Chaos Theory and Randomness from Initial Conditions…So if the initial digit sequence is random, as in the first example, then the subsequent behavior will also be correspondingly random.

Note (e) for Randomness from the Environment…Electronic randomness Since the 1940s a steady stream of electronic devices for producing randomness have been invented, with no single one ever becoming widely used. … (U.S. missiles apparently used a similar method to produce randomly spaced radar pulses for determining altitude.) … Macroscopic thermal diffusion undoubtedly underestimates the time for good microscopic randomization.

Note (e) for The Intrinsic Generation of Randomness…Randomness in Mathematica SeedRandom[n] is the function that sets up the initial conditions for the cellular automaton. The idea of using this kind of system in general and this system in particular as a source of randomness was described in my 1987 U.S. patent number 4,691,291.

Note (b) for The Intrinsic Generation of Randomness…Sources of repeatable randomness In using repeatability to test for intrinsic randomness generation, one must avoid systems in which there is essentially some kind of static randomness in the environment.

Note (b) for Defining the Notion of Randomness…History [of randomness definitions] Randomness and unpredictability were discussed as general notions in antiquity in connection both with questions of free will (see page 1135 ) and games of chance. … With the development of statistical hypothesis testing in the early 1900s various tests for randomness were proposed (see page 1084 ). … Note that definitions of randomness given in dictionaries tend to emphasize lack of aim or purpose, in effect following the common legal approach of looking at underlying intentions (or say at physical construction of dice) rather than trying to tell if things are random from their observed behavior.

Chaos Theory and Randomness from Initial Conditions…[No text on this page] An arrangement of mirrors set up to exhibit randomness arising from sensitive dependence on initial conditions.

Note (b) for Randomness from the Environment…Randomness in computer systems Most randomness needed in practical computer systems is generated purely by programs, as discussed on page 317 . But to avoid having a particular program give exactly the same random sequence every time it is run, one usually starts from a seed chosen on the basis of some random feature of the environment. … Versions of the Unix operating system, for example, began to support a virtual device (typically called /dev/random) to maintain a kind of pool of randomness based on details of the computer system.

Note (b) for Three Mechanisms for Randomness…Definition [of randomness] How randomness can be defined is discussed at length on page 552 .

Note (b) for Three Mechanisms for Randomness…Sources of randomness Two simple mechanical methods for generating randomness seem to have been used in almost every civilization throughout recorded history. … Horse races and other events used as a basis for gambling can be viewed as randomness sources. … Pegboards (as on page 312 ) were used to demonstrate effects of randomness in the late 1800s.

Note (a) for Three Mechanisms for Randomness…Applications of randomness See page 1192 .