Foundations, 1948 to 1987.

Computational mechanics grew out of two questions from the study of chaos: how unpredictable is a system, and can its workings be recovered from measurements alone?

1. 1948

   Information becomes measurable

   Claude Shannon defines information and entropy, the quantities every later step is measured in.

   C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal 27 (1948) 379-423, 623-656.

2. 1950s

   The smallest machine that does the job

   David Huffman's work on minimal machines asks for the simplest automaton that produces a given behaviour. Crutchfield later learned it from Huffman directly, at UC Santa Cruz.

3. 1958 to 1965

   Entropy for dynamical systems

   Kolmogorov and Sinai carry Shannon's entropy into dynamical systems; Kolmogorov and Chaitin define the complexity of a single sequence.

   A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 119 (1958) 861; Ja. G. Sinai, Dokl. Akad. Nauk SSSR 124 (1959) 768; A. N. Kolmogorov, Prob. Info. Trans. 1 (1965) 1.

4. 1963 and 1971

   Order hidden in chaos

   Lorenz shows that simple deterministic equations can be unpredictable; Ruelle and Takens propose strange attractors as the mechanism of turbulence.

   E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130; D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971) 167-192.

5. 1980

   Geometry from a time series

   Packard, Crutchfield, Farmer and Shaw show that a system's state can be reconstructed from a series of its own measurements.

   N. H. Packard, J. P. Crutchfield, J. D. Farmer and R. S. Shaw, Geometry from a time series, Physical Review Letters 45 (1980) 712-716.

6. 1986 and 1987

   From chaos to equations of motion

   The same group explains chaos to a general audience, and Crutchfield and McNamara recover equations of motion directly from data, which shows both the promise and the need for a representation that is not chosen by hand.

   J. P. Crutchfield, J. D. Farmer, N. H. Packard and R. S. Shaw, Chaos, Scientific American 255 (December 1986) 46-57; J. P. Crutchfield and B. S. McNamara, Equations of motion from a data series, Complex Systems 1 (1987) 417-452.