First, recall the concept of spontaneous order, associated with the Austrian School of Economics. Spontaneous order refers to the development of complex social and economic relations arising, not from a guiding intelligence, but from the separate actions of many individuals acting in their own self interest. In mathematics and physics, we frequently see similar phenomena: a system's dynamics are dictated by some energy functional. The system evolves in such a way as to minimize the energy. For the analogy to be apt, however, we do not consider arbitrary energy functionals. We consider only those which are defined by summing (integrating) a local energy. By 'local', I mean that the energy contribution at each point is computed from information that can be measured near that point (e.g. slopes). Then the energy minimizing dynamics at each point require information available at that point, and not global information. A mathematician can construct physically unreasonable systems which do not possess this locality property. Then the analogy with spontaneous order breaks down. So, the relevant correspondence here is
spontaneous order <-> locality of energy.Viewed as the dynamics of a system determined by a local energy, the emergence of spontaneous order is essentially tautological. To recast it as a nontrivial statement, you need to turn it into the assertion that the socioeconomic dynamics are determined by a local energy. Then the emergence of spontaneous order follows from hypotheses such as the absence of a strong central government. Extracting political implications requires an assertion that a local energy functional is better than a nonlocal one, the latter being the natural model for a socioeconomic arrangement based on a guiding state or elite. This returns one to the perpetual conflict between small government conservatives and statists: should individuals or a governing elite determine the choice of energy functional (and thus definition of optimal state) for the society? Should individual happiness or the progress of the collective be regarded as the greater good?
Now we pause to enjoy cheap analogies. In partial differential equations class, we study the most fundamental and ubiquitous energy functional: the local energy of a function f defined on some region M and taking values in some constraint space N is defined to be the square of its derivative, |df|^2 (simply a measure of steepness of slope). If the accessible values, N, are completely unconstrained, then the system evolves to a utopian state where every point has the same value - the resources are equally distributed. When there are constraints on values, however, the dynamics are very complicated. Consider a utopia where every member of M is exactly the same; geometrically we model such a system by assuming M is a sphere (we require it to be a compact connected manifold). If we assume N is a finite size constraint space (a compact manifold), then the dynamics are unstable. Generically, one can always lower the total energy by concentrating all the energy (resources) at a single point. So for my all time cheapest mathematical analogy, we find generically that a completely homogeneous society evolves to one where a single member controls all the resources. Fortunately (think Stalin) these dynamics do not lead to stable equilibria.
Jumping from a cheap analogy to one that I find more instructive, I pass to the numerical approximation of the preceding dynamics. If you wish to model these dynamics on a computer, you have to approximate the infinite number of points on your space M by some finite grid, use the grid to compute approximate energies, and from this data, compute an approximate flow. The finer your grid, the better your approximation is to the optimal energy minimizing flow. The longer you run your approximation, the more you deviate from the optimal dynamics. When designing and executing statist policies, the government divides the population into categories, creating a coarse grid. If we assumed, counterfactually, that Congress understood economics, then we could pretend that it could compute an approximate energy for this grid and a consequent approximate energy minimizing flow. Passage to a grid, however, still entails a loss of data; hence, the approximation deviates from the optimal flow, and the deviation increases the longer the statist policies run, rendering the society ever less prosperous than its optimal state. The primary weakness of this analogy is the pretense that Congress actually includes economic analysis into its policy. With our current Congress, one can discard this pretty dynamics picture, replacing it with a discussion of the downside of pouring sugar into your gas tank.