Transit time definition

Dispersive transport in PVC was investigated. The results of Pfister and Griffits obtained by the transit method are shown in Fig. 6. The hole current forms at temperatures > 400 K clearly show a bend corresponding to the transit time of the holes. At lower temperature the bend is not seen and transit time definition needs special methods. The pulse form shows the broad expansion during transition to the opposite electrodes. This expansion corresponds to the dispersive transport [15]. The super-linear dependence of the transit time versus sample thickness did not hold for pure PVC. This is in disagreement with the Scher-Montroll model. There are a lot of reasons for the discrepancy. One reason may be the influence of the system dimensions. It is quite possible that polymer chains define dimension limits on charge carrier transfer. 

The case of the prescribed material flux at the phase boundary, described in Section 2.5.1, corresponds to the constant current density at the electrode. The concentration of the oxidized form is given directly by Eq. (2.5.11), where K = —j/nF. The concentration of the reduced form at the electrode surface can be calculated from Eq. (5.4.6). The expressions for the concentration are then substituted into Eq. (5.2.24) or (5.4.5), yielding the equation for the dependence of the electrode potential on time (a chronopotentiometric curve). For a reversible electrode process, it follows from the definition of the transition time r (Eq. 2.5.13) for identical diffusion coefficients of the oxidized and reduced forms that... 

However, mathematical evidence of such a definition of characteristic timescale has been understood only recently in connection with optimal estimates [54]. As an example we will consider evolution of the probability, but the consideration may be performed for any observable. We will speak about the transition time implying that it describes change of the evolution of the transition probability from one state to another one. 

It is easy to check that the normalization condition is satisfied at such a definition, wT(t.xoj dt = 1. The condition of nonnegativity of the probability density wx(f,x0) > 0 is, actually, the monotonic condition of the probability Q(t, xq). In the case where c and d are absorbing boundaries the probability density of transition time coincides with the probability density of the first passage time wT(t,x0y. 

This definition completely coincides with the characteristic time of the probability evolution introduced in Ref. 32 from the geometrical consideration, when the characteristic scale of the evolution time was defined as the length of rectangle with the equal square, and the same definition was later used in Refs. 33-35. Similar ideology for the definition of the mean transition time was used in Ref. 30. Analogically to the MTT (5.4), the mean square d2(c,x, d) = (f2) of the transition time may also be defined as... 

Alternatively, the definition of the mean transition time (5.4) may be obtained on the basis of consideration of optimal estimates [54]. Let us define the transition time i) as the interval between moments of initial state of the system and abrupt change of the function, approximating the evolution of its probability Q(t.X(t) with minimal error. As an approximation consider the following function v /(f,xo, ) = flo(xo) + a (xo)[l(f) — l(f — i (xo))]. In the following we will drop an argument of ao, a, and the relaxation time d, assuming their dependence on coordinates of the considered interval c and d and on initial coordinate x0. Optimal values of parameters of such approximating function satisfy the condition of minimum of functional ... 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

Another approach for computing the transition times had been proposed by Malakhov [34,35]. This approach also utilizes the definition of the desired... 

T4 lysozyme 33,497 helix stability of 528, 529 hydrophobic core stability of 533, 544 Tanford j8 value 544, 555, 578, 582-Temperature jump 137, 138, 541 protein folding 593 Terminal transferase 408,410 Ternary complex 120 Tertiary structure 22 Theorell-Chance mechanism 120 Thermodynamic cycles 125-131 acid denaturation 516,517 alchemical steps 129 double mutant cycles 129-131, 594 mutant cycles 129 specificity 381, 383 Thermolysin 22, 30,483-486 Thiamine pyrophosphate 62, 83 - 84 Thionesters 478 Thiol proteases 473,482 TNfn3 domain O-value analysis 594 folding kinetics 552 Torsion angle 16-18 Tbs-L-phenylalanine chloromethyl ketone (TPCK) 278, 475 Transaldolase 79 Tyransducin-o 315-317 Transit time 123-125 Transition state 47-49 definition 55... 

The transitions between the bottom five phases of Fig. 2 may occur close to equilibrium and can be described as thermodynamic first order transitions (Ehrenfest definition 17)). The transitions to and from the glassy states are limited to the corresponding pairs of mobile and solid phases. In a given time frame, they approach a second order transition (no heat or entropy of transition, but a jump in heat capacity, see Fig. 1). 

To normalise this system, the previous definition (2.40) is used for the distance x, and c, the bulk concentration, for the concentration c for the time unit, it is natural to use the transition time r itself. This makes the boundary conditions... 

Figure 29. Representative time-of-flight signals for hole transport, a) tri-p-tolylamine (TTA) (30 wt.%) in polystyrene. (Reprinted with permission from Ref. [60b].) Two operational definitions of the transit time are indicated by t, and t /2. b) p-Diethylamino-benzaldehyde diphenylhydrazone (DEH) (30 wt. /o) in bisphenol-A polycarbonate. (Reprinted with permission from Ref. [60i].) c) A polysiloxane with A-alkylcarbazole pendant groups. (Reprinted with permission from Ref. [72g].)...

Here, Go may be called the multiple source boundary propagator, which describes the water mixture at a location along with the origin/transit times T = (f — f ) of those water masses at the sea surface. It is, by definition, dependent solely on fluid transport properties, and independent of the particular tracer. Since there are no internal sinks or sources of C, we can construct the tracer distribution in a fashion analogous to (46) with... 

Figure 3. Graphical definition of the metabolic transition time, x. The concentration of some metabolic product, P, is plotted as a function of time after the pathway is switched on (e.g., by addition of the initial substrate), with the system being devoid of metabolic intermediates initially. The extrapolation to the time axis, of the linear portion of the product accumulation curve is denoted as the transition time (see Welch and Easterby, 1994).

The power-law decay in Eq. (3) can be regarded as a decay of free carrier density or as a decay of the drift mobility. The tatter interpretation and the usual definition of the transit time tj lead to... 

The standard experimental definition of the transit time corresponds to the extrapolated intersection point of the short- and long-time current decays. By solving Eqs. (A 15) and (A 16) for their intersection point, we find... 

In conclusion, we have introduced a neutral type of linear response experiment for nonlinear kinetics involving multiple reaction intermediates. We have shown that the susceptibility functions from the response equations are given by the probability densities of the transit time in the system. We have shown that a transit time is a sum of different lifetimes corresponding to different reaction pathways, and that in the particular case of a time-invariant system our definition of the transit time is consistent with Easterby s definition [23]. 

It is possible to design the milestones to satisfy the timescale requirement. In contrast to the usual definition of reaction coordinates, which is continuous, milestones are discrete and spatially separated. Therefore the user can control the transition times, moving the hypersurfaces away from each other. Of course, larger separation makes the milestoning procedure less efficient. A compromise must be made between efficiency and accuracy. 

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