Examining the Schelling Model Simulation through an Estimation of Its Entropy

Abstract

The Schelling model of segregation allows for a general description of residential movements in an environment modeled by a lattice. The key factor is that occupants change positions until they are surrounded by a designated minimum number of similarly labeled residents. An analogy to the Ising model has been made in previous research, primarily due the assumption of state changes being dependent upon the adjacent cell positions. This allows for concepts produced in statistical mechanics to be applied to the Schelling model. Here is presented a methodology to estimate the entropy of the model for different states of the simulation. A Monte Carlo estimate is obtained for the set of macrostates defined as the different aggregate homogeneity satisfaction values across all residents, which allows for the entropy value to be produced for each state. This produces a trace of the estimated entropy value for the states of the lattice configurations to be displayed with each iteration. The results show that the initial random placements of residents have larger entropy values than the final states of the simulation when the overall homogeneity of the residential locality is increased.

Incorporating a monetary variable into the Schelling model addresses the issue of a decreasing entropy trace

“…The outline of the methodology will follow a similar notation set used in 25 . The grid in which residents are placed and allowed to move within will be represented by a lattice of .…”

“…What is a common feature of previous work in sociophysics for this modeling development is that the progression of the Schelling model states are monitored by looking at the percentage of the agents which have achieved the homogeneity satisfaction or the related measure of energy of the system due to the local interactions over the grid. An alternative approach is taken in 25 where the entropy over the simulation trace is estimated. It provides an equation set for the Schelling model in which the individual configurations of the agents within the grid at any time point can be considered as a microstate, , belonging to a macrostate at each time point, which is the overall number of satisfied residential agents.…”

“…Subfigure (d) presents the mean entropy values and standard deviation over multiple simulations for each iteration (only the first 3 iterations are shown since there are no samples drawn from the larger macrostate values as the density is tightly focused upon the mode).

Figure 2 The plots here show how the progression of the Schelling model produces a decrease in entropy with the iterations (which is introduced in 25 ). (a) shows the distribution of the grid homogeneity value R across all the agents on the grid at random initializations.

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“…The plots here show how the progression of the Schelling model produces a decrease in entropy with the iterations (which is introduced in25 ). (a) shows the distribution of the grid homogeneity value R across all the agents on the grid at random initializations.…”

Exploring the entropic nature of political polarization through its formulation as a isolated thermodynamic system

“…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.…”