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Hello, bonjour, geia sou, shalom.

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Welcome to the second homework session

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of Statistical Mechanics: Algorithms and Computations

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from the Physics Department of the Ecole Normale Superieure.

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The subject of this week is the simulation of hard disks.

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The aim is twofold. First:

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you will convince yourself

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that the event-driven molecular dynamics

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gives the same results for thermodynamics

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as the direct sampling or the Markov chain Monte Carlo methods.

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You will implement a simple example

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for each method, compare the output,

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and verify yourself the famous ergodic hypothesis.

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Second, you will apply directly this three methods

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event-driven molecular dynamics,

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direct sampling and Markov chain

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to study the phenomenology of hard disks.

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You will observe that their finite radius leads to

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unsuspected and surprising phenomenology.

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For hard disks

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each legal configuration has exactly the same probability,

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but do not let this uniform probability distribution

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mislead you to the assumption

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that the actual behavior of the system will be uniform and boring.

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As you will see in this homework, uniform probability distributions

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can lead to surprisingly non-uniform results.

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This is due to so called entropic forces

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which we will study in more details next week.

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They are everywhere in nature, be assured.

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In the meantime, have fun with Homework session 2.