Mar 17, 2023

Abstract Political polarization has become an alarming trend observed in various countries. In the effort to produce more consistent simulations of the process, insights from the foundations of physics are adopted. The work presented here looks at a simple model of political polarization amongst agents which influence their immediate locality and how a entropy trace of the political discourse can be produced. From this model an isolated system representation can be formulated in respect to the changes in the entropy values across all variables of the system over simulation time. It is shown that a constant entropy value for the system can be calculated so that as the agents coalesce their opinions, the entropy trace in regards to political engagements decreases as the entropy value across non-political engagements increase. This relies upon an intrinsic constraint upon agents imposing a fixed number of activities per time point. As a result the simulation respects the second law of thermodynamics and provides insight into political polarization as a basin of entropy within an isolated system without making assumptions about external activities. Introduction Social physics has laid out an interesting set of goals where natural sciences and social sciences would come together in order to help model human based systems 1 , 2 (that the “verbal reasoning” in the humanities would be assisted by science and technology). From a recent review in the field of social physics 3 , it can be appreciated that the overlap between physics and a myriad of social phenomena has allowed researchers to understand how certain processes are governed. This involves investigating the underlying dynamics between the granular elements, the generative processes, how to visualize complex systems, and other aspects of dynamics around human centered ad-hoc interactivities. The work of 3 , 4 , 5 provide a thorough review of much of the approaches in modeling social activities from a physics perspective highlighting the techniques adopted from physics which have been used successfully in studying datasets arising from human activities. Applications demonstrate new approaches to the statistical physics of crime 6 , climate change dilemmas 7 , social media polarization 8 , and far reaching topics including even the entropy and complexity of the evolution of memes 9 are explored. Ideas along this line can even be traced back earlier publications 10 . One of the most important principles in physics is that of the second law of thermodynamics which states that the the entropy of the system cannot decrease over time 11 . There is notable previous work incorporating this principle in general complex systems such as 12 (global climate) 13 , 14 , (ecosystems) 15 , (entropy pertaining to wealth), and 16 which looks at entropy in the field of economics. In general there is the question, “does the second law of thermodynamics apply to social systems or not?” 17 . Bailey 18 discusses how the 2nd law of thermodynamics is prevalent in all living and nonliving entities regardless of the layers of complexity they rely upon. That regardless of the perspective of the system viewed this law will be acting 19 . In the work of 20 the Schelling model of segregation 21 , 22 has its entropy trace along the simulation produced showing a decrease with the increased agent homogeneity, and in 23 it is shown how with a dual dynamic operating on an income variable that the overall extended Schelling system can display an increasing entropy trace and respect the second law of thermodynamics. The motivation for the inclusion of a monetary variable as influencing the residential movements was inspiration from the work of 24 that observed this from real world data collected. The second law of thermodynamics, also referred to as the arrow of time 25 , provides a direction (gradient) for which the combinatorical nature of a system can be expected to move in. Such work helps establish that social system models can be designed to respect the arrow of time since a decrease produces the Gibbs paradox due to an incomplete system definition 26 , 27 , 28 . There are other notable investigations into the social physics of the Schelling model such as 29 which looks at the phase transitions, and the work of 30 , 31 that are examples of research which makes the connection between the Schelling model and the Ising model of ferromagnetism. From these approaches mentioned, and those discussed in the review articles, the system variables are not modeled as a completely isolated system. In the fields of physics and mechanical engineering it is common for researchers to devise a isolated system where the progression of the simulations result in a constant value of energy as the variables fluctuate according to the underlying dynamics 26 , 32 , 33 . There are many benefits produced by completely defining the system in isolation so that the exchange of entropy contributions between the system variables over simulation time can be examined. Adopting this paradigm into the social physics field can potentially allow deeper insight to be gained and produce a deeper understanding of the results due to constraints on the system variables. In 23 the extension to the Schelling model has a monetary dynamic associated with every move an agent makes on the grid that affects an income variable. This variable and dynamic does have the effect where the overall system entropy can be seen to increase due to the identity entropy decrease that is matched and overcome by the increase in the monetary entropy component. Such a new variable introduction is a plausible proposition given evidence from 24 , 34 , but it can be considered as a modeler selected incorporation since other variables may exist that can provide a similar dynamic which alleviates the physical violation. As an alternative we can take into account the natural constraints that the system agents can be expected to be bounded by in regards to the activities of concern. There is also a wide range of constraints that can be introduced but the most fundamental one (even more than the monetary component), is that of time. The agents can be thought of allocating their constrained time in engaging in a finite set of activities over simulation iterations. This constraint helps in defining the system which will be modeled as an ’isolated system’ with constant entropy. These principles are integrated into a simple model of political polarization 35 , 36 where agents reside in fixed positions within a lattice. Each agent is considered to have a single value for their political affiliation 37 and strength between two choices 38 . Over simulation time the agents have these affiliation values influenced by their immediate neighbors 39 which then can affect their political positions 40 , 41 . The agents are not able to change grid positions and begin with randomized affiliations and are not engaged in political discourse/influence when there is no ideological disparity 42 , 43 with their locality ( 44 , 45 ). As will be shown in the Methodology section the entropy for the distribution of the agent variables in respect to the political affiliation and influence can be calculated using Shannon entropy. The entropy for the system in regards to the political activity can then be produced allowing for the entropy trace to be examined. When the agents’ activity for political actions decreases, the time constrained number of activities increases the agent activity in non-political engagements (peripheral). With this approach the agents are engaged in a constant total number of actions which changes the allocations between political and non-political (peripheral) activities. The peripheral activities introduce a component of uncertainty since the array of different actions (eg. walking, reading etc) are unknown to the modeler and provide a value of entropy as a result. The agents distribute their total activity number between political and peripheral actions for the total system entropy which is shown to be constant (over all time points). This allows the examination of the social model under this constraint to be an isolated system. It should be noted that this model is devised to explore the modeling paradigm for which a social system can be modeled in order to control for the entropy of the system rather than fit it to a real non-isolated system. The agent dynamics do not operate directly on the value of the system entropy or its components, and are agnostic to the values. The entropy trace is calculated in order to describe the system state for the modeler. A required assumption for the system described is for it to be isolated and bounded. By defining the entropy of the system a connection between its behaviors and the second law of thermodynamics is made. This gives the opportunity to acquire greater insight to the expected behaviors of such systems, and also lays the ground work for defining and studying the rest of the thermodynamic variables in social systems like the equivalent of the energy. This approach of using the entropy as the core variable to describe the behaviors of systems has also found applications in pure physics, like in the works of Bekenstein 46 and Susskind 47 considering the holographic principle, utilizing among others the maximum entropy of systems. The Results section begins by showing how the model dynamics drive the agents towards greater political ideological homogeneity, and how this produces a decreasing entropy trajectory along the variable of the political actions. The analysis of the non-political actions (peripheral actions) shows a corresponding increasing entropy trajectory. Together it is shown how this inverse relationship produces a constant entropy value for the system where the simulation and theoretical calculations agree on the constant value. This inverse relationship rooted in a constraint, is reminiscent of the fundamental relationship \(T \propto PV\) which is explored in the work of 48 that investigates demographic distributions based upon this component of thermodynamics. The different values on the number of total actions, that can be taken by agents, is considered to be analogous to the temperature, the political actions to the pressure, and the peripheral actions to the volume. Methodology The simple model of ideological exchange proposed here assumes that there is a square lattice (an \(N\times N\) grid) where in each cell of the grid an agent resides which cannot change its position. Every agent has a ’contained’ political affiliation which is initially sampled uniformly from \(\mathcal {U}[-Cmax,Cmax] \in \mathbb {Z}\) (\(C_{max}=4\) 49 ), and these values are referenced from a matrix C. Agents can influence the contained political affiliation values in those agents found in their immediate adjacency (similar to the locality of the Schelling model and the Ising model of ferromagnetism 50 ). There is a dynamic which governs how the values of an agent’s surrounding neighbors affect its own value. Given the iterations of the simulation, time steps, these values can continue to change as the dynamics are repeatedly applied. At the base of the model is the dynamic of how agents, which contain values \(C_{i,j}\) (representing political affiliation strengths), change this contained political stance over time as agents influence each other from their immediate adjacency. Utilizing the matrix C, 3 more matrices are defined based upon the values in C; M, I, and E. A matrix M, holding binary numbers, is defined to represent the result of how each agent, (i, j), will vote based on the values in C. The matrix I is defined to provide the aggregate value of all the voting decisions taken by agents in adjacent cells with values stored in M, excluding itself. These values in I, \(I_{i,j}\), can be thought of as a bias quantity which an agent at position (i, j) is subjected to when there is not an equal number of neighbors voting for each side of the political spectrum. The matrix, E, holds values indicating whether an agent at position (i, j) will be active engaging in political discourse in an attempt to influence agents in the immediate neighborhood when there lacks an agreement (disparity) in that locality for the voting direction. In a state where each agent agrees on the voting directions stored in M, there are no political actions/engagements being taken by agents and each \(E_{i,j}\) will be zero. Simulations are run with iteration numbers denoted by t and the matrix values change as the agents affect the values contained in C of their neighborhood. Table  1 provides a listing of the matrices proposed and a succinct description of their purpose. Table 1 The matrices proposed and a description of their use.