Backward evolution equations

Obviously, if Plx, Z xo) is known we can compute the mean first passage time from Eq. (8.154). We can also find an equation for this function, by operating with backward evolution operator t (xo) on both sides ofEq. (8.154). Recall that when the operator L is time independent the backward equation (8.122) takes the form (cf. Eq. (8.123b)) 9P(x, Z xo)/9Z = T f (xo)T (x, Z xo) where D is the adjoint... 

Photoionization generates the initial condition for geminate recombination. Now the difference of a few angstroms between Rq and R can be of crucial importance, increasing the survival chances of the newborn ions attracted by the Coulomb potential. The distribution of ions provided by Eq. (9.125) can be used as the initial condition only if ionization is so fast that it is completed before the recombination actually starts. In general, one should consider the backward and forward ET simultaneously, by including a nonlocal source term into the evolution equation for the ion pair probability distribution (r(r, t), as shown by Burshtein et al. [330] and by Dorfman and Payer [331]. As a result, one obtains for the evolution of the total population I t) of ion pairs. 

Solving the forward problem of the isotopic and chemical evolution of n reservoir exchanging a radioactive and its daughter isotope requires the solution of 3n— 1 differential equations (the minus one stems from the closure condition). The parameters are n (n — 1) independent flux factors k for the stable isotope N and n (n — 1) independent M/N fractionation factors D. In addition, the n values of R y the n values of Rh and the n—1 allotments x of the stable isotope among the reservoirs must be assumed at some time, preferably at the beginning of the evolution (e.g., 4.5 Ga ago), or in the modern times, in which case integration is carried out backwards in time. 

Here, the longitudinal evolution of the modes is described by the functions () and () instead of a simple exponential term as in Eq. (4) since we have to take into account both forward and backward propagation modes. Substituting Eq. (11) into Maxwell equations and taking into account the relations described by Eq. (5) we find that the functions Vm (C) and have to satisfy the following very simple set of first-... 

Equation 15.41) explicitly describes how the time evolution of the phase angles must vary so as to minimize the value of (E + population transfer between potential-energy surfaces. Note that control of the field phase is critically influenced by the backward prop-... 

This is an exact system of equations that describes the evolution of modal amplitudes along the z-axis for the forward propagating field. A similar equation holds for the backward propagating component, of course. 

In Equation 1.65, the first part in the parentheses of the equation represents the forward reaction rate, indicating hydrogen oxidation the second part represents the backward reaction rate, indicating the hydrogen evolution reaction. The net hydrogen oxidation rate equals the forward reaction rate minus the backward reaction rate. 

However, this cannot be done The operator U [7 (t —t) is evolution backwards, and it is the backwards solution of the equation we want. This just does not exist... 

The pres it understanding of processes in the interior of stars is the result of combined efforts from many scientific disciplines such as hydrodynamics, plasma physics, nuclear physics, nuclear chemistry and not least astrophysics. To understand what is going on in the inaccessible interior of a star we must make a model of the star which explains the known data mass, diameter, luminosity, surface temperature and surface composition. The development of such a model normally starts with an assumption of how elemental conqx)sition varies with distance from the center. By solving the difierential equations for pressure, mass, tenq>erature, luminosity and nuclear reactions from the surface (where these parameters are known) to the star s center and adjusting the elemental composition model until zero mass and zero luminosity is obtained at the center one arrives at a model for the star s interior. The model developed then allows us to extrapolate the star s evolution backwards and forwards in time with some confidence. Figure 17.4 shows results from such modelling of the sun. 

The mathematical structure of the four MQCB equations (22-24) is such that the time evolution of the combined quantities F t) is uniquely determined by the MQCB equations, and reversibility is given in a strict mathematical sense. Upon propagation r(0)- F(t), changing d/dt— -9/9t and X(t) — X t) propagates the state F(t) backwards to jrield the same initial state F(t) — r(0). 

Consider the evolution of the breakage process as viewed by (7.2.5) from the instant t to t. We regard this time interval to be suitably small in order that the population increases by at most one particle. Thus, if we envisage V particles at time t with masses 2,..., the population at time t must consist of no less than v — 1 particles with masses that must be determined by solving backwards the differential equation... 

The time evolution of these conditional probabilities, P (5, f 15,., ), can also be described by an alternative linear equation instead of the FME, known as the backward ME (BME), which is especially useful in the context of EP problems. The key to deriving the BME equation is to consider the first step out of the initial state S. at time t , rather than the last step of the trajectory, leading to the state S, at time t. Similar to the derivation of the FME, the conditional probability, P (S, 11S , ), can be written down as the sum of the probabiUties of two mutually exclusive events (i) that the system transitioned to a different state S during the time interval dt, with the probability a S, S)dt, and then evolved to a state S by time t, with the probability P(S, 11 S, t +dt), or (ii) that the system was still in state S. at time + with the probability 1-, a (5, 5 )dt, and then by time t evolved to the state 5, with the probability P(S,t S, ti +dt). Together, these terms yield the following equation ... 

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