Maximum Entropy, Information Without Probability and Complex Fractals: Classical and Quantum Approach (Fundamental Theories of Physics, 112)

Every thought is a throw of dice. Stephane Mallarme This book is the last one of a trilogy which reports a part of our research work over nearly thirty years (we discard our non-conventional results in automatic control theory and applications on the one hand, and fuzzy sets on the other), and its main key words are Information Theory, Entropy, Maximum Entropy Principle, Linguistics, Thermodynamics, Quantum Mechanics, Fractals, Fractional Brownian Motion, Stochastic Differential Equations of Order n, Stochastic Optimal Control, Computer Vision. Our obsession has been always the same: Shannon's information theory should play a basic role in the foundations of sciences, but subject to the condition that it be suitably generalized to allow us to deal with problems which are not necessarily related to communication engineering. With this objective in mind, two questions are of utmost importance: (i) How can we introduce meaning or significance of information in Shannon's information theory? (ii) How can we define and/or measure the amount of information involved in a form or a pattern without using a probabilistic scheme? It is obligatory to find suitable answers to these problems if we want to apply Shannon's theory to science with some chance of success. For instance, its use in biology has been very disappointing, for the very reason that the meaning of information is there of basic importance, and is not involved in this approach. This book presents material on three topics, namely the amount of information involved in non-random functions, the amount of information involved in non-probabilistic square matrices (i.e. which are not quantum density matrices), and a new model of complex-valued fractional Brownian motion of order n defined via random walks in the complex plane. These three subjects, which on the surface have no common features, are, in fact, direct consequences of the maximum entropy principle. Moreover, information on non-random functions and complex fractional Brownian motion are directly related to fractals. Thus, a unified framework is constructed which encompasses information with and without probability, quantum information of square matrices with and without probabilistic meaning, and fractals in the complex plane. This volume also features many applications. Audience: This work is intended for theoretical and mathematical physicists, but also for applied mathematicians, experimental physicists, communication engineers, electrical engineers, practitioners in pattern recognition and computer vision, control systems engineers, and theoretical biologists. Front Matter....Pages I-XIX Introduction....Pages 1-8 Summary of Information Theory....Pages 9-28 Path Entropies of Non-Random Functions....Pages 29-59 Path Entropies of Random Functions and of Non-Random Distributed Functions....Pages 60-82 Quantum Entropies of Non-Probabilistic Square Matrices....Pages 83-128 Complex-Valued Fractional Brownian Motion of Order n . Part I....Pages 129-155 Complex-Valued Fractional Brownian Motion of Order n . Part II....Pages 156-181 Information Thermodynamics and Complex-Valued Fractional Brownian Motion of Order n ....Pages 182-204 Fractals, Path Entropy, and Fractional Fokker-Planck Equation....Pages 205-229 Outline of Applications....Pages 230-265 Back Matter....Pages 267-270