Maximal entropy distribution calculation

Note that the coordinatewise optimization method has already found numerous practical applications to optimization of heat, oil, water, and gas supply systems (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumarokov, 1976). As a matter of fact, in the algorithms used for applied problems the flow distribution was calculated not on the base of entropy maximization, but with the help of the closed system of equations of the first and second Kirchhoff laws. However, because of equivalence of approaches that are based on the principle of conservation and equilibrium (extremality) the Kirchhoff equations can be strictly replaced by thermodynamic relations. And the extreme thermodynamic approach in many cases should be preferable owing to the known low sensitivity of the extremal methods to variation of the space of variables. 

One example is the maximum entropy formalism (MEF), which assiunes the final distribution is that which maximizes the entropy production. A detailed discussion is given in [24]. In general, the method is able to calculate the correct shape of the distribution. However, experimental results are needed to obtain the magnitude. At the current time, no method is predictive without some experimental input [4]. 

The statistical ensemble framework of equilibrium statistical mechanics gives us the tools to analyze experimental data and to make theoretical predictions. The concept of entropy, its maximization, and the ensuing definition of intensive quantities such as pressure and temperature reduces the complexity of a statistical system of 10 particles to manageable proportions. In principle, the problem of predicting the collective behavior of equilibrium systems starting from microscopic interactions is solved. In practice, exact calculations are rarities. Computational tools such as the Monte Carlo Metropolis method can, however, fill in this void a priori knowledge of the probability distribution of microstates is at the heart of the Metropolis algorithm. 

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