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Hello, bonjour, namaste, vanakkam,

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welcome to the third session of the homework for

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Statistical Mechanics: Algorithms and Computations,

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from the Physics Department of the Ecole Normale Superieure.

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This week during the lecture and the tutorial, you have studied

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pins on a line, one-dimensional hard spheres.

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We have found a great algorithm and we were able to show analytically that there are no phase transitions.

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This is quite an achievement, for our third week of statistical mechanics.

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We now turn our attention to a case where there exists a transition.

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In fact you are already familiar with this case:

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it is the case of hard disks in two dimensions, that you have already studied last week.

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Historically, this problem is associated with the development of the Monte Carlo method,

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and Molecular Dynamics, and also

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to the discovery that the two-dimensional continuum system

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can turn solid.

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How this takes
place has remained controversial for many decades.

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In this homework, you will follow in the footsteps of the pioneers in this field.

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You will implement a Python program to simulate the transition.

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from the liquid to a dense phase.

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When this transition was discovered in 1962, people were skeptical

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It had been believed for 30 years that no transition can take place

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in a two-dimensional system which is continuous and has only local interactions.

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The phase transition you will see is really a great classic.

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In the coming weeks, we will study other examples of phase transitions,

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ranging from the ferromagnetic phase transition in the Ising model,

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to the Bose-Einstein condensation,

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a type of transition only possible in the miraculous world

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of quantum mechanics.

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So get started with this interesting subject and enjoy homework session 3.