Mean transition time , probability

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 ... 

For the mean transition time, this fact may be explained in the following way If the transition process is going from up to down, then the probability current is large, but it is necessary to fill the lower minimum by the larger part of the probability to reach the steady state if the transition process is going from down to up, then the probability current is small, and it is necessary to fill the upper minimum by the smaller part of the probability to reach the steady state. 

Figure 6.12 Frequency of mean transit times vs. time (min) using the diffusion model II for the blind ant model positions for various concentrations of villi and forward probabilities pf values. Key experimental data solid line, TVvini = 200 and pf = 0.6 dashed line, Nvnn = 200 and pf = 0.5 dotted line, Nvmi = 180 and pf = 0.7 dashed-dotted line, TVviin = 180 and pf = 0.5.

If the experimental data are sufficiently accurate, then from the susceptibility function 3 /(m- t) it is possible to evaluate the transit time probability density, (Puu id-, t), by means of numerical inverse Fourier transformation. If the experimental data are not very accurate, then it is still possible to evaluate the first two or three moments and cumulants of the transit time. Since the mean transit time is nonnegative, it follows that (puu d < 0 r) = 0, and thus the characteristic function of the probability density (Puu id-, t) is identical with the susceptibility function in the frequency domain, Sr/MdsT t), experimentally accessible from eqs. (12.116)-(12.117). It follows that the moments < 0" (r) > , m = 1, 2,..., and the cumulants m = 1,2,...,... 

The RTD quantifies the number of fluid particles which spend different durations in a reactor and is dependent upon the distribution of axial velocities and the reactor length [3]. The impact of advection field structures such as vortices on the molecular transit time in a reactor are manifest in the RTD [6, 33], MRM measurement of the propagator of the motion provides the velocity probability distribution over the experimental observation time A. The residence time is a primary means of characterizing the mixing in reactor flow systems and is provided directly by the propagator if the velocity distribution is invariant with respect to the observation time. In this case an exact relationship between the propagator and the RTD, N(t), exists... 

The transition probabilities depend on the mean squared interaction energy relative to the mechanism which causes the transition, times the value of the spectral density at the required frequencies (Eq. (3.14)). The square of the dipolar interaction energy is, as usual (see Eq. (1.4) and Appendix V), proportional to (p, 1 1x2/r3)2, where p and p2 are the magnetic moments of the two spins. The actual equations are... 

Each point on the curves in Fig. 4.19 corresponds to the mean value of various experimental results. We can notice that, even if we have good trends, the experimental and calculated values do not match well. This can be ascribed to model inadequacies, especially with respect to the liquid exit conditions in that case, we considered that the MWPB output had occurred at x = 0 and at x = H,j when it was experimentally observed that the liquid exit dominantly occurs at x = H,j. This results in a decrease in the mean residence time computed values. If we look at Figs. 4.20 to 4.22, which have been obtained at different operating conditions, we can conclude that we do not have major differences between the computed and experimental values of liquid MWPB hold up then we can consider the equality of the transition probabilities between the individual states of the stochastic model to be realistic. 

Theoretical and experimental results of the gas hold-up inside a MWPB show that the data converge only when the pjj values are greater than 0.7. Figure 4.25 presents a simulation of the presented model, which intends to fit some experimental data [4.82]. In the presented simulation, the initial values of Pi(0,0), P2(0,0) and P3(0,0) injected into the model give an idea about the values of the transition probabilities these are pn = p2i = pji = 0.7, Pn = Pn = P12 = P23 = P22 = P33 = P32 = 0.15. In Fig. 4.25 we can see that we have all the necessary data to begin the computation of the mean residence time of a gas element evolving inside the MWPB. Indeed, relation (4.176) can now be used to calculate the gas hold-up in the bed. 

We can observe that the first and the last cell of the system are in contact with only one cell cell number 2 and number N - 1 respectively. So, in the matrix of the transition probabilities, the values pi3 and Pn-2n will be zero. It is easily noticed that, if we have a complete matrix of the transition probabilities, then we can compute the mean residence time, the dispersion around the mean residence time and the mixing intensity for our cells assembly. The relations (4.324)-(4.326) are used for this purpose. 

The conformational orientation between the excited CNA and CHD should be restricted very much to produce a photocycloadduct in the collision complex indicated in the scheme 1. In the fluid solvents like hexane, the rotational relaxation times of the solute molecules are rather fast compared to the reaction rate, which increases the escape probability of the reactants from the solvent cavity due to the large value of ko. On the other hand, the transit time in the reactive conformation, probably symmetrical face to face, may be longer in the liquid paraffin. This means that the observed kR may be expressed as a function of the mutual rotational relaxation time of reactants and the real reaction rate in the face-to-face conformation. In this sense, it is very important to make precise time-dependent measurements in the course of geminate recombination reaction indicated in Scheme 2, because the initial conformation after photodecomposition of cycloadduct is considered to be close to the face-to-face conformation. The studies on the geminate processes of the system in solution by the time resolved spectroscopy are now progress in our laboratory. 

First, consider the case in which the spin system and the lattice are in equilibrium at temperature T in field Hq. This means that the number of transitions upward must equal the number downward. If Wu is the probability of an upward transition per unit time and Wtransition probability W = (Wu -1- VVd)/2 gives... 

At time t a reaction takes place. According to equation (44) the different reactions that transform configuration a to another configuration f3 have transition probabilities W a- This means that the probability that the system will be in configuration 0 at time t -1- dt is where dt is... 

The observation that the transition time of the trajectory is extremely short, and the existence of a single dominant barrier, can be exploited to design algorithms for the calculation of rate constants of these rare (but rapid) processes. The calculations are based on statistical properties of the transitional (short) trajectories and their weights in the complete ensemble of pathways. In the calculations of rate constants, it is typically assumed that the kinetics follows an exponential law. Let P(t) be the probability that the system remains in the reactant state after time t. Exponential kinetics means that the time evolution of the reactant probability is P(t) = exp(-Kt) where k is a rate constant (with units of the inverse of time). The initial condition is P(0) = 1. This is considered the simplest behavior and is frequently observed experimentally in activated processes. 

The problem of transit-time broadening was recognized many years ago in electric or magnetic resonance spectroscopy in molecular beams [1253]. In these Rabi experiments [1254], the natural linewidth of the radio frequency or microwave transitions is extremely small because the spontaneous transition probability is, according to Vol. 1, (2.22), proportional to co. The spectral widths of the microwave or RF lines are therefore determined mainly by the transit time AT = d/v of molecules with the mean velocity v through the interaction zone in the C field (Fig. 5.10a) with length d. 

Let us come back now to the question of increasing heterogeneity in the local mobilities upon decreasing the temperature. We have already identified a tendency for immediate back jumps after one torsional transition as the reason for the different temperature dependencies of the mean waiting time between torsional transitions (twait) and the torsional autocorrelation time ttacf)- In a homogeneous system, where every chemically identical torsion shows identical dynamics on the time scales of observation the probability distribution of waiting times should be... 

The maximum frequency of the electro-optic effect in Hthium niobate is probably in the hundreds of gigahertz. Thus, in practice the upper limit on the frequency response is set by electrical considerations. The key parameter that divides modulators into one of two types is the ratio of the modulation period of the maximum modulation frequency to the optical trmsit time past the electrodes. If the optical transit time is short relative to the period of the maximum modulation frequency, then the modulator signal will essentially be constant during the optical transit time. In this case the electrodes can be treated as lumped elements, see Fig. 9.54(c), which means that the upper 3-dB frequency is simply determined by the RC rolloff due to the electrode capacitance and resistance as shown in Fig. 9.57. Atypical frequency response of a low-frequency MZM is shown in Fig. 9.59(a). 

As seen from Figs. 15 and 16 theoretical predictions on the mass (p), frequency (co) and temperature (T) dependences of the mean transition probabilities are substantiated by experimental findings. Fig. 15 shows the temperature dependence of Tvib for some diatomic molecules. The vibrational relaxation time is inversely proportional to the deactivation probability (Pi,o) (see Eq. (8.45) and Section IV. 15), so that the temperature dependence of Tyib is mainly due to that of (Pi,o)-Thus, according to Eq. (14.1)... 

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