Hello, bonjour, namaste, vanakkam, welcome to the third session of the homework for Statistical Mechanics: Algorithms and Computations, from the Physics Department of the Ecole Normale Superieure. This week during the lecture and the tutorial, you have studied pins on a line, one-dimensional hard spheres. We have found a great algorithm and we were able to show analytically that there are no phase transitions. This is quite an achievement, for our third week of statistical mechanics. We now turn our attention to a case where there exists a transition. In fact you are already familiar with this case: it is the case of hard disks in two dimensions, that you have already studied last week. Historically, this problem is associated with the development of the Monte Carlo method, and Molecular Dynamics, and also to the discovery that the two-dimensional continuum system can turn solid. How this takes
place has remained controversial for many decades. In this homework, you will follow in the footsteps of the pioneers in this field. You will implement a Python program to simulate the transition. from the liquid to a dense phase. When this transition was discovered in 1962, people were skeptical It had been believed for 30 years that no transition can take place in a two-dimensional system which is continuous and has only local interactions. The phase transition you will see is really a great classic. In the coming weeks, we will study other examples of phase transitions, ranging from the ferromagnetic phase transition in the Ising model, to the Bose-Einstein condensation, a type of transition only possible in the miraculous world of quantum mechanics. So get started with this interesting subject and enjoy homework session 3.