Why do we need a model? And how do we create and use it?
A model is not the same as the reality that it represents. Particles on two dimensional lattice obviously are totally different from a group of people. But, then, the model has been proven to be quite useful. How can that be? A group of people is much more complex than a group of particles.
People are using equations (often called “models”) to explain collective behaviors in a complex system without taking into account the underlying mechanisms that generate the collective behavior. While the equations are quite useful and powerful they may fail to explain the microscopical mechanisms behind them.
The collective behavior are changing from time to time. The equations that work for some collective behaviors may not work for some other collective behaviors. If a collective behavior persists for a significant period of time, the equations that describe the collective behavior will be very useful to predict the future behavior of the system. However, the equations themselves do not predict the change in collective behavior that have to be explained by diﬀerent equations. The situation can get worse when the collective behavior has a stochastic element in it.
In this study we propose a method to connect the microscopic dynamics of a system with its collective behavior. While the reality is much more complex than the model system that we use, our model is useful to high-light the dominant microscopical features that would lead to the collective behavior. The purpose of the simulation is, therefore, to explain the collective behavior in term of its underlying microscopic behavior. We employ a simple lattice-gas model where it is simple enough to measure the collective behavior on one hand and to describe the microscopical behavior on other hands, and to make a direct connection between the two.
Lattice-gas model has been used extensively in statistical physics, surface physics, and electrochemistry. Here we propose to apply it to complex economical systems.
In this is study we examine the dynamics of distributions of particles ad-sorbed on a surface. The dynamics of particle distributions can be viewed as the change of configurations with time. The configurations are created by adsorption, desorption, or diﬀusion. This kind of study is not only important for surface-science studies, but it is also very useful in the study of complex system in general. The average properties of the lattice-gas surface as a function of time are used to model the collective behavior of complex economical systems. We measure the average properties such as coverage and correlation length to represent the collective behavior. The microscopic configurations are representated by snapshots that includes the histogram of size distributions and configuration snapshots, coupled with diversity measurements.
In our modeling of an economic system, a particle is a representative of an agent. When a particle occupies a site, it is basically representing a contribution to economic growth. When a particle leave a site, it is representing a contribution to economic shrinkage. The total growth is proportional to the change of coverage, and the rate would depend on whether the particles are interacting with each other or not. And if they are interacting with each other, how exactly the interactions manifest themselves. The interaction (or non-interaction) creates the dynamics of surface morphology, which later can be shown by diversity and size distribution dynamics. By measuring this morphological dynamics and correlates it with the average properties of the system as a function of time, we eﬀectively connect the microscopical dynamics to the collective behavior.
In the simplest approach, the particles are not mutually interacting. The only limitation is that each lattice site can be occupied by at most one particle at a time. In this case, the way to adsorb, desorp, or diﬀuse particles on the surface is through random adsorption, desorption, or diﬀusion. Any application of additional rules for the particle distribution requires interactions.
In a complex economical system, the interaction between the agents are very important. The diﬀerence between classical and modern economical model is basically the diﬀerence between incorporating or not-incorporating this interaction. In our lattice-gas model, particles are being adsorbed or desorbed with a certain probability. The interactions, therefore, are manifested through these probabilities.
In conclusion, the model is indeed different from reality. But SOME important behavior of the real system have the same manifestation as that of the model. This manifestations in the model can be explained well thus explaining the corresponding phenomena in the real system.