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Hello, bonjour, sabah el kheir, pozdrav,

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welcome to the third tutorial of Statistical Mechanics: Algorithms and Computations,

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from the Physics Department of Ecole Normale Superieure.

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This week's subject is a model of random clothes-pins

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randomly positioned on a line between two poles, our boundaries.

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This models is equivalent to one-dimensional hard spheres.

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This model allows us to move forward

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very far, along three directions

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First: it is prime example for the Asakura-Oosawa depletion interaction

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which is of great importance for macromolecules, polymers, blood cells and other systems.

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This is what we studied in the lecture using the halo picture.

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Second: the random clothes-pin model can be simulated very efficiently

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we started out in the lecture with a naive algorithm

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but Michael will show a providential method

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that allows us to simulate hundred, thousand and millions of clothes-pin on a line

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without rejection and without any effort.

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Third: this fantastic algorithm will show us the way to solve the model analytically.

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Alberto will show us how to compute the partition function,

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how to sum up the virial expansion

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and how to prove that this system in fact does not have a phase transition.

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Then Vivien will investigate

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the relationship between pins on a line with walls or poles

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and pins with periodic boundary conditions on a ring.

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He will characterize the correlations in the system

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and show that they are short range and that this characterizes a liquid.

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In what we will do, we will see that analytical results are strong

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and the algorithms that Michael will present are superb.

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But don't worry, the Python implementation will be about 5 to 15 lines long.

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In this week's lecture, Werner presented the random clothes-pin model,

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a model with hard-core interactions, just like the hard-disks we discussed earlier.

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We consider N clothes-pins of width (2 sigma)

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on a washing line, that is a line segment,

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between boundaries at x=0 to x=L

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as is shown here.

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The pins are placed one after the other

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but if an overlap is generated, we take them all off the line and start anew.

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This is again our familiar tabula rasa principle

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and once more it gives us a flat probability distribution.

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It samples N-pins configurations

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with a distribution pi(x_0,..,x_(N-1))

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=1 if the configuration is legal

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and =0 if the configuration is illegal.

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This is in agreement with the equiprobability principle:

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all legal configurations have the same energy

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E=0.

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Our pins are in one dimension what hard disks are in two dimensions and hard spheres are in three dimensions.

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A configuration is illegal if two pins overlap.

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This happens if their centers are less than (2 sigma) away from each other.

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Likewise, pins are not allowed to overlap with the boundaries.

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This happens if their centers are away less than sigma from x=0 or x=L.

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Here is how the program can be implemented in Python

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Of course this algorithm works and obeys the equiprobability principle

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as it should.

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but - just like in the case of two-dimensional hard-disks - it has a very high rejection rate.

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Output of the program is shown here.

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Is this the best algorithm for this problem?

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Or can we be a bit smarter? Of course we can.

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In fact, we are able to sample N-pins configurations without producing any illegal configuration on the way.

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This gives us a rejection rate of exactly zero.

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What you will see now is an example of a rejection free algorithm

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and it is way more sophisticated of all the algorithms we have presented so far in this course.

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But how can we possibly achieve a zero rejection rate?

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This works because we already learned from our experience with hard-disks

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that in the full configuration space

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of legal and illegal configurations

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the regions of illegal configurations are like the holes in the Swiss cheese.

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For the clothes-pin problem, we can actually remove the holes from the cheese.

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To illustrate the idea, let us consider this example

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of 5 clothes-pins on a washing line.

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The first pin could not be placed within the halo of the left wall.

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Likewise, the second pin cannot be placed

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within a distance of (2 sigma) from the first pin's center,

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and so on, for the other pins.

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Finally, no pin can be placed within the halo of the right wall.

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We can therefore deflate the washing line by cutting out these locked regions.

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If we focus on the pin centers, and glue the remaining line together,

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we obtain a shorter washing line of length (L - 2 N sigma).

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Let us call the pin centers on this shorter washing line y_0, y_1, ...

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Then these y's are related to the original positions x_0, x_1, x_2, ...

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on the longer washing line

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by x_i = y_i + 2 (i+1) sigma.

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Since every legal configuration can be deflated in the same way and gives a shorter washing line

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of the same length,

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we realize that the legal configurations

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distribute the pin centers not on the long washing line
we realize that the legal configurations

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distribute the pin centers not on the long washing line

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but on a shorter washing line of length L - 2 N sigma.

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This becomes even clearer if we invert this procedure.

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Any chosen five points on the shorter washing line, for example this five points here,

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can be inflated back to a configuration of five pins on a longer washing line of length L.

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Let us turn this into a Python program.

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Start by randomly choosing N points between 0 and L - 2 N sigma

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then sort them so that y_0 < y_1 < y_2 < ..

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and then finally take care of the inflation in one step.

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In Python, this gives the following program.

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First we choose the y's randomly between 0 and L - 2 N sigma,

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then we sort them, and this line here

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takes care of the inflation,

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that is: it related the positions y_0, y_1, y_2, ... to the positions x_0, x_1, x_2, ...

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Notice that the program is called noreject

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in honor of its remarkable rejection rate of exactly zero.

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Finally you can see here the comparison of outputs of the programs direct_pins.py

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and direct_pins_noreject.py, for 10 pins

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at a density of 75%.

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Visibly they are the same, but direct_pins.py

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generated more than 500000 illegal and therefore useless configurations

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on the way, while every single configuration generated by direct_pins_noreject.py is legal.

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Before moving on, please take a moment or two to download, run and modify the two algorithms discussed in this section.

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On the Coursera website there is the algorithm direct_pins.py

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which mirrors last week's direct_disks_box.py

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There's also the algorithm that Alberto will discuss in a few moments

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and this algorithms is really beautiful: it allows you to sample configurations

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with thousand, millions, billions clothe-pins without any rejection.

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Unfortunately, it is very difficult to generalize this algorithm to 2 or even 3 dimensions

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mainly because the crucial sorting steps then becomes impossible.

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Michael just presented a rejection-free direct-sampling Monte Carlo algorithm

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for a many-body interacting system.

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This algorithm is rooted in the fact

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that we can actually compute the exact partition function of the system

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namely the number of legal configurations

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given by the following integral,

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where pi as usual is 0 if there is an overlap and 1 if there is no overlap.

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In Michael's derivation, the sorting step played a crucial role.

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We can do the sorting step in the above integral

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and obtain the following result

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where the theta function selects between the N factorial possible orderings of the pins' centers

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the sorted one with x_0 < x_1 < x_2 < ..

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We can now apply Michael's deflation step

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and change the x variables to the y variables.

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We obtain this integral, where there is no longer overlap condition

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but only the ordering of the y variables imposed by the theta function.

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Remember: the theta function picks out one of the N factorial permutations of the integrand

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so we can remove both the theta function and the N factorial,

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compute the integral and get the final result for the partition function.

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This is a beautiful formula.

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Which allows you to compute the acceptance probability

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for the program direct_pin.py, that was introduced in this week's lecture.

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The acceptance probability is the ratio of the volume of legal configurations to the volume of legal and illegal configurations.

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To test your understanding on this concept, please let me ask you a few questions

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Question 1: select from the following three possibilities

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the correct expression for the acceptance probability of direct_pins.py.

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The correct answer is option A, because in direct_pin.py

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the centers are generated between sigma and L - sigma

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and because the volume of legal configurations

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is (L - 2 N sigma)^N

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(the partition function that we have just computed).

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Question 2: select from the following three possibilities

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and estimate of the average number of rejected configurations

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that should be generated before having a single legal configuration.

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The correct answer is option B,

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because the number of configurations that we have to wipe-out before generating a single legal configuration is on average the inverse of the acceptance probability

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Question 3: compute this number for a density of eta = 0.75 and 15 pins

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Is this number 5000000,

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50000000 or 500000000?

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As anticipated by Werner in the lecture, the correct answer is option C.

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The acceptance probability is 2 times 10^(-9)

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which means 500000000 rejected configurations.

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Now, let us recall the virial expansion introduced by Vivien in Tutorial 2.

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For clothes-pins, we can actually compute exactly the derivative of log(Z)

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with respect to the volume, or rather with respect to the length L of the system.

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And we obtain this exact expression

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that can be written in an even simpler way 1- 1/eta, where eta is the pin density.

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Note that this quantity can be expressed as an infinite virial expansion

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1 + eta - eta^2 + eta^3 ...

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This series converges for all densities smaller than one

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this means that there is no phase transition.

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However, we not the divergence at the point eta=1

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a special point where we have long-range order.

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Knowing the partition function, we can actually compute exactly other quantities

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like the distribution pi(x) that shows the oscillatory behavior that you have seen in this figure.

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In fact, pi(x) is the probability of having a pin center at x

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which is also the probability, given the pin center at x, to put the other pins

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on its left and its right.

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Let's try to place k pins on the left, this is possible only if we have enough space.

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The number of possibilities of doing so is the partition function of k pins in the interval x - sigma.

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In the same way, we have to put N - 1 - k pins on the right.

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Putting these pieces together gives the following formula,

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where a combinatorial factor gives the number of choices of picking

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k left pins out of the bucket of the N - 1 pins

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that remain once we have placed the first one at x.

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In Python, this gives the following program

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If you have doubts on our calculation, you can compare this result with the result that I have just shown from the numerical simulation.

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The function pi(x) increases steeply close to the poles

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Our formula allows us to see that the distribution pi(x) for x=sigma

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is a factor 1/(1 - eta) larger than in the center.

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This becomes very large at high densities, where the pins are strongly attracted by the walls.

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In a few moments, Vivien will show us that the attraction of the wall is not a simple boundary effect:

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the same attraction acts also between two clothes-pins.

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Before doing that, please take a moment

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to download, run and modify the program we discussed in this section.

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On the Coursera website you will find direct_pins_density.py

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Its output should give you the same results as the numerical simulations performed with

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direct_pins_noreject.py, discussed by Michael.

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Several times throughout the lectures and the tutorials we have generated an annoying situation

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by putting particles in a box rather than allowing for periodic boundary conditions.

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For instance, in the adults' game in the heliport,

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we could have allowed a particle which exits the system from the right

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to enter back the system from the left.

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Nevertheless, such strict but realistic rule of the wall has allowed us to learn about the Metropolis algorithm.

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That is, how to treat rejections.

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The same objection can be made for our system of interest: the pins on a line.

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If the probability distribution pi(x) is oscillatory close to the boundaries, this must be due to the poles.

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Now it is true that the probability depends on the position

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because of the poles, because of the boundaries.

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This raises the following question

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with periodic boundary conditions, this probability is uniform?

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So, does it mean that our discussion above was useless?

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Not at all, as we now  discuss.

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by examining and comparing the two situations

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where the pins are put on a line

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to the situation where the pins are put on a circle

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that is to say: with periodic boundary conditions.

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Let us first write a program to simulate our periodic system

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we start by putting a particle or a pin

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at position L - sigma.

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Its right side provides a pole at position x=0

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due to the periodic boundary conditions

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this leaves us with N - 1 pins

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that we have to put between 0 and L - 2 sigma

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where L - 2 sigma is the left side of the pin we have put initially.

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All in all, this means that our system N pins on a circle of length L

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is completely equivalent to a system of N - 1 pins that we put on a line of length L - 2 sigma.

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Besides, to ensure the invariance by translations, we shift the centers of the pins

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by the same number which is drawn uniformly between 0 and L.

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In Python, this gives the following program

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direct_pins_noreject_periodic.py

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By running this program, we observe that the distribution pi(x) converges to a flat function

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as the number of samples increases from 1000 to 10000 to 100000.

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You notice a sharp difference compared to the previous boundary conditions:

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the oscillations of the distribution close to the poles have vanished.

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But beware, this does not mean that there are no correlations in the system between the pins.

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To study such possible correlations, a good tool is provided by the pair correlation function

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pi(x,x'), which represents the probability 

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of having a pins at position x and another pin at position x'.

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This correlation function is a useful tool.

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For instance it is trivially equal to a constant

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if there are no correlations in the system.

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Now let us analyze the pair correlation function using the following program

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direct_pins_noreject_periodic_pair.py

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Let us fix the density to eta=0.9

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and increase the system size at fixed pin width 2 sigma = 1.

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We observe that the pair correlation function quickly converges to its thermodynamic limit

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in the example displayed here

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for system size equal 100, 500, 1000 and 2000.

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You notice that this pair correlation function presents oscillations in the same way as for the distribution

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for the system on a line.

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In fact, those two functions are the same

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because the pin which is at position x acts as a pole for the pin at position x'.

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Indeed, remember now how we devised our simulation

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we first put a pin at position L - sigma

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using the same reasoning you will convince yourself that the following formula holds

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it relates the pair correlation function

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for a system of N pins on a circle of length L

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to the probability distribution pi for N - 1 pins

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in a line of length L - 2 sigma.

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Besides, Alberto explained to you previously

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that this function can be analytically determined

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Hence, to summarize, we have constructed a system which is exactly solvable, in which you can compute

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in an analytical way the pair correlation function

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in one dimension and for a system where there are non-trivial interactions between pins.

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Let us now physically interpret the features of this correlation function.

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As you see here, the oscillations are embedded in a decaying envelope

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and the correlation goes to a constant at large distances.

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To be more quantitative, let us plot this decay in a logarithmic scale.

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As you now observe, the linear behavior indicated that the correlation function

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decays exponentially as exp(- x / xi)

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xi has a physical meaning

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it represents the correlation lengths between the pins in the system.

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At pin density eta=0.8, its value is approximately 2

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Increasing to a density 0.9, we observe the same behavior

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now with xi=6

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And a dramatic increase to xi = 26 when eta=0.95.

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In fact, one can show that the correlation length diverges as the density eta goes to one.

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Alberto has shown that the partition function of our system of pins in one dimensions

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can be computed and is analytic for all densities, which means that there are no phase transitions.

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This is now supplemented by the fact

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that the pair correlation function, which can also be computed,

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decays exponentially on a length xi

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This length xi, which is the length of the correlations between the pins,

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remains finite even if the system size goes to infinite.

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At distances larger than xi, the system is homogeneous

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This is what defines a liquid state.

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Long-range positional order is possible in our system only at density 1.

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that is for L = 2 N sigma

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that is for configurations where the pins are close-packed.

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What we have shown conclude our study of this one-dimensional model

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but in fact what we have found

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(the exponential decay of correlation functions and the absence of phase transition)

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is in fact generic in one-dimensional systems with local interactions.

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Naturally many subjects remain to be studied, especially if one wants to carry out

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the insights we arrived to from dimension 1 (realizing wires and tubes)

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to dimension 2 (surfaces, interfaces) and even to dimension 3

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(bulk materials).

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In this tutorial, we have gotten quite far in our study of the random clothes-pin model

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in other words: of one dimensional hard spheres.

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We again obtain a beautiful relationship

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between the algorithms and the analytical solutions.

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The providential direct sampling Monte Carlo algorithm

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showed us the way to the analytical solution of the model,

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the calculation of the partition function, of the pair correlation function and of density distribution.

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From a physical point of view, we saw that the system of random clothes-pin is a liquid

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There are some correlations close to the position of one pin, but these correlations decay exponentially.

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On large length-scales, the system is homogeneous

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So again, at small distances we have structure, like in a crystal

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but on large distances it is homogeneous, like in water.

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So this system is a liquid.

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This week and the previous week we studied hard sphere models

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and maybe you wander why we never mentioned velocities or the temperature.

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This was possible in fact because the systems we study are athermal

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they are the same at all temperatures

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Wait until next week, when we will discuss the Maxwell and Boltzmann distribution

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and take up what was missing today and last week:

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the discussion of velocities, of temperature and of finite interactions.

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Until then, have fun with the homework session 3 of Statistical Mechanics: Algorithms and Computations.