the volume of mobile phase that is required to elute the analyte from the column, is given by Equation 17.4 ... 

The difference in movement rates of various compounds through a column is attributed to differential migration in HPLC. This can be related to the equilibrium distribution of different compounds such as X, Y, and Z between the stationary phase and the flowing solvent(s), or mobile phase. The speed with which each compound moves through the column (ux) is determined by the number of molecules of that compound in the moving phase, at any moment, since sample molecules do not move through the column while they are in the stationary phase. The molecules of the solvent or mobile phase move at the fastest possible rate except in size exclusion chromatography, where molecular... 

Equation [19] ensures that the thermodynamic equilibrium distribution of Eq. [20] is the stationary (long-time) limit of the Markov chain generated by Eq. [18]. It does not specify the transition rates uniquely, however. Let us write them in the following way ... 

The overloading of the stationary phase is related to the maximum solute concentration. Cm, at which the sorption isotherm associated with equilibrium distribution underlying chromatographic retention ceases to be linear. That deviation results in a broadening and deformation of the peak profile. Since this review deals with chromatographic phenomena and optimization we consider thermodynamics as beyond its scope. 

Such in situ coated columns with an equilibrium distribution of liquid phase have long operational lifetimes because the amount of liquid stationary phase remains constant in the column as long as its concentration in the mobile phase and the temperature remain constant. The column can be coated successively with different liquid phases after removing the stationary liquid phase with a suitable solvent and reactivating by treatment with dry eluent before recoating. 

Many problems that arise in the application of adsorption chromatography can be related to the slow attainment of equilibrium distribution of the omnipresent water and other modulators tetween eluent and stationary phase. With suitable precautions, such as moisture control, reproducible work is possible with both silica and alumina as the stationary phase. 

With fairly few exceptions, all discussion of computed molecular properties up to this point has proceeded under the assumption that the value computed for the stationary equilibrium structure is relevant in comparison to experiment. However, the experimental population is in constant vibrational motion, even at 0 K, so the experimental measurement actually samples structures having a distribution dictated by the molecular vibrational wave function. Thus, for some property A, the measured value is the expectation value given by... 

This difficulty does not arise in the case of closed, isolated physical systems, because there the stationary solution is known to be the thermal equilibrium distribution Pe(x), as given by ordinary statistical mechanics. This knowledge implies some information about At and Bij9 but more information is available if also detailed balance (V.6.1) or extended detailed balance (V.6.14) holds. In the following we shall therefore examine the situation specified by the following stipulations. 

In the case of the quasilinear Fokker-Planck equation (2.4), the free energy U defined in terms of the stationary solution by (2.6) is identical with the potential in the deterministic equation (5.2). That identity is often taken for granted when time-dependent solutions have to be constructed for systems of which only the equilibrium distribution is known. We shall now show, however, that it holds only for systems of diffusion type whose Fokker-Planck equation is quasilinear, i.e., of the form (2.4). 

T he partitioning of a solute between the stationary and mobile phases of a gel permeation column is a function of the molecular size and shape of the solute and the size distribution of gel pores separating the two phases. For a gel permeation column operating under conditions in which an equilibrium distribution of solute between the phases is ob-... 

In discussing the equilibria between a monomer and its living polymer, two aspects of the problem should be considered (1) establishment of an equilibrium, or more correctly of a stationary state, between monomer and growing ends (2) establishment of an equilibrium between all the growing polymers. The first process approaches its equilibrium state fairly rapidly, whereas the other is usually very slow and requires a long time to produce the ultimate equilibrium distribution. 

When the ideas of symmetry and of microscopic reversibility are combined with those of probability, statistical mechanics can deal with many stationary state nonequilibrium problems as well as with equilibrium distributions. Equations for such properties as viscosity, thermal conductivity, diffusion, and others are derived in this way. 

The separation of compounds by their differential partition between two immiscible phases is the basis for partition chromatography. The system consists of a stationary liquid phase coated on an inert solid support, and an immiscible mobile phase. Chromatographic separations are based on the different equilibrium distributions of the samples between these two phases. The greater the quantity of substance in the stationary phase at equilibrium the dower is the migration. For analyses, this equilibrium must remain constant over a suitable concentration range. Thus an increase in the concentration of solute results in a linear increase in the concentration of solute in the mobile and stationary phase, respectively. Under these conditions, the retention time, tR, is independent of the amount of sample chromatographed and a symmetrical peak (gaussian band) is observed. 

Table 1 shows that with the equilibrium stationary flow distribution that corresponds to the maximum entropy of an isolated system... 

See for example the stationary state distributions in Eqs. (3) and (3 ) which correspond to thermal equilibrium. 

This GME, being of infinite age, fits the Onsager principle. Therefore, we can identify the stationary correlation function with the out of equilibrium distribution, namely,... 

Dissipative structures can sustain long-range correlations. The temperature and chemical potential are well defined with the assumption of local equilibrium, and the stationary probability distribution is obtained in the eikonal approximation so the fluctuation-dissipation relation for a chemical system with one variable is... 

In this subsection we put a particle at an origin in a velocity field u(r) and study effects of flow on a stationary density profile n,i(r). When there is no flow u(r) = o,n (r) is obviously given by niq(r), with g(r) a radial distribution function. Due to the flow u(r), this equilibrium distribution is distorted and from the distortion we can calculate some transport coefficients, like viscosity g and friction constant C, as we show below. 

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