Monte Carlo - Examples and Simulations

The Monte Carlo method is used to create a model of air flow past an ellipse.  The varying density of the air flow (indicated by the color changes) is calculated using the Boltzmann equation.

These simulations show some of the ways that the Monte Carlo method can be used to model phenomena in the real world: A model of the molecules in a gas flow hitting a barrier.  [Explanation of model] Estimating π using Monte Carlo simulation.  [Explanation of model] Model the spread of a forest fire. Several Monte Carlo demonstrations, including one that uses the Monte Carlo method to distribute chocolate chips among cookies.  In a room with 30 people, what is the probability that at least two people share the same birthday?  This can be found using Monte Carlo simulation. Simulate the position of a population of stars and see how they move over time. Even model a rainbow! Below are some examples of random walks using the Monte Carlo method for the data source.  Let the one-dimensional random walks run for a while, and note the shape of the data distribution. One-dimensional random walks: Example of one-dimensional random displacement.  Note that in spite of steps being taken in a random direction to either the left or right, the average number of steps taken for each distance converges to a stable value. In this example, a random walk is taken for sixty seconds from the origin.  This applet graphs the distance covered in that time. Who would have thought that the old mechanical Pachinko machines could be modeled as a one-dimensional random walk?  Strictly speaking, this applet models Pascal's triangle although it has other obvious applications. Two-dimensional random walks: This applet shows the path for a random walk of a certain number of steps.  Each time the applet is run, it will generate a different path.  Another applet shows just how much the random walks can differ:  each walk is plotted on the same graph.  And yet another colorful version can be found here. This applet shows the paths of particles diffusing from a source. What is the difference between one-dimensional and two-dimensional random walks? The case of Brownian motion: Einstein explained Brownian motion as the jostling of a large particle by many small but energetic particles.  The small particles' motion can be modeled using the Monte Carlo method. Einstein's Explanation of Brownian Motion Further illustration of the mechanics of Brownian Motion  [and yet another example!] A more sophisticated model of Brownian motion with several modifiable parameters In the above examples, the large particle is essentially passive, and the Monte Carlo method is only applied to the small particles.  So what happens when an energetic large particle is introduced into the system? What is the difference between large dust grains and small dust grains in air?

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