A Mathematical Analogue
What is happening when a system self-organises? And why? One way of explaining the behaviours of real-world dissipative systems is to map them into (draw analogies with) the behaviours of isomorphic (similarly structured) systems of mathematical relationships. For so modelling dissipative systems, the relevant mathematics appears to be the study of trajectories through time of solutions to systems of differential equations—what is known as the theory of non-linear dynamic systems, meaning systems which change through time but far from smoothly.80 More popularly, this body of theory is called chaos theory. Drawing on this theory and its vocabulary, when a system self-organises (restructures itself) from one network of paths for cycling its component materials to another network, it is being pushed out of one basin of attraction and into another. Once inside such a new basin of attraction, the system spontaneously moves, under the impetus of (predominantly) positive feedback processes, along a trajectory towards a restricted part of the basin called the attractor.
Francis Heylighen defines an attractor as a region in state space (this being the set of all conceivable system configurations) that a system can enter but not leave; not leave easily perhaps.81 Once a dissipative system enters an attractor region, its trajectory—the sequence of states it subsequently cycles through—will tend to stay inside that region. That is, it will be in a steady state of dynamic equilibrium and following an equilibrium trajectory from which, left to itself, it will show no tendency to depart. However, if it is nudged or pushed off this equilibrium trajectory by ‘noise’, meaning small random fluctuations in its material-energy throughflows, or by a modest change in rates of material-energy inflows to the system, negative feedback processes will start up and take the system back to the attractor trajectory. A system which returns to its former equilibrium after being thus moderately disturbed is defined to be in stable equilibrium. In contrast, a system which is in unstable equilibrium would continue to deviate from its equilibrium trajectory once such deviation is initiated. Unlike most human-designed systems, self-organising systems have a strong capacity to restore themselves after disturbance.
The extent to which a dissipative system’s trajectory can be displaced from an attractor region and still return to the attractor when the disturbance ceases is a measure of the system’s resilience or homeostatic capacity to absorb and recover from disturbance. Commonly, the larger the initial displacement, the faster the return to the dynamic equilibrium of the attractor.82
So, a basin of attraction can be looked at in two ways. One is to see it as the set of all initial system configurations such that, starting from any of them, the system will spontaneously move towards one specific attractor or sequence of states. The other is to see it as setting the limits to the system’s homeostatic capacity, namely the set of states beyond which, after disturbance, it cannot move and still return to the basin’s attractor.
The theory of non-linear dynamic systems recognises at least four types of state trajectories that systems can follow once they have entered an attractor region:
Point attractors are ‘one-state trajectories’. That is, the system appears to remain the same even though its materials are continually turning over and it is degrading energy. A system reaching a point attractor is said to be in a stationary state.
Cyclic or periodic attractors (also called limit cycles or stable oscillations) are trajectories in which the system passes through a fixed sequence of states and then repeats the same sequence indefinitely.
Strange attractors (also called chaotic attractors) are trajectories which, without leaving a bounded region of the current basin of attraction, traverse an infinite number of states without ever returning to a previously visited state. A system following such a trajectory is said to be behaving chaotically and behaviour can vary from nearly periodic to apparently random.
Developmental attractors (also called homeorhetic attractors) are sequences of states (i.e. trajectories) corresponding to developmental stages in classes of systems which have well-defined life cycles, e.g. organisms. More generally, a homeorhetic system is regulated around ‘set points’ as in homeostasis, but those set points change with time, e.g. migrate across the basin.
When observing real-world dissipative systems, it is difficult to confidently detect point and cyclic attractors, partly because such only occur in simple systems with a few degrees of freedom (i.e. unlike global cycles) and, also, random (inherently unpredictable) disturbances obscure the underlying form of the attractor. More bluntly, cyclic and point attractors exist only approximately and for limited periods. It seems that global cycles and other physical dissipative systems are generally better described as having an underlying tendency to behave chaotically and, in the presence of frequent external disturbances, more or less randomly, at least within the confines of the attractor associated with the system’s current basin of attraction. Developmental attractors, on the other hand, seem to be more the province of biological dissipative systems. A dynamic system will stay within its attractor, transporting materials, making static structures perhaps, and degrading energy, until it is pushed into a different basin by a sufficiently prolonged and sufficiently large perturbation or disturbance, meaning a change in the pattern of availability of energy/material inputs from the system’s environment. The caveat here is that the disturbance should not be so large as to overload and destroy the system.
More completely, disturbances which trigger such self-organisation do not have to be exogenous, meaning ‘from the outside’. They could be endogenous. What does that mean? An endogenous disturbance in a self-organising system is a fluctuation in the external environment of a component sub-system which is itself self-organising. That is, an endogenous disturbance is a fluctuation which is external to the reorganising sub-system but internal to the total system. A small change in one sub-system triggers a large change and, from there, a reorganisation in other sub-systems. The energy to drive this sort of reorganisation from inside will normally be energy which has been stored within the system itself after being captured from the energy flowing into the system from the environment. So, even though it is lagged, the reorganisation is still being powered by environmental energy flow.
If a dissipative system is located in an environment which is not variable enough to spill it out of its current basin of attraction, the system is said to be stable—at least in that environment. Putting that another way, stability in a real system means staying within some basin of attraction. As well as being a function of its environment’s variability, a system’s stability will also be function of its own resilience. Greater stability goes with a system’s greater tendency to pull in (through positive feedbacks), concentrate and dissipate thermodynamic potential, i.e. organised materials and high-quality (free) energy. Other things being equal, a system which is supplying its own feedstock materials through recycling is more likely to be stable, and hence persist. To take a global example, when the energy and moisture load of the atmosphere above a tropical sea of the appropriate temperature becomes too large to transport moisture aloft in the normal way, the transport system may spontaneously re-organise (self-organise) itself to include a new attractor called a cyclone. A cyclone, because it comprises fast-moving material, can dissipate a much greater quantity of energy per unit time for every gram of water it contains than the normal evaporation-rainfall cycle (doing things faster consumes more energy). When the energy load on the tropical sea drops back to more normal levels, there will no longer be a tendency for cyclones to form.
Recapitulating then, each of the global cycles, networks, through which energy and stuff passes, dissipating as it goes, reservoir by reservoir, link by link, is, in the language of non-linear dynamic systems theory, an attractor in a basin of attraction. It is to this state of dynamic equilibrium that the system returns, quickly or slowly, after disturbances from outside or noise from inside—provided, as noted above, these disturbances are not too large. If the degree of noise/disturbance is above some critical threshold level, part of the global cycle will shut down (collapse) through lack of feedstock. Alternatively, it will react to the changes by self-organising, by spontaneously jumping into another of the system’s latent basins, one containing, perhaps, an attractor ‘trajectory’ that can successfully process the post-disturbance flows of materials and energy as they enter the global cycle in question.
A system which has been driven away from its dynamic equilibrium state towards a point, a bifurcation point, where it is close to undergoing a self-organising change is said to be in a critical state. A self-organising system which has reached a bifurcation point is unstable in the sense that it requires only a small pulse of energy to push it into one of several adjacent basins of attraction. Which of the available basins will be entered is quite unpredictable; the outcome is effectively random and, in this sense, evolving physical self-organising systems display the same ‘blind variation’ as biological systems evolving in accordance with the Darwinian ‘variation and selective retention’ model. To complete the parallel with natural selection, evolving physical dissipative systems are also selected in the sense that successive variations will be ‘rejected’ until the system reaches a basin where it is stable, i.e. where it can persist within its attractor trajectory without being rapidly nudged or jolted, endogenously or exogenously, into a new basin.
Maximum Entropy Production
Something which cannot be proven, but which is strongly suspected by many, and which can be mathematically modelled with some degree of confidence,83 is that the global dissipative system, the stable self-organising Earth, is behaving in accordance with the maximum entropy production principle introduced above (see page 7). That is, at all times and places, within the bounds of what is kinetically (materially) possible, the Earth is spontaneously attracted to that mix (network) of material-cycling energy-dissipating pathways which produces more entropy, dissipates more energy per gram, than any other feasible organisation.
There are normally many alternative pathways potentially available to the material-energy passing through any global cycle. To the extent that these alternative paths are incompatible, i.e. cannot proceed simultaneously, only one can emerge, can be selected. For example, a cloud can produce rain in Belgium or in England, but not both. Storms can blow up anywhere. At any time a particular path may be blocked or open, depending on what is happening in other global cycles or indeed in that same cycle, e.g. clouds can reduce evaporation. What is being suggested, at least in regard to the physical dissipative systems of the pre-biotic Earth, is that the particular mix of paths adopted for moving stuff around will always be changing in the direction of increasing entropy production. This capacity to spontaneously readjust the operating mix, the active network of paths, in a dissipative system in order to better meet, at least locally, the cosmic imperative to maximise entropy production is at the heart of the self-organisation-reorganisation process. Let me explain further.
Energy can only be degraded in the presence of matter, basically (but not exclusively) as a corollary of moving it around, which means doing work to overcome its inertia—matter’s tendency to resist such movement. It is just not possible for a dissipative process to occur without producing some transient non-equilibrium structuring of its material constituents. What Rod Swenson has usefully observed is that, in such systems, ordered (internally correlated) flows of disaggregated matter—kinetic structures—produce local entropy faster than disordered flows which rely mainly on friction and conduction to produce entropy.84 So, to the extent that different paths are feasible, the ‘most ordered’ mix of paths, the one which produces the largest amount of entropy and degrades the largest amount of exergy will be spontaneously selected. Exergy is a useful term for high-quality freely available energy which when used to do work is degraded to a lower quality unsuitable for doing further similar work. Exergy lost always equals entropy produced.
This thermodynamic selection is the behaviour which locally satisfies the cosmological (thermodynamic) imperative, namely the universe’s tendency to eventually eliminate all its own energy and material gradients. Distinguishing it from the additional sorts of selection processes which occur in chemical, biological, social and psychological dissipative systems, it is helpful to define thermodynamic selection as the process wherein a self-organising dissipative system ‘gropes’ its way towards the physically feasible set of kinetic paths which degrade energy at the maximum rate possible for that system.
In the language of non-linear dynamic systems, thermodynamic selection is the process whereby a dissipative system passes through a sequence of basins in each of which it is in unstable equilibrium until it reaches a basin-attractor where it is in stable equilibrium. It is stable because all the incoming exergy is being used to build and maintain kinetic structures rather than to perturb existing structures. And, in a situation where all the incoming available energy is being actively processed through kinetic structures, entropy production from that system-environment combination will necessarily be as high as it could be. Expressed in this way, ‘thermodynamic rejection’, i.e. sequential rejection of the unstable, is perhaps more descriptive than thermodynamic selection. Of course both words have connotations of ‘purpose’ which are not intended.85
So, while energy can be degraded by moving material around in a disordered way, more energy can be degraded by moving the same material around in an ordered or structured way such as a convection current. For example, when a vortex forms around the plughole of an emptying bathtub, the rate of emptying accelerates. Moving any material faster uses more exergy and produces more entropy. From the observer’s perspective, a structure here is a persistent macro-pattern of stuff, one in which the finer-scale material components in each part of the pattern are being turned over (imported and exported) incessantly, more or less regularly and more or less accurately in terms of reproducing the macro-pattern. The system will tend to be kinetically unstable (tend to keep changing) as long as a still more ordered structure, a new attractor requiring still more energy throughput to maintain it, can be constructed from the available materials. It is only when this is no longer the case that the system will be kinetically stable, will be maximally ordered, will be producing entropy at the maximum feasible rate and will be degrading a maximum amount of usable energy.
Succinctly then, the more structure there is in a dissipative system, the more entropy it is producing, the more free energy (exergy) it is degrading, the more energy it is storing, the more outside energy it is using to maintain itself and the more rapidly the cosmological drive towards equilibrium is being satisfied. And, to complete the mutually causal loop here, the more energy it is degrading, the more structure it is producing and the further it is from equilibrium.
Not only is such behaviour illustrating Swenson’s principle of maximum entropy production, it is consistent with Chaisson’s view of evolution as a grand self-organising process in which ‘islands’ of increased complexity (increased structure) and increased free energy rate density emerge from a falling ‘sea’ of pre-existing less complex systems and persist when conditions are right, i.e. when constraints on what is possible are relaxed.86 Free energy rate density is the rate at which, per gram of material, a system is processing energy. As we shall see, Chaisson’s process is most dramatically illustrated by biological evolution but, as noted earlier, it is also evident in the pre-biotic universe. Self-reorganisation in the direction of maximum entropy production is also self-organisation in the direction of a system in which free energy rate density has been increased. Chaisson’s work has focussed on the jump in free energy rate density which occurs when a radically new sort of dissipative system emerges but, less spectacularly, the same is happening when any dissipative system self-reorganises in response to an increased energy throughput.
A final idea which neatly links the ideas of self-organisation and maximum entropy production is that a self-organising system which has settled into a strange attractor region in a state of dynamic equilibrium is also, plausibly, producing entropy at the maximum rate possible for that system.