A common theme in thermodynamics is the balance between entropy and order, a never-ending fluctuation of particles between organized configurations. While the energy-dissipating tendency of spacetime yields to this disorder, localized interactions between groups of particles allow the formation of large structures and the macroscopic world we are familiar with.
At the center of this balance is the Ising Model, created by German physicist Wilhelm Lenz in 1920 to study ferromagnetism. In this model, just as they would in the real world, atoms in a grid take on random magnetic spins. Interactions between close neighbors, however, favor the formation of magnetically-aligned regions. The mathematical laws governing this stochastic process can be used to predict the physical state of large assemblies of atoms. Specifically, this model is represented by a probability distribution of states that changes over a time-dependent function called a Hamiltonian (1). While this model is an enormous simplification for even magnets, it still sheds light on phenomena like the conversion between solids, liquids, and gases (2).
Interestingly enough, however, when we remove the thermodynamic premise of this model, the mathematics is applicable to a vast and diverse array of phenomena much more familiar to us. Whereas previously, the Hamiltonian and resulting probability distribution of states specifically described magnetic spins of atoms, they can now more generally represent properties of individuals within arbitrary networks. With an eye for local interactions prevailing against a backdrop of disorder, let us travel from the molecular foundations of life to the largest byproducts of human society, and finally to the technological and material surroundings that support the growth of said society.
The proteins in our cells are composed of one-dimensional strings of amino acids; the electrostatic interactions between neighboring amino acids are the main forces that mold these proteins’ stochastic process of folding (3). Furthermore, the cyclic reactions that these proteins often participate in, sometimes referred to as “biological switches”, are themselves stochastic systems but are integrated into much larger networks that allow them to influence their neighboring reactions. Unsurprisingly, Ising Models offer a whole host of mathematical conclusions that can be drawn from these systems (4).
Next, over the course of millennia of evolution, fragments of genetic material and their mutations are randomly propagated through generations, with the DNA of consecutive generations being highly correlated, but the evolution of taxonomically distant species being virtually uncoordinated (5). Fast forward hundreds of thousands of years, and the terms “propagation” and “functional connectivity” take on a new meaning. At the centers of our nervous systems, cortical networks prioritize direct signals from their neighbors to facilitate ordered communication amidst a background of electrical chaos (6). Scientists even believe that larger neurophysiological products like emotions are the product of dynamic networks that can be modelled with Ising Models (7).
Even at the societal level, in a way, we are each individual atoms in our own vast geographical grid. We often conform to those around us and aggregate among familiar communities, yet we often have almost no control over those that preside across the world from us. In this scenario, the Ising Model is powerful enough to foreshadow the likes of racial segregation, economic disparities, and language evolution (8). Even the disorder of the stock market is ground on which the Ising Model can yield insight into the social interactions, the pricing decisions, and the transfer of resources that occur between adjacent organizations in the complex network of our global economy (9).
So is the Ising Model a be-all-end-all mathematical description of the entire universe? It may seem so by now, but in reality, it is simply one of many ways to observe changes in a lattice over time. The generality of its premises is what lends it to the modelling of such different phenomena; all that is required is a large entropic network in which neighbors can influence each other. Abstract patterns are a foundation of mathematics, and as this discussion indicates, can often be found in unassuming places. If interpreted correctly, the underlying structures of nature and society can offer us a slew of practical applications, and for those who are willing to look deeper, a moment of awe is just the icing on the cake.
- Onsager, L. (1944). Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Physical Review, 65(3-4), 117-149. doi:10.1103/physrev.65.117
- Viswanathan, G. M. (2011). Chapter 2.4. In The physics of foraging: An introduction to random searches and biological encounters (pp. 19-21). Cambridge: Cambridge University Press.
- Bakk, A., & Høye, J. S. (2003). One-dimensional Ising model applied to protein folding. Physica A: Statistical Mechanics and Its Applications, 323, 504-518. doi:10.1016/s0378-4371(03)00018-9
- Merchan, L., & Nemenman, I. (n.d.). Ising models of strongly coupled biological networks. Retrieved from http://q-bio.org/w/images/c/c6/Poster_MerchanLina.pdf
- Matsuda, H. (1981). The Ising Model for Population Biology. Progress of Theoretical Physics, 66(3), 1078-1080. doi:10.1143/ptp.66.1078
- Roudi, Y., Tyrcha, J., & Hertz, J. (2009). Ising model for neural data: Model quality and approximate methods for extracting functional connectivity. Physical Review E, 79(5). doi:10.1103/physreve.79.051915
- Loossens, T., Mestdagh, M., Dejonckheere, E., Kuppens, P., Tuerlinckx, F., & Verdonck, S. (2020). The Affective Ising Model: A computational account of human affect dynamics. PLOS Computational Biology, 16(5). doi:10.1371/journal.pcbi.1007860
- Stauffer, D. (2008). Social applications of two-dimensional Ising models. American Journal of Physics, 76(4), 470-473. doi:10.1119/1.2779882
- Zhao, L., Bao, W., & Li, W. (2018). The stock market learned as Ising model. Journal of Physics: Conference Series, 1113, 012009. doi:10.1088/1742-6596/1113/1/012009