Diffusion, definition linear

The "upward" curvature in the SV plot (in our case the equivalent plot Q vs. c) which is also "not uncommon" in liquid solution and probably an indication for down chain exciton diffusion according to Webber (62), has been found in Nos. 5-8, 10,11 and 13. Only in No. 12,Q(c) is definitely linear up to Q=5 (7 ) ... 

Equation (7.15) shows that at small times the track develops as a collection of m isolated spheres, whereas at long times, V(f) increases linearly with t, which is characteristic of cylindrical diffusion. Using the definition of V(t) the solution of Eq. (7.13) may be given as... 

H is the plate height (cm) u is linear velocity (cm/s) dp is particle diameter, and >ni is the diffusion coefficient of analyte (cm /s). By combining the relationships between retention time, U, and retention factor, k tt = to(l + k), the definition of dead time, to, to = L u where L is the length of the column, and H = LIN where N is chromatographic efficiency with Equations 9.2 and 9.3, a relationship (Equation 9.4) for retention time, tt, in terms of diffusion coefficient, efficiency, particle size, and reduced variables (h and v) and retention factor results. Equation 9.4 illustrates that mobile phases with large diffusion coefficients are preferred if short retention times are desired. 

The diffusion-layer concept is an artifice for handling the flux arising from what would be, if treated in a proper hydrodynamic way, a complicated space variation of concentration at the interface. There is always some gradient of concentration at the interface there is an initial region in which the concentration changes linearly with distance, but there is, in the real case, no sharply defined layer of definite thickness, even when convection (natural or forced) produces a steady-state concentration... 

Summary. The special class of master equations characterized by (1.1) will be said to be of diffusion type. For such master equations the -expansion leads to the nonlinear Fokker-Planck equation (1.5), rather than to a macroscopic law with linear noise, as found in the previous chapter for master equations characterized by (X.3.4). The definition of both types presupposes that the transition probabilities have the canonical form (X.2.3), but does not distinguish between discrete and continuous ranges of the stochastic variable. The -expansion leads uniquely to the well-defined equation (1.5) and is therefore immune from the interpretation difficulties of the Ito equation mentioned in IX.4 and IX.5. 

The dimensionless distance parameter may be found by substituting the definition of Ax into Equation 20.7. In the case of semi-infinite linear diffusion, Equation 20.17 is used to define Ax this substitution yields... 

A characteristic property of surface migration is that ((Ar)2 varies linearly with time. Note that the very definition of a hopping frequency Th tacitly implies statistic averaging over many hopping events. The time difference between the individual jumps of a specific particle varies stochastically. The corresponding tracer diffusion coefficient is defined as ... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

In the Sumi-Marcus (SM) model 13a], the perspective is changed, with a TST rate constant based on a low-frequency molecular mode (m) as the reaction coordinate, and with G dependent on a diffusive solvent coordinate X. For ease of comparison with other models, we transform X (relative to its definition in [13a]) so as to correspond to a continuous charging parameter (X = 0 for the bottom of the reactant well, and X = 1 for the bottom of the product well for the case of parabolic free energy profiles the transformation is linear the more general situation is dealt with in [98]). Also, 7.ci = A + z , and /.d = a, , where 7, , is the reorganization energy associated with the low-frequency mode m. These definitions lead to the following equation ... 

These redox cells can operate on a number of scales that depend on the length of the diffusion path from the point that the oxidised form becomes reduced to the point where it reduces another sediment constituent. In some pelagic cores these diffusion paths can be observed in linear portions of the pore-water profiles (e.g. Sawlan Murray, 1983). Here the sedimentation rate and the carbon burial rate are sufficiently low, relative to diffusion, to extend the processes of early diagenesis over tens of metres into the sediment. In coastal environments the sedimentation rate and the concentration and reactivity of the organic matter is often high, which results in a much more complex pattern. In this case, the distances between the cells are much shorter, since by definition the adjustment must occur more rapidly. Like laminar and turbulent flow, there may come a point where the flow of electrons downwards is better dispersed through eddies , which in this case are transitory micro-environments with small-scale three dimensional diffusion, rather than more stable... 

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