vol 2: Synopsis
part III: Modern physics
page 22: Probability
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Probability
One of the most startling features of quantum mechanics from the point of view of a classical physicist is that it does not always make definite predictions. Often it merely assigns probabilities to the different possible outcomes of an initial situation. This has led physicists to believe that the quantum world is not completely deterministic, as the Newtonian world it replaced was believed to be. Instead, as Max Born described wave mechanics: 'The motion of particles follows probability laws but the probability itself propagates according to the law of causality.' Pais Inward 258. The theory of probability is fundamental, and shows us how to understand systems that are partially controlled by the nature of their constituents and partially free. It enables us to measure constraint and freedom, a first step toward modelling an optimally constrained (and free) system.
Probability is the mathematics of tolerance. A probability is a measure or estimate of the degree of confidence one may have in the occurrence of a particular event, measured on a scale running from 0 (the event is impossible) to 1 (the event is certain).
The theory of probability is of particular interest to gamblers, and its first use was to calculate the probability of various outcomes in games of chance as a guide to rational investment. Since then probability theory has become an important branch of mathematics concerned with the relationship between abstract mathematical models and the real concrete world.
As with other mathematical theories, the theory of probability begins with a set of points and axioms (rules) rules governing the relationships of those points. In the case of probability theory, we begin with a set of elementary events. Such a set may contain, for instance, two events: a coin falling head up and a coin falling head down. It is a feature of axiomatic (or abstract) mathematical theories that each may have unlimited number of concrete interpretations. So the theoretical structure which governs the tossing of coins may be applied to any situation whose elementary events comprise p and not-p.
The formal study of probability seeks to establish relationships between the probabilities of elementary events and the probability of random events which comprise certain sets of elementary events. Thus we can build up a theory which relates the outcome of tossing a coin one hundred times to the outcome of tossing it once.
Probability theory is particularly important in physics. The classical application of probability theory gives us statistical mechanics. Classical statistical mechanics is a development of thermodynamics. Thermodynamics is concerned with the macroscopic behaviour of large numbers of individuals such as atoms or molecules. Classical statistical mechanics faces a similar task, of linking large numbers of microscopic elementary events to macroscopic phenomena such as the freezing and melting of water, or the formation of raindrops.
In the deterministic Newtonian world, classical statistical mechanics is seen as a method of compressing the vast amount of information available at the microscopic level into a smaller volume of information meaningful at the macroscopic level. In quantum mechanics, however, we find that definite information is often not available, and that the outcomes of quantum mechanical calculations are probabilities rather than certainties.
The theory of probability is intimately connected to the theory of communication. It plays an important part in the development of the theory of the transfinite network.
Books
| Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' Amazon back |
| Khinchin, A Y, The Mathematical Foundations of Quantum Statistics, Dover 1998 'In the area of quantum statistics, I show that a rigorous mathematical basis of the computational formulas of sttistical physics ... may be obtained from an elementary application of the well-developed limit theorems of the theory of probability' Amazon back |
| Kolmogorov, A N , and Nathan Morrison (Translator) (With an added bibliography by A T Bharucha-Reid), Foundations of the Theory of Probability, Chelsea 1956 Preface: 'The purpose of this monograph is to give an axiomatic foundation for the theory of probability. ... This task would have been a rather hopeless one before the introduction of Lebesgue's theories of measure and integration. However, after Lebesgue's publication of his investigations, the analogies between measure of a set and mathematical expectation of a random variable became apparent. These analogies allowed of further extensions; thus, for example, various properties of independent random variables were seen to be incomplete analogy with the corresponding properties of orthogonal functions ... 'back |
| Pais, Abraham, Inward Bound: Of Matter and Forces in the Physical World, Clarendon Press, Oxford University Press 1986 Preface: 'I will attempt to describe what has been discovered and understood about the constituents of matter, the laws to which they are subject and the forces that act on them [in the period 1895-1983]. ... I will attempt to convey that these have been times of progress and stagnation, of order and chaos, of belief and incredulity, of the conventional and the bizarre; also of revolutionaries and conservatives, of science by individuals and by consortia, of little gadgets and big machines, and of modest funds and big moneys.' AP Amazon back |
| Tanenbaum, Andrew S, Computer Networks, Prenctice Hall International 1996 Preface: 'The key to designing a computer network was first enunciated by Julius Caesar: Divide and Conquer. The idea is to design a network as a sequence of layers, or abstract machines, each one based upon the previous one. ... This book uses a model in which networks are divided into seven layers. The structure of the book follows the structure of the model to a considerable extent.' Amazon back |
| Wax, Nelson, and (editor), Selected Papers on Noise and Stochastic Processes, Dover 1954 These six papers on stochastic processes have been selected by Nelson Wax ... to meet the needs of American physicists, applied mathematicians and engineers. ... The papers ... are reproduced unabridged.' Authors are: Chandrasekhar, Doob, Kac, Ming, Ornstein, Rice and Uhlenbeck. Amazon back |
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