Energy Distribution of Atoms

Wealth Distributions

Before discussing the energy distribution of atoms, we will investigate wealth distributions in a very simple model of an economy.

Exercise 1.10.1: Simple Economy Prediction
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NetLogo model 1.10.1 is a very simple model of an economy. There are 500 people who each start with \$100. Each time step, each one picks another person randomly and gives them \$1. That's it. There are two visualizations of wealth in the model. In the black rectangle, each green dot represents a person and the x-coordinate of the dot represents their wealth. The histogram below that shows a the wealth distribution (how many people there are of each wealth amount). Predict what will happen in the question below, then run the model and see what happens.

  1. What do you think the steady-state distribution of wealth will be after the Simple Economy model runs for a long time?

Exercise 1.10.2: Simple Economy Result
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What wealth distribution actually occurs after the model is run for a long time?

  1. What wealth distribution actually occurs after the Simple Economy model is run for a long time?

Energy Distributions

Now, we will investigate an analogous phenomenon: the energy distribution of atoms in a closed system. NetLogo model 1.10.2 is a molecular dynamics (MD) simulation with a visualization of the distribution of kinetic energies (KE) in the model. The base MD simulation is the same as what we have seen earlier in the course, but the visualization has been altered for the current purpose of investigating energy distributions. The atoms themselves are colored based on whether their KE is low (blue), medium (green), or high (red). The the right of the atomic visualization is a histogram of the distribution of kinetic energies, also colored the same way. This distribution is built up over the course of the whole simulation run; so it shows the average number of atoms at each energy across the whole run up until that point. Dividing by the total number of atoms would give the probability distribution for each atom having each KE.

Exercise 1.10.3: KE energy distribution
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Use NetLogo model 1.10.2 to answer the questions below. Remember, that kinetic energy (KE) is related to velocity as $KE = \frac{1}{2}mv^2$. That is, KE is the energy associated with the motion of a mass.

  1. Run the model for a few dozen ticks and describe what the KE distribution looks like and how KE of individual atoms changes.

  2. The KE distribution looks similar to the wealth distribution in the Simple Economy model. Try to explain the analogy between the two models.

  3. What happens to the KE distribution has temperature is varies?