A guiding question of recent work in dynamic systems is how order – that is, the sophisticated, stable patters which are readily apparent – are able to emerge despite the second law of thermodynamics, which states that order tends to move to chaos, or that systems in disequilibrium tend to move to equilibrium.[…] To take a frequently cited example, oil, when heated, will suddenly exhibit convection rolls and vortices as it is heated and before it boils. Before the oil is heated, the oil is in an equilibrium state in which entropy is at a maximum; in other words, the oil molecules are randomly scattered throughout the container such that no order or consistency is present. One section of the container would be indistinguishable from another. We thus have chaos, or a random set of points with no identifiable order, what is called ‘equilibrium thermal chaos.’ When heated, however, the oil moves away from equilibrium, and it is under these conditions that the convection rolls and vortices appear. Once the oil is in a full boil, chaos reappears, or ‘non-equilibrium thermal chaos,’ and subsequently one section of the boiling oil is indistinguishable from any other. Dynamic systems and chaos theorists will pay particular attention to far-from-equilibrium conditions, and more precisely to the order which emerges at the critical threshold between equilibrium and non-equilibrium chaos.

The far-from-equilibrium conditions which give rise to spontaneous order most often occur during what is called phase transition. A phase transition is a transition between two steady and stable equilibrium states, such as liquid and gas, or liquid and solid. As these systems approach a phase transition, they enter a far-from-equilibrium state wherein self-organized patterns tend to emerge, and at a critical point (e.g., of temperature), there is a discontinuous jump to the new phase. Related to these phase transitions, and also occurring in far-from-equilibrium conditions, is the phenomenon of bifurcations. As Ilya Priogogine and Isabelle Stengers discuss bifurcations in their well-known book Order out of Chaos, a bifurcation point arises at a critical point where a system is poised to transition and when not just one stable state but, rather, ‘two new stable solutions emerge.’ For example, at the critical point where the stable solution of a convection roll appears in the heated oil the rolling motion may assume either a clockwise or counterclockwise direction – both solutions are possible. Which solution, or which branch of the bifurcation the system will ‘choose,’ is impossible to predict: ‘How will the system choose between left and right? There is an irreducible random element; the macroscopic equation cannot predict the path the system will take … We are faced with the chance events very similar to the fall of dice.’ As the oil is heated, further bifurcations appear, rolls within rolls, in what is called a process of ‘cascading bifurcations,’ which then leads to turbulent chaos. A bifurcation diagram of such a process between ‘equilibrium thermal chaos’ and ‘non-equilibrium thermal chaos’ is surprisingly ordered, or ‘order or coherence is sandwiched between thermal chaos and non-equilibrium turbulent chaos.

Jeffrey Bell (Philosophy at the Edge of Chaos)