Infinitesimal transition probability

The infinitesimal transition probabilities Pj it) have Puit) = 0 and Y.Pjk(t) = U ye 1 "-Using the Chapman-Kolmogorov equation we can derive... 

The two models are equivalent when the infinitesimal transition probabilities are linear in every co-ordinate of the indices. 

The methods start from an evolution equation describing the internal fluctuation. In the next step a fixed value of an external parameter, namely the infinitesimal transition probability, is substituted by a stochastic process. This procedure can be done in the master equation or in the equation for the generating function. The first case leads to an integrodifferential equation, while the second model leads to a stochastic partial differential equation. 

The previous results for the transition probability held over intermediate time scales. On infinitesimal time scales the adiabatic evolution of the steady-state probability has to be accounted for. The unconditional transition probability over an infinitesimal time step is given by... 

We have shown that the transition probability to states of different energy is always an infinitesimal quantity which tends to zero with k under very general con-... 

The concept of infinitesimal operator is frequently used when the random evolutions are the generators of stochastic models from a mathematical point of view. This operator can be defined with the help of a homogeneous Markov process X(t) where the random change occurs with the following transition probabilities ... 

We have to notice that, for different X(t) values, we associate different values for the elements of the matrix of transition probabilities. When the movement randomly changes the value of X into a value around A, Eq. (4.91) is formulated with expressions giving the probability of process X(t) at different states. The infinitesimal operator [Qf] ([Qf] = Q by function f) is defined as the temporary derivative of the mean value of the stochastic process for the case when the process evolves randomly... 

The simulated spectrum is computed by the use of eq. (25), wherein the integrals are converted into discrete sums. It is clear from (25) that, in particular, one needs to know the resonant field values for the various transitions, as well as their transition probabilities for numerous orientations of the external magnetic field over the unit sphere over the unit sphere. A considerable saving of computer time can be accomplished if one uses numerical techniques to minimize the number of required diagonalizations of the SH matrix in the brute-force method. That is, when one uses the known resonant-field value at angle (0,(p) to calculate the one at an infinitesimally close orientation, (0 -i- 80, (p + 8(p), known as the method of homo-... 

Where A, Ax, and Av are normalization constants. Since the potential and kinetic energies are additive, the probability of finding the system in the configuration interval ([3c, 3c + dx], [v, v + dv]) turns out to be separable into probabilities of position and velocity. We can now determine the probability of finding the system in the infinitesimal vicinity of thickness, Sx, around the transition state to be ... 

We first review fundamentals of the theory of stochastic processes. The system dynamics are specified by the set of its states, 5, and the transitions between them, S -> S, where S,S e 5. For example, the state S can denote the position of a Brownian particle, the numbers of molecules of different chemical species, or any other variable that characterizes the state of the system of interest. Here we restrict ourselves to processes for which the transition rates depend only on the system s instantaneous state, andnotontheentirety of its history. Such memoryless processes are known as Markovian and are applicable to a wide range of systems. We also assume that the transition rates do not explicitly depend on time, a condition known as stationarity. In this review we make the standard assumption that the transitions between the states are Poisson distributed random processes. In other words, the probability of transitioning from state S to state 5 in an infinitesimal interval, dt, is a S,S )dt, where a(S,S ) is the transition rate. 

---