Condensed matter physics and sociology are both about emergence. Phenomena in sociology that are intellectually fascinating and important for public policy often involve qualitative change, tipping points, and collective effects. One example is how social networks influence individual choices, such as whether or not to get vaccinated. In my previous post, I briefly introduced some Ising-type models that allow the investigation of fundamental questions in sociology. The main idea is to include heterogeneities and interactions in models of decision.

What follows is drawn from Sections 2 and 3 of the following paper from the *Journal of Statistical Physics. *

Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges by Jean-Philippe Bouchaud

Bouchaud first considers a homogeneous population which reaches an equilibrium state. This is then described by an Ising model with an interaction (between agents) J, in an external field, F that describes the incentive for the agents to make one of the choices. The state of the model (in the mean-field approximation) is then found by solving the Curie-Weiss equation. In the sociological context, this was first derived by Weidlich and in the economic context re-derived by Brock and Durlauf. (Aside: The latter paper is in one of the "top-five" economic journals, was published five years after submission, and has been cited more than 2000 times.)

As first noted by Weidlich, a

spontaneous “polarization” of the populationoccurs in the low noise regime β>β c , i.e. [the average equilibrium value of S_z] ϕ ∗≠1/2 even in the absence of any individually preferred choice (i.e. F=0). When F≠0, one of the two equilibria is exponentially more probable than the other, and in principle the population should be locked into the most likely one: ϕ ∗>1/2 whenever F>0 and ϕ ∗<1/2 whenever F<0.

Unfortunately,the equilibrium analysis is not sufficient to draw such an optimistic conclusion.A more detailed analysis of the dynamics is needed, which reveals that the time needed to reach equilibrium is exponentially large in the number of agents, and as noted by Keynes, "in the long run, we are all dead."This situation is well-known to physicists, but is perhaps not so well appreciated in other circles—for example, it is not discussed by Brock and Durlauf.

Bouchaud then discusses the meta-stability associated with the two possible polarisations, as occurs in a first-order phase transition. From a non-equilibrium dynamical analysis, based on a Langevin equation,

one finds that the time τ needed for the system, starting around ϕ=0, to reach ϕ ∗≈1 is given by: 𝜏 ∝ exp[𝐴𝑁(1−𝐹/𝐽)], where A is a numerical factor. This means that whenever 0<F<J, the system should really be in the socially good minimum ϕ ∗≈1, but the time to reach it is exponentially large in the population size.The important point about this formula is the presence of the factor N(1−F/J) in the exponential.

In other words, it has no chance of ever getting there on its own for large populations. Only when F reaches J, i.e. when the

adoption cost C becomes zero will the population be convinced to shift to the socially optimal equilibrium...This is

very different from the standard model of innovationdiffusion, based on a simple differential equation proposed by Bass in 1969 [cited more than 10,000 times].In physics, the existence of mutually inaccessible minima with different potentials is a pathology of mean-field models that disappears when the interaction is short-ranged. In this case, the

transition proceeds through “nucleation”, i.e. droplets of the good minimum appear in space and then grow by flipping spins at the boundaries.

This suggests an interesting policy solutionwhen social pressure resists the adoption of a beneficial practice or product:subsidize the cost locally, or make the change compulsory there, so that adoption takes place in localized spots from which it will invade the whole population.The very same social pressure that was preventing the change will make it happen as soon as it is initiated somewhere.

This analysis provides concepts to understand wicked problems. Societies get "trapped" in situations that are not for the common good and outside interventions, such as providing incentives for individuals to make better choices, have little impact.

In the next post, I hope to discuss the role of heterogeneity (i.e. the role of a random field in the Ising model). A seminal paper published in the *American Journal of Sociology* in 1978 is Threshold models of collective behavior by Mark Granovetter. It has been cited more than 6000 times. The central idea is how changes in heterogeneity can induce a transition between two different collective states.

Aside: The famous Keynes quote was in his 1923 publication, *The Tract on Monetary Reform*. The fuller quote is “But this long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task, if in tempestuous seasons they can only tell us, that when the storm is long past, the ocean is flat again.”