Mathematical methods complex notation

All the derivations of the Boltzmann equation are based on a number of assumptions and hypotheses, making the analyzes somewhat ad hoc irrespective of the formal mathematical rigor and complexity accomplished. So in this book a heuristic theory, which is physically revealing and equally ad hoc to the more fundamental derivations of the Boltzmann equation, is adopted. It is stressed that the notation used resembles that introduced by Boltzmann [6] and is not strictly in accordance with the formal mathematical methods of classical mechanics. However, some aspects of the formal formulations and vocabulary outlined in sect. 2.2 are incorporated although somewhat based on intuition and empirical reasoning rather than fundamental principles as discussed in sect. 2.3. 

Manipulations needed to relate heat flow hr to temperature changes through theoretical models for transitions, or through properties of calorimeters, are usually more conveniently done via such complex qualities. The value C can then lead directly to evaluations of real specific heat and parameters controlling kinetics. However, the use of complex notation does not imply a different theoretical treatment or method. It is simply a more convenient mathematical formalism. The terms real and imaginary heat capacity and phase-corrected reversing and kinetic heat capacities are interchangeable. 

These methods essentially follow the rote procedure outlined above. The important difference is that the mass action, mass balance, and charge balance equations are written in generalized mathematical notation. They can then be applied to any chemical system by specifying the reactions and species of interest. The approach we outline here is described in detail by Crerar (1975). However, we include a change in the mass action equations which was not in the original paper this improves the speed of the method and its chances of success with very complex systems. Consider an bitrary system of c components containing N chemical species. Equilibrium constants are known for M independent reactions relating some or all of these species. 

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