Characteristic decay time lengths

In the above equations, y is a variable related to the transport mean free path /, which is the length that a photon must travel before its direction is completely randomized, and / is the sample thickness. Eq. 5.39 has a characteristic decay time In neither setup, can Orbe resolved directly unless / is known. 

The decay on the rear side can be fitted to an exponential function y = A exp [(x — Xo)/Ao] + yo with amplitude A and offsets Xq and yo- As expected by theory [42,44,46,47], the characteristic decay length Aq did not vary with annealing time. Fits to the profiles shown in Fig. 16 all resulted in Aq 0.5 pm. We note, however, that Aq could vary between 0.3 pm and several micrometers for samples with slightly varying thicknesses of the PDMS coating. The hole opening velocity was faster for larger Aq. 

The equation for turbulent dispersion was based on the classical development of Chen and Middleman (1967) (see Section 12-2), with the energy dissipation term calculated for drag on a cylinder. Two cases were assumed for the dissipation volume in the wake region behind the cylindrical impeller blade. The first was that an eddy length proportional to the cylinder diameter determined the dissipation volume. The second was that this volume was proportional to the velocity of the cylinder (tip speed) and a characteristic eddy decay time. Equation (12-74) results from the second case. It showed reasonable agreement with data taken at higher speeds. 

The thickness dependence of v is given in Fig. 10.9. Remarkably, this quantity presents an exponential trend similarly to Aen , with a characteristic decay length of 7 nm. Interestingly, such strong correlation to Ae is also found for the conversion time, defined as tc = 1/v, as revealed by the close similarity between the temperature dependence of the two parameters, see Fig. 10.7. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

The following argument was used first by E. Y. Schweidler in 1905 to describe radioactive decay but it applies to all similar kinetic processes. The fundamental assumption is that the probability p of an event occurring over a time interval dt is independent of past history of a molecule it depends only on the length of time represented by dt and for sufficiently short intervals is just proportional to dL Thus, p — kdt where k is a constant of proportionality characteristic of the process being awaited. In fluorescence decay it is characteristic of the kind of molecule in chemical terms. 

Figure 9. Fluorescence spectra and decay characteristics of POS containing polystyrene. M, styrene monomer region, dual decay kinetics. D, styrene excimer region, triple decay characteristics (double fit shovm does not correlate with other wave lengths, thus meaningless). P is POS fluorescence, triple decay characteristics when styrene excited (see box), but single, t = 1.68 ns when excited directly. EGS is early-gated time-resolved spectrum which matches closely spectrum of P excited directly, and difference between late-gated spectrum LGS and known spectrum of D. ...

We wish to determine the velocity distribution in the fluid at large times, after any initial transients have decayed, so that the velocity field is strictly periodic. Normally we would proceed as in the related problem, 3 9, by first nondimensionalizing and then solving the problem in dimensionless form. Here, however, there is not an immediately obvious characteristic length scale, so we follow a different procedure. 

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