Maximal entropy distribution determination

Consider first the limit where no dynamical input is given. Then only the value of C(t) at / = 0 is imposed as a constraint in determining P(yf) by the procedure of maximal entropy. In this case all the Lagrange multipliers have exactly the same value, namely (x = 1/X.o and so all the distributions P(yf) are the same. One can therefore explicitly carry out the sum over final states in (102), with the result... 

The details of determining a distribution of maximal entropy are spelled out in a number of reviews. Here we just give an example suppose the dynamic constraint is the final vibrational energy. This means that the dynamics imposes on the distribution of final states a mean value of the final vibrational energy... 

For a dimension-bearing variable, one generally expects SS oc Sy/y or S(y) oc lny. Hence we can impose as a constraint on the maximization of the entropy. This will lead to a y-square distribution where now (v - l)/2 is the Lagrange multiplier for this additional constraint. The value of v is now to be determined as usual, by equating the value of as determined from (2.25)... 

In Section 2, we introduce model identification to interpret dynamic PCA, and find that the selection of the time window length can be accordingly solved with the approach of determination the order of AR model. Identification of a system model may be regarded as confirming the probability distribution of a stochastic process. From this understanding, the principle of information criteria is to select appropriate model structure to maximize the approximate extent between the real probability distribution and the estimated probability distribution based on observation data. Shannon entropy is always used to measure the approximate extent, which is represented as ... 

The second law of thermodynamics states that, under equilibrium conditions, the entropy is at maximum. This law can be used to determine equilibrium expressions for thermodynamic properties. However particular constraints also have to be satisfied. In a closed system of interacting molecules at equilibrium, entropy is maximized under the condition that the total number of particles as well as the total energy are constant. As will be shown this determines the energy distribution function of the particles in the system. The entropy is computed from W, the number... 

---