Survival probability, time evolution

Figure 10. Evolution of the survival probability for the potential Tf.v) — ax2 - for3 for different values of noise intensity the dashed curve denoted as MFPT (mean first passage time) represents exponential approximation with MFPT substituted into the factor of exponent.

Equation (164) describes the evolution with time t0 of the survival probability of an ion-pair formed at time t with a separation r. In the general, case this equation cannot be solved, but if no long-range transfer occurs and the transport coefficients are constant, this reduces to... 

Usually, experiments are performed with steady-state photolysis or radiolysis of the solution and the yield of scavenger products determined optically or by ESR methods. There is no direct interest in the actual time evolution of the density or recombination (survival) probability. Consequently, the creation of ion-pairs may be pictured as occurring at a constant rate, say 1 s 1, from time t0 = 0 to infinity. The steady-state ion-pair density distribution, which arises when dp/dt = 0, is the balance between continuous creation of ion pairs at a rate Is-1, recombination and scavenging. Removing the instantaneous creation of an ion-pair at time t = t0 (i.e. removing the 6(f — f0) in the source term), means that ion-pairs were continuously formed from time t = — 00 to t. At long times, f > — oo the density distribution is independent of t and, of course, t0. Let pss(r cs r0) = /i p(r, t cs t0, 0)d 0 be the steady-state ion-pair density distribution for ion pairs continuously formed at r0, and note d/dt J" f pd 0 = 0. The diffusion equation (169) becomes... 

As an example we treat the decay process of IV.6 in terms of the master equation. The decay probability y per unit time is a property of the radioactive nucleus or the excited atom, and can, in principle, be computed by solving the Schrodinger equation for that system. To find the long-time evolution of a collection of emitters write P(n, t) for the probability that there are n surviving emitters at time t. The transition probability for a... 

Figure 8. Time evolution, obtained via a numerical diagonalization of the effective Hamiltonian M (adapted from Ref. 45) when N = K and N > K for two different initial states. Shown is the survival probability of the initial state (dashed line) and the probability to remain bound (full line) vs. time. The reason for the difference between these two is due to the system sampling the rest of the bound phase space. At higher number N of bound states, for a given value of K, the delayed decay would be shifted to even longer times while the survival probability will remain essentially unchanged, showing that the delay is due to sampling of the bound phase space.

G. Garci-Calderon, J.L. Mateos, M. Moshinsky, Resonant spectra and the time evolution of the survival and nonescape probabilities. Phys. Rev. Lett. 74 (1995) 337. 

Figure 7 Time evolution of the initial 6v,) state, computed using the Chebyshev propagation scheme (a) survival probability (lower curve) and overall chromophore population (upper curve, defined in Eq. (123). (b), (c), and (d) Vibrational mode populations, defined in Eq. (124).

Figure 2.11 gives an example of such a comparison, considering specifically the evolution of the molecule out of the v = 7 state, under a k = 420 nm, I = 0.05 X lO W/cm, 56 fs long pulse of the same form as shown in Figure 2.10. An excellent agreement is note. In fact, this agreement holds not only for the final value of the probability for the molecule to remain bound after the pulse is over but also for the survival probability evaluated at any time during the pulse.

By definition, (t) represents the probabihty amphtude that, after a finite evolution time t, the system is stiU in the same state as at time t = 0. Referring back to Eq. (32), we see that (t) and the spectral intensity distribution P E) are related by Fourier transformation and thus contain the same information on the system dynamics. The square (t)P is also known as survival probability and provides a measure of noiuadiative decay processes occurring on picosecond or nanosecond time scales. ... 

Figure 1.9 Time evolution of the squared norm (survival probabilities) for onedimensional wave packets started at TS1 or TS2 for the [CpCo(L)H(C2H4)] complexes. The fitted norm is based on y=(1 — c)e H-c.

Computation of transition probabilities. In addition to survival amplitudes, the RRGM can also be used to compute state-to-state time-dependent transition amplitudes. If we denote the initial state at t = 0 as i), then the state that evolves from this initial state is i(t)) = U(t) i), where U t) is again the evolution operator. At time t, the amplitude for finding state /) in this evolved state is given by A,f i) = (f U(t) i). If we know the eigenvectors, i / ), and eigenvalues, Ea, for this Hamiltonian, then the transition amplitude can be written... 

To take into account the evolution of the living conditions, the generic failure rate A,j that represents the probability density that the component fails to mode k in the interval t,t + dt knowing that it has survived with no failures up to the time t, should be continuously updated to account for its dependence from the evolving IPs. In the proposed modeling approach, Xj is updated at time steps of length at most Dt and then remains constant within the time step. In practice, the duration of the time step must be chosen so as to satisfy the hypothesis. 

The evolution of human food consumption happened alongside the development of a vast bacterial ecosystem resident within the human body this gut microbiome comprises a cargo of commensals and symbionts that has collectively evolved to survive in the distal part of the human intestine. This community of microorganisms is thought to harbour more than 100-fold the number of genes within the human genome (BSckhed et al 2004). This genetic payload is carried by an excess of lO microbial cells with over 1000 bacterial types (Wallace et al 2011) - the evolution of which probably happened in response to the particular diet available to humans at a particular time. 

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