Thermodynamics is about temperature and heat, both of which are irrelevant to the dynamics of these complex, non–equilibrium systems. But all of the thermodynamic variables that we can measure experimentally in an equilibrium system can be calculated from the Boltzmann distribution, and hence statements about thermodynamics are equivalent to statements about this underlying probability distribution. It is then only a small jump to realize that [*all*]{} probability distributions over $N$ variables can have an associated thermodynamics in the $N\rightarrow\infty$ limit. This link between probability and thermodynamics is well studied by mathematical physicists [@ruelle], and has been a useful guide to the analysis of experiments on dynamical systems [@halsey+al_86; @feigenbaum+al_86], although perhaps still is not as widely appreciated as it might be.

To be concrete, we consider a system with $N$ elements, and each element is described by a state $\sigma_{\rm i}$; the state of the entire system is given by ${\mathbf \sigma} \equiv \{\sigma_1 ,\, \sigma_2, \,\cdots , \, \sigma_N\}$. We are interested in the probability $P({\mathbf\sigma})$ that we will find the system in any one of its possible states. There is one state ${\mathbf\sigma}_0$ that is the most likely state, and we can measure all probabilities relative to the 