Non-Extensive Thermodynamics of Algorithmic Processing - the Case of Insertion Sort Algorithm

In this chapter it will be shown that there can exist possible connections of Tsallis nonextensive definition of entropy (Tsallis, 1988) with the statistical analysis of simple insertion sort algorithm behaviour. This will be done basing on the connections between the idea of Turing machines (Turing, 1936) as a basis of considerations in computer science and especially in algorithmic processing and the proposal of non-equilibrium thermodynamics given by Constatino Tsallis (Tsallis, 1988; Tsallis, 2004) for indication of the possible existence of non-equilibrium states in the case of one sorting algorithm behaviour. Moreover, it will be also underlined that a some kind of paradigm change (Kuhn, 1962) is needed in the case of computer systems analysis because if one considers the computers as physical implementations of Turing machines should take into account that such implementations always need energy for their work (Strzalka, 2010) – Turing machine as a mathematical model of processing does not need energy. Because there is no (computer) machine that have the efficiency η = 100%, thus the problem of entropy production appears during their work. If we note that the process of sorting is also the introduction of order (obviously, according to a given appropriate relation) into the processed set (sometimes sorting is considered as an ordering (Knuth, 1997)), thus if one orders it must decrease the entropy in sorted set and increase it somewhere else (outside the Turing machine – in physical world outside its implementation). The connections mentioned above will be given basing on the analysis of insertion sorting, which behaviour for some cases can lead to the levels of entropy production that can be considered in terms of non-extensivity. The presented deliberations can be also related to the try of finding a new thermodymical basis for important part of authors' interest, i.e., the physics of computer processing.