# Hard disks and the liquid-solid transition in two dimensions_Questions

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You can insert math formulæ using LaTeX code: e.g. typing
[[math]]
\pi_C(t+1)=\sum_{C'} \pi_{C'}(t) p_{C' \to C}
[[math]]
yields:
$\pi_C(t+1)=\sum_{C'} \pi_{C'}(t) p_{C' \to C}$

## Questions and Remarks:

• Q: in the first variant we should replace a=choice(L) - actually there are two line which read "a = choice(L)": one two choose a random disc to propose a move, and one to calculate the pair correlation. Which lines should be replaced? the first, the second or both?
• A: you have to replace both instances.

• Q: in the question about the two variants of the MC algorithm, the second variant includes both the replacements or only the second one?
• A: the second variant include only the replacement for b.

• Q: In the question 2 of the very first section ("why can we forget about the velocities?"), I'm not sure what the question is about: should we disscuss the fact that we are sampling equilibrium, distribution or compare the algorithms to molecular dynamics?
• A: The dynamics of hard spheres is described by their individual uniform motion punctuated by collisions. It occurs in phase space, whose coordinates consists of the set of their positions and velocities. As discussed in the lectures and in the class sessions, according to the ergodic hypothesis, their stationnary distribution is uniform in phase space (which includes positions and velocities). However, in the proposed programs, the sampling is performed uniquely on positions. We ask to justify why we can forget about the velocities.

• Q: In the questions about the algorithm A, what is exactly the property ? I read it as "the probability to find a disk at position (x,y) is uniform in space" but this is clearly false (as the histogram of the x-coordinate of the disks shows), so I do not understand the following questions.
• A: The hard spheres are distributed with the microcanonical distribution: their probability density in phase space is uniform. The property that we ask you to decide whether it is true of false indeed reads: "the probability to find the center of a disk at spatial position (x,y) is uniform". In question 2, you have to justify theoretically why you think this property is true or false. In the following questions (3 and 4), you have to test your choice numerically.
Besides, we cannot comment on the rest of your remark « but this is clearly false (as the histogram of the x-coordinate of the disks shows) », as it would partially answer to the question. However, we advice you to clearly identify and rethink of the premises and implications of your reasoning.

• Q: I think %1 is missing in the last question (second variant).
• A: Yes. The question is now reformulated in a clearer way.

• Q: It seems that the definition of N is missing for a = L[iter%N] (for the last part of the homework).
Should it be a = L[iter%len(L)]?
• A: Yes, here N is the number of disks in the specific problem (the way you propose should work).

• Q: in the part about Version A, are we expected to compare our results with the results from Version B, or not?
• A: Your question is not very precise. If you refer to this question: "Explain why (and in which limit) direct_disks.py and markov_disks_box_B.py give the same results." the answer is yes. If you refer to the following questions, the answer is no.

• Q: The last part of the question is not clear to me. Is the detailed balance referred to the detailed balance of a markov chain as discussed in lecture 1?
• A: yes
• It is not obvious to me how the pair-correlation function can give information on the balance properties (e.g. what are the transition probabilities in this case)? [[user:yiteng]* A: This is not the question. You have two things to do: 1) Answer the question about the detailed balance being respected or not. 2) Calculate the pair-correlation function and check that the results make sense. We do not ask you to find the link between 1) and 2).

• Q: In the question of drawing the histogram of x-coordinates, is it about running the program many times and getting a large number of realizations of L, and drawing the histograms of x-coordinates in all these realizations?
• Q: About the super bonus: do we have to explain simply by reasoning or by explicit calculation?