Transition probability canonical form

Summary. The special class of master equations characterized by (1.1) will be said to be of diffusion type. For such master equations the -expansion leads to the nonlinear Fokker-Planck equation (1.5), rather than to a macroscopic law with linear noise, as found in the previous chapter for master equations characterized by (X.3.4). The definition of both types presupposes that the transition probabilities have the canonical form (X.2.3), but does not distinguish between discrete and continuous ranges of the stochastic variable. The -expansion leads uniquely to the well-defined equation (1.5) and is therefore immune from the interpretation difficulties of the Ito equation mentioned in IX.4 and IX.5. 

The high a-reactivity probably results principally from stabilisation of the canonical form 20-11 in the transition state by the electron release from the tin, and the P-reactivity from stabilisation of the resulting P-stannylalkyl radical by hyperconjugation (Section 3.1). 

In the Metropolis MC, the reciprocal of the maximum displacement allowed for an MC move during time interval At may be taken as the parameter Cl. Then the transition probability has the canonical form with... 

In the particular case of a naive algorithm, i.e., the case in which displacements are proposed at random, one can see that the transition probabilities T X Y) and T Y X) are the same. For a canonical, NVT ensemble, the function / takes the form... 

The relative enei gies of n and k bases are given in Table lO. It is obvious that the canonical amino and keto forms of the new bases are exceptionally stable with respect to the amino/imino and ketofenol tautomeric transitions. It was suggested that this property makes both bases proper candidates for the species, reducing the probability of spontaneous mutations. However, calculations of the interaction energy for the jt-ac base pair reveal its notable propeller twist... 

There are mainly historical reasons why the canonical approach, as discussed in the previous section, has been used most frequently for the analysis of structural and phase transitions. This is because the exponential form of the canonical microstate probability (2.22) in suitable for the development of approximation methods and field-theoretical formulations which enable analytic calculations of thermodynamic quantities. 

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