Between order and chaos

Journal name:
Nature Physics
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Published online
Corrected online


What is a pattern? How do we come to recognize patterns never seen before? Quantifying the notion of pattern and formalizing the process of pattern discovery go right to the heart of physical science. Over the past few decades physics’ view of nature’s lack of structure—its unpredictability—underwent a major renovation with the discovery of deterministic chaos, overthrowing two centuries of Laplace’s strict determinism in classical physics. Behind the veil of apparent randomness, though, many processes are highly ordered, following simple rules. Tools adapted from the theories of information and computation have brought physical science to the brink of automatically discovering hidden patterns and quantifying their structural complexity.

At a glance


  1. [epsi]-machines for four information sources.
    Figure 1: ε-machines for four information sources.

    a, The all-heads process is modelled with a single state and a single transition. The transition is labelled p|x, where p[0;1]is the probability of the transition and x is the symbol emitted. b, The fair-coin process is also modelled by a single state, but with two transitions each chosen with equal probability. c, The period-2 process is perhaps surprisingly more involved. It has three states and several transitions. d, The uncountable set of causal states for a generic four-state HMM. The causal states here are distributions Pr(A,B,C,D) over the HMM’s internal states and so are plotted as points in a 4-simplex spanned by the vectors that give each state unit probability. Panel d reproduced with permission from ref. 45.

  2. Structure versus randomness.
    Figure 2: Structure versus randomness.

    a, In the period-doubling route to chaos. b, In the two-dimensional Ising-spinsystem. Reproduced with permission from: a, ref. 36, © 1989 APS; b, ref. 61, © 2008 AIP.

  3. Complexity-entropy diagrams.
    Figure 3: Complexity–entropy diagrams.

    a, The one-dimensional, spin-1/2 antiferromagnetic Ising model with nearest- and next-nearest-neighbour interactions. Reproduced with permission from ref. 61, © 2008 AIP. b, Complexity–entropy pairs (hμ,Cμ)for all topological binary-alphabet ε-machines with n=1,…,6 states. For details, see refs 61 and 63.

Change history

Corrected online 17 May 2013
In the version of this Review Article originally published online there were several errors. In the section entitled 'Complicated yes, but is it complex?', seventh paragraph, the subscripts should appear as follows: past X:t; future Xt:; and blocks Xt:t'. All instances of x:l should appear as x0:l. In the section entitled 'Applications', in the third and seventh paragraphs, x: should appear as x0:. In the caption of Fig. 1d, the distributions should be listed as Pr(A, B, C, D), and panel d should have been attributed to ref. 45. These errors have now been corrected in the PDF and HTML versions of the Review Article.


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  1. Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, California 95616, USA

    • James P. Crutchfield

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