Conditional transition probabilities

The transition x —> x" is determined by one-half of the external change in the total first entropy. The factor of occurs for the conditional transition probability with no specific correlation between the terminal states, as this preserves the singlet probability during the reservoir induced transition [4, 8, 80]. The implicit assumption underlying this is that the conductivity of the reservoirs is much greater than that of the subsystem. The second entropy for the stochastic transition is the same as in the linear case, Eq. (71). In the expression for the second entropy... 

Fig. 8. Representatives of the four conformations obtained in the M = 4 analysis and the conditional transition probabilities between them (lag time r = O.lps). Fat numbers indicating the statistical weight of each conformation, numbers in brackets the conditional probability to stay within a conformation. Flexibility in peptide angles is marked with arrows, cf. Fig. 7. Note that the transition matrix relating to this picture is not symmetric but reversible. Top left For the helix conformation the backbone is colored blue for illustrative purpose. It should be obvious from Fig. 5 that for significantly larger lag time r only two eigenvalues will correspond to metastability such that only the helical conformation and a mixed flexible and partially unfolded one remain with significantly high conditional probability to stay within...

At this point, it is usually assumed that the average of the classical variables over the conditional transition probabilities, so-called stochastic averaging, is equivalent to averaging the classical equations of motion over the equilibrium distribution of the bath. More specifically,... 

Pratt [43] made the innovative suggestion that transition pathways could be determined by maximizing the cumulative transition probability connecting the known reactant and product states. That is, the most probable transition pathways would be expected to be those with the largest conditional probability. 

Since the equilibrium probability Ed.s, t) contains the Boltzmann factor with an energy Tid.s, ), the condition (12) leads to the ratio of transition probabilities of the forward and backward processes as... 

As long as the condition (13) is satisfied, any choice of the transition probability is possible. For the lattice-gas model with the Hamiltonian (2), a simple choice is the following ... 

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a ... 

To illustrate this, we shall start with 2500 A ingredients and set the transition probabilities to Pi (A B) = 0.01, Pi (B A) = 0.02, Pi (A C) = 0.001, and Pi (C A) = 0.0005. Note that these values yield a situation favoring rapid initial transition to species B, since the transition probability for A B is 10 times than that for A C. However, the formal equilibrium constant eq[C]/[A] is 2.0, whereas eq[B]/[A] = 0.5, so that eventually, after the establishment of equilibrium, product C should predominate over product B. This study illustrates the contrast between the short run (kinetic) and the long run (thermodynamic) aspects of a reaction. To see the results, plot the evolution of the numbers of A, B, and C cells against time for a 10,000 iteration run. Determine the average concentrations [A]avg, [B]avg, and [C]avg under equilibrium conditions, along with their standard deviations. Also, determine the iteration Bmax at which ingredient B reaches its maximum value. 

Matsuda and Hata [287] have argued that the species that are detectable using OES only form a very small part (<0.1%) of the total amount of species present in typical silane deposition conditions. From the emission intensities of Si and SiH the number density of these excited states was estimated to be between 10 and 10 cm", on the basis of their optical transition probabilities. These values are much lower than radical densities. lO " cm . Hence, these species are not considered to partake in the deposition. However, a clear correlation between the emission intensity of Si and SiH and the deposition rate has been observed [288]. From this it can be concluded that the emission intensity of Si and SiH is proportional to the concentration of deposition precursors. As the Si and SiH excited species are generated via a one-electron impact process, the deposition precursors are also generated via that process [123]. Hence, for the characterization of deposition, discharge information from OES experiments can be used when these common generation mechanisms exist [286]. 

Here g has been set to zero, as is justified later.) This shows that the fluctuations in the-transition probability are determined by a symmetric matrix, g, in agreement with previous analyses [35, 82]. Written in this form, the second entropy satisfies the reduction condition upon integration over x to leading order (c.f. the earlier discussion of the linear expression). One can make it satisfy the reduction condition identically by writing it in the form... 

The final equality follows from the normalization of the conditional stochastic transition probability. This is the required result, which shows the stationarity of the steady-state probability under the present transition probability. This result invokes the preservation of the steady-state probability during adiabatic evolution over intermediate time scales. 

The evolution of the probability distribution over time consists of adiabatic development and stochastic transitions due to perturbations from the reservoir. As above, use a single prime to denote the adiabatic development in time A, r —> r, and a double prime to denote the final stochastic position due to the influence of the reservoir, T —> T . The conditional stochastic transition probability may be taken to be... 

Based on the results obtained in the investigation of the effects of modulation of the electron density by the nuclear vibrations, a lability principle in chemical kinetics and catalysis (electrocatalysis) has been formulated in Ref. 26. This principle is formulated as follows the greater the lability of the electron, transferable atoms or atomic groups with respect to the action of external fields, local vibrations, or fluctuations of the medium polarization, the higher, as a rule, is the transition probability, all other conditions being unchanged. Note that the concept lability is more general than... 

The expression for the frequency factor A in the mean transition probability depends on the values of Wlf(q) and Wfl(q). If they are sufficiently small so that the conditions... 

The above method enables us to calculate the transition probability at various initial nonequilibrium conditions. As an example, we will consider the transition from the state in which the initial values of the coordinate and velocity of the reactive oscillator are equal to zero.85 In this case, the normalized distribution function has the form... 

The activation factor in the first case is determined by the free energy of the system in the transitional configuration Fa, whereas in the second case it involves the energy of the reactive oscillator U(q ) = (l/2)fi(oq 2 in the transitional configuration. The contrast due to the fact that in the first case the transition probability is determined by the equilibrium probability of finding the system in the transitional configuration, whereas in the second case the process is essentially a nonequilibrium one, and a Newtonian motion of the reactive oscillator in the field of external random forces in the potential U(q) from the point q = 0 to the point q takes place. The result in Eqs. (171) and (172) corresponds to that obtained from Kramers theory73 in the case of small friction (T 0) but differs from the latter in the initial conditions. 

Eq. (174) gives the well-known expression for the transition probability [see Eqs. (9) and (10)]. If the condition opposite to Eq. (175) holds, the transition probability for the adiabatic process takes the form... 

Transition-matrix estimators are typically more accurate than their histogram counterparts [25,26,46], and they offer greater flexibility in accumulating simulation data from multiple state conditions. This statistical improvement over histograms is likely due to the local nature of transition probabilities, which are more readily equilibrated than global measures such as histograms [25], Fenwick and Escobedo... 

In this chapter we continue our journey into the quantum mechanics of paramagnetic molecules, while increasing our focus on aspects of relevance to biological systems. For each and every system of whatever complexity and symmetry (or the lack of it) we can, in principle, write out the appropriate spin Hamiltonian and the associated (simple or compounded) spin wavefunctions. Subsequently, we can always deduce the full energy matrix, and we can numerically diagonalize this matrix to obtain the stable energy levels of the system (and therefore all the resonance conditions), and also the coefficients of the new basis set (linear combinations of the original spin wavefunctions), which in turn can be used to calculate the transition probability, and thus the EPR amplitude of all transitions. 

Formula (2.2) contains only one-dimensional probability density W(xi, t ) and the conditional probability density. The conditional probability density of Markov process is also called the transition probability density because the present state comprehensively determines the probabilities of next transitions. Characteristic property of Markov process is that the initial one-dimensional probability density and the transition probability density completely determine Markov random process. Therefore, in the following we will often call different temporal characteristics of Markov processes the transition times, implying that these characteristics primarily describe change of the evolution of the Markov process from one state to another one. 

The transition probability density satisfies the following conditions ... 

The Transition Probability. Suppose we have a Brownian particle located at an initial instant of time at the point xo, which corresponds to initial delta-shaped probability distribution. It is necessary to find the probability Qc,d(t,xo) = Q(t,xo) of transition of the Brownian particle from the point c 0 Q(t,xo) = W(x, t) dx + Jrf+ X W(x, t) dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary. Here we suppose that c and d are arbitrary chosen points of an arbitrary potential profile (x), and boundary conditions at these points may be arbitrary W(c, t) > 0, W(d, t) > 0. 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

Presently Monte Carlo calculations are based on the technique proposed by Metropolis [22] in 1953 which involves selecting the successive configurations in such a way that they build up a Markov chain [23], The one-step transition probabilities pij are defined as the probability that beginning from the i configuration with qj(N), the configuration j with qj,N> is reached in one step. These probabilities are the elements of the one-step probability matrix associated to the Markov chain and they must fulfill the following conditions ... 

If the n-steps transition probability elements are defined as the probability to reach the configuration j in n steps beginning from the configuration i and Ilj, = n (qjMarkov chain is ergodic (the ergodicity condition states that if i and j are two possible configurations with 0 and Ilj 0, for some finite n, pij(nl 0 ) and aperiodic (the chain of configurations do not form a sequence of events that repeats itself), the limits... 

The interaction of forced and natural convective flow between cathodes and anodes may produce unusual circulation patterns whose description via deterministic flow equations may prove to be rather unwieldy, if possible at all. The Markovian approach would approximate the true flow pattern by subdividing the flow volume into several zones, and characterize flow in terms of transition probabilities from one zone to others. Under steady operating conditions, they are independent of stage n, and the evolution pattern is determined by the initial probability distribution. In a similar fashion, the travel of solid pieces of impurity in the cell can be monitored, provided that the size, shape and density of the solids allow the pieces to be swept freely by electrolyte flow. 

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